{"_id": "8570", "text": "Irrational Numbers form G-Delta Set in Reals Let $\\R \\setminus \\Q$ denote the set of irrational numbers. Let $\\struct {\\R, \\tau}$ denote the real number line with the usual (Euclidean) topology. Then $\\R \\setminus \\Q$ forms a $G_\\delta$ set in $\\R$."}
{"_id": "18855", "text": "Automorphism Maps Generator to Generator Let $G$ be a cyclic group. Let $g$ be a generator of $G$. Let $\\phi$ be an automorphism on $G$. Then $\\map \\phi g$ is also a generator of $G$."}
{"_id": "918", "text": "Hermitian Matrix has Real Eigenvalues Every Hermitian matrix has eigenvalues which are all real numbers."}
{"_id": "3887", "text": "Fortissimo Space is not Weakly Countably Compact Let $T = \\struct {S, \\tau_p}$ be a Fortissimo space. Then $T$ is not weakly countably compact."}
{"_id": "16445", "text": "Equivalent Statements for Congruence Modulo Subgroup/Right Let $x \\equiv^r y \\pmod H$ denote that $x$ is right congruent modulo $H$ to $y$. Then the following statements are equivalent: {{begin-eqn}} {{eqn | n = 1       | l = x       | o = \\equiv^r       | r = y \\pmod H }} {{eqn | n = 2       | l = x y^{-1}       | o = \\in       | r = H }} {{eqn | n = 3       | l = \\exists h \\in H: x y^{-1}       | r = h }} {{eqn | n = 4       | l = \\exists h \\in H: x       | r = h y }} {{end-eqn}}"}
{"_id": "5291", "text": "Existence-Uniqueness Theorem for Homogeneous First-Order Differential Equation Let $P \\left({x}\\right)$ be a continuous function on an open interval $I \\subseteq \\R$. Let $a \\in I$. Let $b \\in \\R$. Let $f \\left({x}\\right) = y$ be a function satisfying the differential equation: :$y' + P \\left({x}\\right) y = 0$ and the initial condition: :$f \\left({a}\\right) = b$ Then there exists a unique function satisfying these initial conditions on the interval $I$. That function takes the form: :$f \\left({x}\\right) = b e^{-A \\left({x}\\right)}$ where: :$\\displaystyle A \\left({x}\\right) = -\\int_a^x P \\left({t}\\right) \\mathrm d t$"}
{"_id": "612", "text": "Non-Zero Integers are Cancellable for Multiplication Every non-zero integer is cancellable for multiplication. That is: :$\\forall x, y, z \\in \\Z, x \\ne 0: x y = x z \\iff y = z$"}
{"_id": "11740", "text": "Linearly Independent Solutions of y'' - y = 0 The second order ODE: :$(1): \\quad y'' - y = 0$ has solutions: :$y_1 = e^x$ :$y_2 = e^{-x}$ which are linearly independent."}
{"_id": "12985", "text": "Characterization of Prime Filter by Finite Suprema Let $L = \\struct {S, \\vee, \\preceq}$ be a join semilattice. Let $F$ be a filter in $L$. Then :$F$ is a prime filter {{iff}}: :for all non-empty finite subset $A$ of $S: \\paren {\\sup A \\in F \\implies \\exists a \\in A: a \\in F}$"}
{"_id": "4236", "text": "Upper Semilattice on Classical Set is Semilattice Let $\\struct {S, \\vee}$ be an upper semilattice on a classical set $S$. Then $\\struct {S, \\vee}$ is a semilattice."}
{"_id": "2512", "text": "Set of Division Subrings forms Complete Lattice Let $\\struct {D, +, \\circ}$ be a division ring. Let $\\mathbb K$ be the set of all division subrings of $K$. Then $\\struct {\\mathbb K, \\subseteq}$ is a complete lattice."}
{"_id": "9719", "text": "Primitive of x over 1 plus Cosine of a x :$\\displaystyle \\int \\frac {x \\rd x} {1 + \\cos a x} = \\frac x a \\tan \\frac {a x} 2 + \\frac 2 {a^2} \\ln size {\\cos \\frac {a x} 2} + C$"}
{"_id": "18084", "text": "Principle of Stationary Action with Standard Lagrangian implies Newton's Laws of Motion Let $\\MM$ be an $n$-dimensional Euclidean manifold. Let $P$ be a physical system composed of a countable number of classical particles with inertial masses $m_i$ with $i \\in \\N$. Let $\\mathbf x = \\map {\\mathbf x} t$ be twice-differentiable vector-valued function embedded in $\\MM$. Suppose ${\\mathbf x}_i$ represents the position of the $i$-th particle of $P$. Suppose the action of $P$ is of the following form: :$\\displaystyle S = \\int_{t_0}^{t_1} L \\rd t$ where $L$ is the standard Lagrangian. Suppose, all (internal or external) forces ${\\mathbf F}_i$ acting upon $P$ are of the form: :${\\mathbf F}_i = - \\dfrac {\\partial U} {\\partial \\mathbf x_i}$ where: :$\\displaystyle U = \\map U {t, \\set{\\mathbf x_i} }$ is a differentiable real function :$\\set {\\mathbf x_i}$ denotes dependence on the positions of all the particles. Then the stationary point of $S$ implies Newton's Second Law of Motion."}
{"_id": "16945", "text": "Addition of Fractions Let $a, b, c, d \\in \\Z$ such that $b d \\ne 0$. Then: :$\\dfrac a b + \\dfrac c d = \\dfrac {a D + B c} {\\lcm \\set {b, d} }$ where: :$B = \\dfrac b {\\gcd \\set {b, d} }$ :$D = \\dfrac d {\\gcd \\set {b, d} }$ :$\\lcm$ denotes lowest common multiple :$\\gcd$ denotes greatest common divisor."}
{"_id": "17208", "text": "Median of Exponential Distribution Let $X$ be a continuous random variable of the exponential distribution with parameter $\\beta$ for some $\\beta \\in \\R_{> 0}$.  Then the median of $X$ is equal to $\\beta \\ln 2$."}
{"_id": "10922", "text": "Minus One is Less than Zero :$-1 < 0$"}
{"_id": "12576", "text": "Equivalence of Definitions of Semantic Equivalence for Predicate Logic Let $\\mathbf A, \\mathbf B$ be WFFs of predicate logic. {{TFAE|def = Semantic Equivalence (Predicate Logic)|semantic equivalence}}"}
{"_id": "13706", "text": "Sums of Partial Sequences of Squares Let $n \\in \\Z_{>0}$. Consider the odd number $2 n + 1$ and its square $\\paren {2 n + 1}^2 = 2 m + 1$. Then: :$\\displaystyle \\sum_{j \\mathop = 0}^n \\paren {m - j}^2 = \\sum_{j \\mathop = 1}^n \\paren {m + j}^2$ That is: :the sum of the squares of the $n + 1$ integers up to $m$ equals: :the sum of the squares of the $n$ integers from $m + 1$ upwards."}
{"_id": "17978", "text": "Mapping is Bijection iff Composite with Direct Image Mapping with Complementation Commutes Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Then: :$f$ is a bijection {{iff}}: :$f^\\to \\circ \\complement_S = \\complement_T \\circ f^\\to$ where: :$f^\\to: \\powerset S \\to \\powerset T$ denotes the direct image mapping of $f$ :$\\complement_S: \\powerset S \\to \\powerset S$ denotes the complement relative to $S$ :$\\complement_T: \\powerset T \\to \\powerset T$ denotes the complement relative to $T$ :$\\powerset S$ and $\\powerset T$ denote the power sets of $S$ and $T$ respectively."}
{"_id": "8620", "text": "Interior of Intersection may not equal Intersection of Interiors Let $T$ be a topological space. Let $\\mathbb H$ be a set of subsets of $T$. That is, let $\\mathbb H \\subseteq \\powerset T$ where $\\powerset T$ is the power set of $T$. Then it is not necessarily the case that: :$\\displaystyle \\paren {\\bigcap_{H \\mathop \\in \\mathbb H} H}^\\circ = \\bigcap_{H \\mathop \\in \\mathbb H} H^\\circ$ where $H^\\circ$ denotes the interior of $H$."}
{"_id": "17621", "text": "Excess Kurtosis of Chi-Squared Distribution Let $n$ be a strictly positive integer.  Let $X \\sim \\chi^2_n$ where $\\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the excess kurtosis $\\gamma_2$ of $X$ is given by:  :$\\gamma_2 = \\dfrac {12} n$"}
{"_id": "19040", "text": "Set together with Omega-Accumulation Points is not necessarily Closed Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. Let $\\Omega$ denote the set of $\\omega$-accumulation points of $H$. Then it is not necessarily the case that $H \\cup \\Omega$ is a closed set of $T$."}
{"_id": "17814", "text": "Equivalence of Definitions of Convergent Sequence in Metric Space Let $M = \\struct {A, d}$ be a metric space or a pseudometric space. Let $\\sequence {x_k}$ be a sequence in $A$. {{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}}"}
{"_id": "16076", "text": "Laplace Transform of Generating Function of Sequence Let $\\left\\langle{a_n}\\right\\rangle$ be a sequence which has a generating function which is convergent. Let $G \\left({z}\\right)$ be the generating function for $\\left\\langle{a_n}\\right\\rangle$. Let $f \\left({x}\\right)$ be the step function: :$f \\left({x}\\right) = \\displaystyle \\sum_{k \\mathop \\in \\Z} a_k \\left[{0 \\le k \\le x}\\right]$ where $\\left[{0 \\le k \\le x}\\right]$ is Iverson's convention. Then the Laplace transform of $f \\left({x}\\right)$ is given by: :$\\mathcal L \\left\\{ {f \\left({s}\\right)}\\right\\} = \\dfrac {G \\left({e^{-s} }\\right)} s$"}
{"_id": "1598", "text": "Existence of Local Coordinates Let $X^n$ be an $n$-dimensional manifold. Let $U$ be a neighborhood of a point $p \\in X^n$. Then there exist local coordinates on $U$."}
{"_id": "17502", "text": "Laplace Transform of Half Wave Rectified Sine Curve Consider the half wave rectified sine curve: :$\\map f t = \\begin {cases} \\sin t & : 2 n \\pi \\le t \\le \\paren {2 n + 1} \\pi \\\\ 0 & : \\paren {2 n + 1} \\pi \\le t \\le \\paren {2 n + 2} \\pi \\end {cases}$ The Laplace transform of $\\map f t$ is given by: :$\\laptrans {\\map f t} = \\dfrac 1 {\\paren {1 - e^{-\\pi s} } \\paren {s^2 + 1} }$"}
{"_id": "9846", "text": "Primitive of Arccosecant of x over a :$\\displaystyle \\int \\operatorname{arccsc} \\frac x a \\rd x = \\begin{cases} \\displaystyle x \\operatorname{arccsc} \\frac x a + a \\ln \\left({x + \\sqrt {x^2 - a^2} }\\right) + C & : 0 < \\operatorname{arccsc} \\dfrac x a < \\dfrac \\pi 2 \\\\ \\displaystyle x \\operatorname{arccsc} \\frac x a - a \\ln \\left({x + \\sqrt {x^2 - a^2} }\\right) + C & : -\\dfrac \\pi 2 < \\operatorname{arccsc} \\dfrac x a < 0 \\\\ \\end{cases}$"}
{"_id": "7416", "text": "Dedekind-Complete Bounded Ordered Set is Complete Lattice Let $\\left({L, \\preceq}\\right)$ be an ordered set. Let $L$ have a lower bound $\\bot$ and an upper bound $\\top$. Let $\\left({L, \\preceq}\\right)$ be Dedekind-complete. Then $\\left({L, \\preceq}\\right)$ is a complete lattice."}
{"_id": "11796", "text": "Linear Second Order ODE/y'' - 2 y' + 5 y = 25 x^2 + 12 The second order ODE: :$(1): \\quad y'' - 2 y' + 5 y = 25 x^2 + 12$ has the general solution: :$y = e^x \\paren {C_1 \\cos 2 x + C_2 \\sin 2 x} + 2 + 4 x + 5 x^2$"}
{"_id": "6971", "text": "Non-Equivalence as Conjunction of Disjunction with Disjunction of Negations :$\\neg \\paren {p \\iff q} \\dashv \\vdash \\paren {p \\lor q} \\land \\paren {\\neg p \\lor \\neg q}$"}
{"_id": "2403", "text": "Increasing Sum of Binomial Coefficients :$\\displaystyle \\sum_{j \\mathop = 0}^n j \\binom n j = n 2^{n - 1}$"}
{"_id": "7702", "text": "Equivalence of Axiom Schemata for Groups/Warning Suppose we build an algebraic structure with the following axioms: {{begin-axiom}} {{axiom | n = 0         | lc= Closure Axiom         | q = \\forall a, b \\in G         | m = a \\circ b \\in G }} {{axiom | n = 1         | lc= Associativity Axiom         | q = \\forall a, b, c \\in G         | m = a \\circ \\paren {b \\circ c} = \\paren {a \\circ b} \\circ c }} {{axiom | n = 2         | lc= Right Identity Axiom         | q = \\exists e \\in G: \\forall a \\in G         | m = a \\circ e = a }} {{axiom | n = 3         | lc= Left Inverse Axiom         | q = \\forall x \\in G: \\exists b \\in G         | m = b \\circ a = e }} {{end-axiom}} Then this does '''not''' (necessarily) define a group (although clearly a group fulfils those axioms)."}
{"_id": "11126", "text": "Laplace Transform of Dirac Delta Function by Function Let $\\map f t: \\R \\to \\R$ or $\\R \\to \\C$ be a function. Let $\\map \\delta t$ denote the Dirac delta function. Let $c$ be a positive constant real number. Let $\\laptrans {\\map f t} = \\map F s$ denote the Laplace transform of $f$. Then: :$\\laptrans {\\map \\delta {t - c} \\, \\map f t} = e^{- s c} \\, \\map f c$"}
{"_id": "7548", "text": "Generating Function for Binomial Coefficients Let $\\left \\langle {a_n}\\right \\rangle$ be the sequence defined as: : $\\displaystyle \\forall n \\in \\N: a_n = \\begin{cases} \\binom m n & : n = 0, 1, 2, \\ldots, m \\\\ 0 & : \\text{otherwise}\\end{cases}$ where $\\displaystyle \\binom m n$ denotes a binomial coefficient. Then the generating function for $\\left \\langle {a_n}\\right \\rangle$ is given as: : $\\displaystyle G \\left({z}\\right) = \\sum_{n \\mathop = 0}^m \\binom m n z^n = \\left({1 + z}\\right)^m$"}
{"_id": "1857", "text": "Equivalence of Definitions of Second Chebyshev Function {{TFAE|def = Second Chebyshev Function}} :$(1): \\quad \\displaystyle \\map \\psi x = \\sum_{p^k \\mathop \\le x} \\ln p$ :$(2): \\quad \\displaystyle \\map \\psi x = \\sum_{1 \\mathop \\le n \\mathop \\le x} \\map \\Lambda n$ :$(3): \\quad \\displaystyle \\map \\psi x = \\sum_{p \\mathop \\le x} \\floor {\\log_p x} \\ln p$ where: :$p$ is a prime number :$\\map \\Lambda n$ is the von Mangoldt function :$\\floor {\\, \\cdot \\, }$ denotes the floor function."}
{"_id": "14263", "text": "3-Digit Numbers forming Longest Reverse-and-Add Sequence Let $m \\in \\Z_{>0}$ be a positive integer expressed in decimal notation. Let $r \\left({m}\\right)$ be the reverse-and-add process on $m$. Let $r$ be applied iteratively to $m$. The $3$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are: :$187, 286, 385, 583, 682, 781, 880$ all of which need $23$ iterations. :$869$ and $968$ are the result of the first iteration for each of these, and so take $22$ iterations to reach a palindromic number."}
{"_id": "8595", "text": "Rational Number Space is Sigma-Compact Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is $\\sigma$-compact."}
{"_id": "14304", "text": "Smallest Numbers with 240 Divisors The smallest integers with $240$ divisors are: :$720 \\, 720, 831 \\, 600, 942 \\, 480, 982 \\, 800, 997 \\, 920, \\ldots$"}
{"_id": "12596", "text": "Signed Stirling Number of the First Kind of 0 :$s \\left({0, n}\\right) = \\delta_{0 n}$"}
{"_id": "9435", "text": "Primitive of x cubed by Root of x squared plus a squared :$\\displaystyle \\int x^3 \\sqrt {x^2 + a^2} \\rd x = \\frac {\\paren {\\sqrt {x^2 + a^2} }^5} 5 - \\frac {a^2 \\paren {\\sqrt {x^2 + a^2} }^3} 3 + C$"}
{"_id": "10536", "text": "L1 Metric on Closed Real Interval is Metric Let $S$ be the set of all real functions which are continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Let $d: S \\times S \\to \\R$ be the $L^1$ metric on $\\left[{a \\,.\\,.\\, b}\\right]$: :$\\displaystyle \\forall f, g \\in S: d \\left({f, g}\\right) := \\int_a^b \\left\\vert{f \\left({t}\\right) - g \\left({t}\\right)}\\right\\vert \\ \\mathrm d t$ Then $d$ is a metric."}
{"_id": "18141", "text": "Ambiguous Case for Spherical Triangle Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the sides $a$ and $b$ be known. Let the angle $\\sphericalangle B$ also be known. Then it may not be possible to know the value of $\\sphericalangle A$. This is known as the '''ambiguous case (for the spherical triangle)'''."}
{"_id": "17230", "text": "Double of Antiperiod is Period Let $f: \\R \\to \\R$ be a real antiperiodic function with an anti-period of $A$. Then $f$ is also periodic with a period of $2A$."}
{"_id": "17563", "text": "Set of Strictly Positive Real Numbers has no Smallest Element Let $\\R_{>0}$ denote the set of strictly positive real numbers. Then $\\R_{>0}$ has no smallest element."}
{"_id": "8451", "text": "Subgroup of Order 1 is Trivial Let $\\struct {G, \\circ}$ be a group. Then $\\struct {G, \\circ}$ has exactly $1$ subgroup of order $1$: the trivial subgroup."}
{"_id": "13861", "text": "Number of Odd Entries in Row of Pascal's Triangle is Power of 2 The number of odd entries in a row of Pascal's triangle is a power of $2$."}
{"_id": "16142", "text": "Sextuple Angle Formula for Cosine : $\\cos \\paren {6 \\theta} = 32 \\cos^6 \\theta - 48 \\cos^4 \\theta + 18 \\cos^2 \\theta - 1$"}
{"_id": "8578", "text": "Irrational Number Space is Paracompact Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is paracompact."}
{"_id": "11579", "text": "Existence of Infinitely Many Integrating Factors Let the first order ordinary differential equation: :$(1): \\quad \\map M {x, y} + \\map N {x, y} \\dfrac {\\d y} {\\d x} = 0$ be such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree. Suppose $(1)$ has an integrating factor. Then $(1)$ has an infinite number of integrating factors"}
{"_id": "3711", "text": "Ring Monomorphism from Integers to Rationals Let $\\phi: \\Z \\to \\Q$ be the mapping from the integers $\\Z$ to the rational numbers $\\Q$ defined as: :$\\forall x \\in \\Z: \\map \\phi x = \\dfrac x 1$ Then $\\phi$ is a (ring) monomorphism, but specifically not an epimorphism."}
{"_id": "13535", "text": "Generator for Almost Isosceles Pythagorean Triangle Let $P$ be a Pythagorean triangle whose sides correspond to the Pythagorean triple $T = \\tuple {a, b, c}$. Let the generator for $T$ be $\\tuple {m, n}$. Then: :$P$ is almost isosceles {{iff}} :$\\tuple {2 m + n, m}$ is the generator for the Pythagorean triple $T'$ of another almost isosceles Pythagorean triangle $P'$."}
{"_id": "14230", "text": "Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors The following sequence of integers have the property that both the harmonic mean and arithmetic mean of their divisors are integers: :$1, 6, 140, 270, 672, \\ldots$ {{OEIS|A007340}}"}
{"_id": "18420", "text": "ODE/(D^4 - 1) y = sin x The second order ODE: :$(1): \\quad \\paren {D^4 - 1} y' = \\sin x$ has a general solution: :$y = C_1 e^x + C_2 e^{-x} + C_3 \\sin x + C_4 \\cos x + \\dfrac {x \\cos x} 4$"}
{"_id": "12330", "text": "Value of b for b by Logarithm Base b of x to be Minimum Let $x \\in \\R_{> 0}$ be a (strictly) positive real number. Consider the real function $f: \\R_{> 0} \\to \\R$ defined as: :$f \\left({b}\\right) := b \\log_b x$ $f$ attains a minimum when :$b = e$ where $e$ is Euler's number."}
{"_id": "17147", "text": "Order of Automorphism Group of Prime Group Let $p$ be a prime number. Let $G$ be a group of order $p$. Let $\\Aut G$ denote the automorphism group of $G$. Then: :$\\order {\\Aut G} = p - 1$ where $\\order {\\, \\cdot \\,}$ denotes the order of a group."}
{"_id": "13640", "text": "Smallest Odd Number not of form 2 a squared plus p $17$ is the smallest odd number $n$ greater than $3$ which cannot be expressed in the form: :$n = 2 a^2 + p$ where: :$p$ is prime :$a \\in \\Z_{>0}$ is a (strictly) positive integer."}
{"_id": "10834", "text": "Complex Numbers under Multiplication do not form Group The algebraic structure $\\struct {\\C, \\times}$ consisting of the set of complex numbers $\\C$ under multiplication $\\times$ is not a group."}
{"_id": "5930", "text": "Numbers of Type Rational a plus b root 2 form Field/Corollary The field $\\struct {\\Q \\sqbrk {\\sqrt 2}, +, \\times}$ is a subfield of $\\struct {\\R, +, \\times}$."}
{"_id": "13386", "text": "Pi Squared is Irrational Pi squared ($\\pi^2$) is irrational."}
{"_id": "17466", "text": "Bessel Function of the First Kind of Negative Integer Order :$\\map {J_{-n} } x = \\paren {-1}^n \\map {J_n} x$"}
{"_id": "9130", "text": "Derivative of Arcsecant Function/Corollary 2 :$\\dfrac {\\mathrm d \\left({\\operatorname{arcsec} x }\\right)} {\\mathrm d x} = \\dfrac 1 {x^2 \\sqrt {1 - \\frac 1 {x^2}}}$"}
{"_id": "5051", "text": "Multiplication of Cross-Relation Equivalence Classes on Natural Numbers is Well-Defined Let $\\struct {\\N, +}$ be the semigroup of natural numbers under addition. Let $\\struct {\\N \\times \\N, \\oplus}$ be the (external) direct product of $\\struct {\\N, +}$ with itself, where $\\oplus$ is the operation on $\\N \\times \\N$ induced by $+$ on $\\N$. Let $\\boxtimes$ be the cross-relation defined on $\\N \\times \\N$ by: :$\\tuple {x_1, y_1} \\boxtimes \\tuple {x_2, y_2} \\iff x_1 + y_2 = x_2 + y_1$ Let $\\eqclass {x, y} {}$ denote the equivalence class of $\\tuple {x, y}$ under $\\boxtimes$. Let $\\otimes$ be the binary operation defined on these equivalence classes as: :$\\forall \\eqclass {a, b} {}, \\eqclass {c, d} {} \\in \\N \\times \\N: \\eqclass {a, b} {} \\otimes \\eqclass {c, d} {} = \\eqclass {\\tuple {a \\cdot c} + \\tuple {b \\cdot d}, \\tuple {a \\cdot d} + \\tuple {b \\cdot c} } {}$ where $a \\cdot c$ denotes natural number multiplication between $a$ and $c$. {{WIP|Introduce the language of the Definition:Quotient Set.}} The operation $\\otimes$ on these equivalence classes is well-defined, in the sense that: {{begin-eqn}} {{eqn | l = \\eqclass {a_1, b_1} {}       | r = \\eqclass {a_2, b_2} {}       | c =  }} {{eqn | l = \\eqclass {c_1, d_1} {}       | r = \\eqclass {c_2, d_2} {}       | c =  }} {{eqn | ll= \\leadsto       | l = \\eqclass {a_1, b_1} {} \\otimes \\eqclass {c_1, d_1} {}       | r = \\eqclass {a_2, b_2} {} \\otimes \\eqclass {c_2, d_2} {}       | c =  }} {{end-eqn}}"}
{"_id": "3249", "text": "Excluded Point Topology is T4 Let $T = \\left({S, \\tau_{\\bar p}}\\right)$ be an excluded point space. Then $T$ is a $T_4$ space."}
{"_id": "19008", "text": "De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Corollary :$T_1 \\cup T_2 = \\overline {\\overline T_1 \\cap \\overline T_2}$"}
{"_id": "18673", "text": "Determinant with Columns Transposed If two columns of a matrix with determinant $D$ are transposed, its determinant becomes $-D$."}
{"_id": "12440", "text": "Summation of Powers over Product of Differences/Example {{begin-eqn}} {{eqn | l = \\frac 1 {\\paren {a - b} \\paren {a - c} } + \\frac 1 {\\paren {b - a} \\paren {b - c} } + \\frac 1 {\\paren {c - a} \\paren {c - b} }       | r = 0 }} {{eqn | l = \\frac a {\\paren {a - b} \\paren {a - c} } + \\frac b {\\paren {b - a} \\paren {b - c} } + \\frac c {\\paren {c - a} \\paren {c - b} }       | r = 0 }} {{eqn | l = \\frac {a^2} {\\paren {a - b} \\paren {a - c} } + \\frac {b^2} {\\paren {b - a} \\paren {b - c} } + \\frac {c^2} {\\paren {c - a} \\paren {c - b} }       | r = 1 }} {{eqn | l = \\frac {a^3} {\\paren {a - b} \\paren {a - c} } + \\frac {b^3} {\\paren {b - a} \\paren {b - c} } + \\frac {c^3} {\\paren {c - a} \\paren {c - b} }       | r = a + b + c }} {{end-eqn}}"}
{"_id": "930", "text": "Z-Module Associated with Abelian Group is Unitary Z-Module Let $\\struct {G, *}$ be an abelian group with identity $e$. Let $\\struct {G, *, \\circ}_\\Z$ be the $Z$-module associated with $G$. Then $\\struct {G, *, \\circ}_\\Z$ is a unitary $Z$-module."}
{"_id": "14394", "text": "Equivalence of Definitions of Nilradical of Ring {{TFAE|def = Nilradical of Ring}} Let $A$ be a commutative ring."}
{"_id": "16439", "text": "Multiplicative Group of Reduced Residues Modulo 7 is Cyclic Let $\\struct {\\Z'_7, \\times_7}$ denote the multiplicative group of reduced residues modulo $7$. Then $\\struct {\\Z'_7, \\times_7}$ is cyclic."}
{"_id": "2589", "text": "Multiplicative Group of Rationals is Subgroup of Complex Let $\\struct {\\Q, \\times}$ be the multiplicative group of rational numbers. Let $\\struct {\\C, \\times}$ be the multiplicative group of complex numbers. Then $\\struct {\\Q, \\times}$ is a normal subgroup of $\\struct {\\C, \\times}$."}
{"_id": "8648", "text": "Trace of Unit Matrix Let $\\mathbf I_n$ be the unit matrix of order $n$. Then: :$\\map \\tr {\\mathbf I_n} = n$ where $\\map \\tr {\\mathbf I_n}$ denotes the trace of $\\mathbf I_n$."}
{"_id": "763", "text": "Even Order Group has Order 2 Element Let $G$ be a group whose identity is $e$. Let $G$ be of even order. Then: :$\\exists x \\in G: \\order x = 2$ That is: :$\\exists x \\in G: x \\ne e: x^2 = e$"}
{"_id": "15590", "text": "Power Series Expansion for Logarithm of Sine of x {{begin-eqn}} {{eqn | l = \\ln \\size {\\sin x}       | r = \\ln \\size x - \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^n 2^{2 n - 1} B_{2 n} \\, x^{2 n} } {n \\paren {2 n}!}       | c =  }} {{eqn | r = \\ln \\size x - \\frac {x^2} 6 - \\frac {x^4} {180} - \\frac {x^6} {2835} + \\cdots       | c =  }} {{end-eqn}} for all $x \\in \\R$ such that $0 < \\size x < \\pi$."}
{"_id": "5675", "text": "Equal Powers of Group Element implies Finite Order Let $\\struct {G, \\circ}$ be a group. Let $g \\in G$ such that $g^r = g^s$ where $r, s \\in \\Z: r \\ne s$. Then there exists $m \\in \\Z_{>0}$ such that: :$(1): \\quad g^m = e$ :$(2): \\quad 0 \\le i < j < m \\implies g^i \\ne g^j$"}
{"_id": "2600", "text": "Free Commutative Monoid is Commutative Monoid The free commutative monoid on a set $\\family {X_j: j \\in J}$ is a commutative monoid."}
{"_id": "8953", "text": "Complement of Open Set in Complex Plane is Closed Let $S \\subseteq \\C$ be an open subset of the complex plane $\\C$. Then the complement of $S$ in $\\C$ is closed."}
{"_id": "2472", "text": "Quotient Subgroup of Semigroup Induced on Power Set Let $\\struct {G, \\circ}$ be a group. Let $\\struct {H, \\circ}$ be a normal subgroup of $\\struct {G, \\circ}$. Then $\\struct {G / H, \\circ_H}$ is a subgroup of $\\struct {\\powerset G, \\circ_\\mathcal P}$, where: :$\\struct {G / H, \\circ_H}$ is the quotient group of $G$ by $H$ :$\\struct {\\powerset G, \\circ_\\mathcal P}$ is the semigroup induced by the operation $\\circ$ on the power set $\\powerset G$ of $G$."}
{"_id": "395", "text": "Quotient Mapping of Inverse Completion Let $\\struct {T, \\circ'}$ be an inverse completion of a commutative semigroup $\\struct {S, \\circ}$, where $C$ is the set of cancellable elements of $S$. Let $f: S \\times C: T$ be the mapping defined as: :$\\forall x \\in S, y \\in C: \\map f {x, y} = x \\circ' y^{-1}$ Let $\\RR_f$ be the equivalence relation induced by $f$. Then: :$\\tuple {x_1, y_1} \\mathrel {\\RR_f} \\tuple {x_2, y_2} \\iff x_1 \\circ y_2 = x_2 \\circ y_1$"}
{"_id": "9099", "text": "Beta Function is Defined for Positive Reals Let $x, y \\in \\R$ be real numbers. Let $\\Beta \\left({x, y}\\right)$ be the Beta function: :$\\displaystyle \\Beta \\left({x, y}\\right) := \\int_{\\mathop \\to 0}^{\\mathop \\to 1} t^{x - 1} \\left({1 - t}\\right)^{y - 1} \\ \\mathrm d t$ Then $\\Beta \\left({x, y}\\right)$ exists provided that $x, y > 0$."}
{"_id": "14370", "text": "Continuous implies Increasing in Scott Topological Lattices Let $T_1 = \\struct {S_1, \\preceq_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\preceq_2, \\tau_2}$ be up-complete topological lattices with Scott topologies. Let $f: S_1 \\to S_2$ be a continuous mapping. Then $f$ is an increasing mapping."}
{"_id": "12810", "text": "Powers of Group Elements/Sum of Indices/Additive Notation :$\\forall m, n \\in \\Z: m g + n g = \\paren {m + n} g$"}
{"_id": "17146", "text": "Automorphism Group of Cyclic Group is Abelian Let $G$ be a cyclic group. Let $\\Aut G$ denote the automorphism group of $G$. Then $\\Aut G$ is abelian."}
{"_id": "17988", "text": "Direct Image of Intersection with Inverse Image Then: :$\\forall A \\in \\powerset S, B \\in \\powerset T: \\map {f^\\to} {A \\cap \\map {f^\\gets} B} = \\map {f^\\to} A \\cap B$"}
{"_id": "16783", "text": "Odd Integers under Multiplication do not form Group Let $S$ be the set of odd integers: :$S = \\set {x \\in \\Z: \\exists n \\in \\Z: x = 2 n + 1}$ Let $\\struct {S, \\times}$ denote the algebraic structure formed by $S$ under the operation of multiplication. Then $\\struct {S, \\times}$ is not a group."}
{"_id": "8496", "text": "Identity Mapping is Idempotent Let $S$ be a set. Let $I_S: S \\to S$ be the identity mapping on $S$. Then $I_S$ is idempotent: :$I_S \\circ I_S = I_S$"}
{"_id": "9745", "text": "Primitive of Reciprocal of Sine of a x by 1 minus Cosine of a x :$\\displaystyle \\int \\frac {\\mathrm d x} {\\sin a x \\left({1 - \\cos a x}\\right)} = \\frac {-1} {2 a \\left({1 - \\cos a x}\\right)} + \\frac 1 {2 a} \\ln \\left\\vert{\\tan \\frac {a x} 2}\\right\\vert + C$"}
{"_id": "17028", "text": "Difference of Two Fourth Powers :$x^4 - y^4 = \\paren {x - y} \\paren {x + y} \\paren {x^2 + y^2}$"}
{"_id": "18419", "text": "Linear Second Order ODE/y'' - 2 y' + y = 1 over 1 + e^x The second order ODE: :$(1): \\quad y'' - 2 y' + y = \\dfrac 1 {1 + e^x}$ has a particular solution: :$y = 1 + e^x \\displaystyle \\int \\map \\ln {1 + e^{-x} } \\rd x$"}
{"_id": "15557", "text": "Power Series Expansion for General Exponential Function Let $a \\in \\R_{> 0}$ be a (strictly) positive real number. Then: Then: {{begin-eqn}} {{eqn | ll= \\forall x \\in \\R:       | l = a^x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {\\left({x \\ln a}\\right)^n} {n!}       | c =  }} {{eqn | r = 1 + x \\ln a + \\frac {\\left({x \\ln a}\\right)^2} {2!} + \\frac {\\left({x \\ln a}\\right)^3} {3!} + \\cdots       | c =  }} {{end-eqn}}"}
{"_id": "7376", "text": "Polynomials Closed under Addition/Polynomial Forms Let: : $\\displaystyle f = \\sum_{k \\mathop \\in Z} a_k \\mathbf X^k$ : $\\displaystyle g = \\sum_{k \\mathop \\in Z} b_k \\mathbf X^k$ be polynomials in the indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$ over the ring $R$. Then the operation of polynomial addition on $f$ and $g$: Define the sum: :$\\displaystyle f \\oplus g = \\sum_{k \\mathop \\in Z} \\left({a_k + b_k}\\right) \\mathbf X^k$ Then $f \\oplus g$ is a polynomial. That is, the operation of polynomial addition is closed on the set of all polynomials on a given set of indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$."}
{"_id": "17309", "text": "Coprimality Relation is Symmetric :$\\perp$ is symmetric."}
{"_id": "12990", "text": "Bound for Difference of Irrational Number with Convergent Let $x$ be an irrational number. Let $\\sequence {C_n}$ be the sequence of convergents of the continued fraction expansion of $x$. Then $\\forall n \\ge 1$: :$C_n < x < C_{n + 1}$ or $C_{n + 1} < x < C_n$ :$\\size {x - C_n} < \\dfrac 1 {q_n q_{n + 1} }$"}
{"_id": "15135", "text": "Infinite Product of Weakly Locally Compact Spaces is not always Weakly Locally Compact Let $I$ be an indexing set with infinite cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be weakly locally compact-compact. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also weakly locally compact."}
{"_id": "2023", "text": "Equivalence of Definitions of Continuous Mapping between Topological Spaces/Point Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $f: S_1 \\to S_2$ be a mapping from $S_1$ to $S_2$. Let $x \\in S_1$. {{TFAE|def = Continuous at Point of Topological Space|view = continuity at a point of a topological space}} === Definition using Open Sets === {{Definition:Continuous Mapping (Topology)/Point/Open Sets}} === Definition using Filters === {{Definition:Continuous Mapping (Topology)/Point/Filters}}"}
{"_id": "2480", "text": "Dual of Total Ordering is Total Ordering Let $\\preccurlyeq$ be a total ordering. Then its dual ordering $\\succcurlyeq$ is also a total ordering."}
{"_id": "554", "text": "Ordered Group Equivalences Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered group whose identity is $e$. Let $x, y, z, \\in S$. Then the following are all equivalent: : $(1): \\quad x \\prec y$ : $(2): \\quad x \\circ z \\prec y \\circ z$ : $(3): \\quad z \\circ x \\prec z \\circ y$ : $(4): \\quad y^{-1} \\prec x^{-1}$ : $(5): \\quad e \\prec y \\circ x^{-1}$ : $(6): \\quad e \\prec x^{-1} \\circ y$"}
{"_id": "18026", "text": "Integration by Inversion :$\\displaystyle \\int_0^{+\\infty} \\map f x \\rd x = \\int_0^{+\\infty} \\dfrac {\\map f {\\frac 1 x} } {x^2} \\rd x$"}
{"_id": "10915", "text": "Compact Hausdorff Space is Locally Compact Let $T = \\left({S, \\tau}\\right)$ be a Hausdorff space which is compact. Then $T$ is locally compact."}
{"_id": "1500", "text": "Real Addition is Well-Defined The operation of addition on the set of real numbers $\\R$ is well-defined."}
{"_id": "12524", "text": "Power Function is Completely Multiplicative/Integers Let $c \\in \\Z$ be an integer. Let $f_c: \\Z \\to \\Z$ be the mapping defined as: :$\\forall n \\in \\Z: \\map {f_c} n = n^c$ Then $f_c$ is completely multiplicative."}
{"_id": "12481", "text": "Ceiling Function/Examples/Ceiling of Minus One Half :$\\ceiling {- \\dfrac 1 2} = 0$"}
{"_id": "10027", "text": "Primitive of x by Inverse Hyperbolic Sine of x over a :$\\displaystyle \\int x \\sinh^{-1} \\frac x a \\rd x = \\paren {\\frac {x^2} 2 + \\frac {a^2} 4} \\sinh^{-1} \\frac x a - \\frac {x \\sqrt {x^2 + a^2} } 4 + C$"}
{"_id": "8211", "text": "Categories of Elements of Ring Let $\\left({R, +, \\circ}\\right)$ be a ring. The elements of $R$ are partitioned into three classes: :$(1): \\quad$ the zero :$(2): \\quad$ the units :$(3): \\quad$ the proper elements."}
{"_id": "12398", "text": "Permutation of Indices Let $R: \\Z \\to \\set {\\T, \\F}$ be a propositional function on the set of integers. Let the fiber of truth of $R$ be finite. Let  $\\pi$ is a permutation on the fiber of truth of $R$. Then:"}
{"_id": "9595", "text": "Primitive of Half Integer Power of a x squared plus b x plus c Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\left({a x^2 + b x + c}\\right)^{n + \\frac 1 2} \\ \\mathrm d x = \\frac {\\left({2 a x + b}\\right) \\left({a x^2 + b x + c}\\right)^{n + \\frac 1 2} } {4 a \\left({n + 1}\\right)} + \\frac {\\left({2 n + 1}\\right) \\left({4 a c - b^2}\\right)} {8 a  \\left({n + 1}\\right)} \\int \\left({a x^2 + b x + c}\\right)^{n - \\frac 1 2} \\ \\mathrm d x$"}
{"_id": "16415", "text": "Rule of Association/Disjunction/Formulation 2/Forward Implication :$\\vdash \\left({p \\lor \\left({q \\lor r}\\right)}\\right) \\implies \\left({\\left({p \\lor q}\\right) \\lor r}\\right)$"}
{"_id": "13686", "text": "Sum of Squares of Divisors of 24 and 26 are Equal The sum of the squares of the divisors of $24$ equals the sum of the squares of the divisors of $26$: :$\\map {\\sigma_2} {24} = \\map {\\sigma_2} {26}$ where $\\sigma_\\alpha$ denotes the divisor function."}
{"_id": "17973", "text": "Successor Mapping on Natural Numbers is not Surjection Let $f: \\N \\to \\N$ be the successor mapping on the natural numbers $\\N$: :$\\forall n \\in \\N: \\map f n = n + 1$ Then $f$ is not a surjection."}
{"_id": "17036", "text": "Sum of Fourth Powers with Product of Squares :$x^4 + x^2 y^2 + y^4 = \\paren {x^2 + x y + y^2} \\paren {x^2 - x y + y^2}$"}
{"_id": "11304", "text": "Primitive of Arcsecant of x over a/Formulation 2 :$\\displaystyle \\int \\arcsec \\frac x a \\rd x = x \\arcsec \\frac x a - a \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$ for $x^2 > 1$. $\\displaystyle \\arcsec \\frac x a$ is undefined on the real numbers for $x^2 < 1$."}
{"_id": "9772", "text": "Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Cosine :$\\displaystyle \\int \\frac {\\cos^m a x} {\\sin^n a x} \\rd x = \\frac {\\cos^{m - 1} a x} {a \\paren {m - n} \\sin^{n - 1} a x} + \\frac {m - 1} {m - n} \\int \\frac {\\cos^{m - 2} a x} {\\sin^n a x} \\rd x + C$"}
{"_id": "17085", "text": "Magnitude and Direction of Equilibrant Let $\\mathbf F_1, \\mathbf F_2, \\ldots, \\mathbf F_n$ be a set of $n$ forces acting on a particle $B$ at a point $P$ in space. The '''equilibrant''' $\\mathbf E$ of $\\mathbf F_1, \\mathbf F_2, \\ldots, \\mathbf F_n$ is: :$\\mathbf E = -\\displaystyle \\sum_{k \\mathop = 1}^n \\mathbf F_k$ That is, the magnitude and direction of $\\mathbf E$ is such as to balance out the effect of $\\mathbf F_1, \\mathbf F_2, \\ldots, \\mathbf F_n$."}
{"_id": "18975", "text": "Finite Union of Closed Sets is Closed/Normed Vector Space Let $M = \\struct {X, \\norm {\\, \\cdot \\, }}$ be a normed vector space. Then the union of finitely many closed sets of $M$ is itself closed."}
{"_id": "12273", "text": "Set of Integers is not Well-Ordered by Usual Ordering The set of integers $\\Z$ is not well-ordered under the usual ordering $\\le$."}
{"_id": "11512", "text": "Terminal Velocity of Body under Fall Retarded Proportional to Square of Velocity Let $B$ be a body falling in a gravitational field. Let $B$ be falling through a medium which exerts a resisting force of magnitude $k v^2$ upon $B$ which is proportional to the square of the velocity of $B$ relative to the medium. Then the terminal velocity of $B$ is given by: :$v =  \\sqrt {\\dfrac {g m} k}$"}
{"_id": "5596", "text": "Poset Elements Equal iff Equal Weak Lower Closure Let $\\left({S, \\preccurlyeq}\\right)$ be an ordered set. Let $s, t \\in S$. Then $s = t$ {{iff}}: :$s^\\preccurlyeq = t^\\preccurlyeq$ where $s^\\preccurlyeq$ denotes weak lower closure of $s$. That is, {{iff}}, for all $r \\in S$: :$r \\preccurlyeq s \\iff r \\preccurlyeq t$"}
{"_id": "12069", "text": "Existence of Prime-Free Sequence of Natural Numbers Let $n$ be a natural number. Then there exists a sequence of consecutive natural numbers of length $n$ which are all composite."}
{"_id": "2468", "text": "Left and Right Operation Closed for All Subsets Let $S$ be a set. Let: : $\\leftarrow$ be the left operation on $S$ : $\\rightarrow$ be the right operation on $S$. That is: : $\\forall x, y \\in S: x \\leftarrow y = x$ : $\\forall x, y \\in S: x \\rightarrow y = y$ Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Then for all $T \\in \\mathcal P \\left({S}\\right)$, both $\\leftarrow$ and $\\rightarrow$ are closed on $T$. Thus, for all $T \\in \\mathcal P \\left({S}\\right)$: : $\\left({T, \\leftarrow}\\right)$ is a subsemigroup of $\\left({S, \\leftarrow}\\right)$ : $\\left({T, \\rightarrow}\\right)$  is a subsemigroup of $\\left({S, \\rightarrow}\\right)$."}
{"_id": "5843", "text": "Coproduct of Free Monoids Let $\\mathbf {Mon}$ be the category of monoids. Let $\\map M A$ and $\\map M B$ be free monoids on sets $A$ and $B$, respectively. Let $A \\sqcup B$ be the disjoint union of $A$ and $B$. Then the free monoid $\\map M {A \\sqcup B}$ on $A \\sqcup B$ is the coproduct of $\\map M A$ and $\\map M B$ in $\\mathbf {Mon}$."}
{"_id": "12445", "text": "Sum of Elements in Inverse of Vandermonde Matrix Let $V_n$ be the Vandermonde matrix of order $n$ given by: :$V_n = \\begin{bmatrix}   x_1 & x_2 & \\cdots & x_n \\\\   x_1^2 & x_2^2 & \\cdots & x_n^2 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\   x_1^n & x_2^n & \\cdots & x_n^n \\end{bmatrix}$ Let $V_n^{-1}$ be its inverse, from Inverse of Vandermonde Matrix: :$b_{i j} = \\begin{cases} \\paren {-1}^{j - 1} \\paren {\\dfrac{\\displaystyle \\sum_{\\substack {1 \\mathop \\le m_1 \\mathop < \\ldots \\mathop < m_{n - j} \\mathop \\le n \\\\ m_1, \\ldots, m_{n - j} \\mathop \\ne i} } x_{m_1} \\cdots x_{m_{n - j} } } {x_i \\displaystyle \\prod_{\\substack {1 \\mathop \\le m \\mathop \\le n \\\\ m \\mathop \\ne i} } \\paren {x_m - x_i} } } & : 1 \\le  j < n \\\\ \\qquad \\qquad \\qquad \\dfrac 1 {x_i \\displaystyle \\prod_{\\substack {1 \\mathop \\le m \\mathop \\le n \\\\ m \\mathop \\ne i} } \\paren {x_i - x_m} } & : j = n \\end{cases}$ The sum of all the elements of $V_n^{-1}$ is: :$\\displaystyle \\sum_{1 \\mathop \\le i, \\ j \\mathop \\le n} b_{i j} = 1 - \\prod_{k \\mathop = 1}^n \\paren {1 - \\dfrac 1 {x_k} }$"}
{"_id": "797", "text": "Intersection of Quotient Groups Let $N \\lhd G$ be a normal subgroup of $G$. Let: :$N \\le A \\le G$ :$N \\le B \\le G$ For a subgroup $H$ of $G$, let $\\alpha$ be the bijection defined as: :$\\map \\alpha H = \\set {h N: h \\in H}$ Then: :$\\map \\alpha {A \\cap B} = \\map \\alpha A \\cap \\map \\alpha B$"}
{"_id": "1855", "text": "Existence of Non-Standard Models of Arithmetic There exist non-standard models of arithmetic."}
{"_id": "12722", "text": "Beta Function of Real Number with 1 :$\\Beta \\left({x, 1}\\right) = \\Beta \\left({1, x}\\right) = \\dfrac 1 x$"}
{"_id": "893", "text": "Group Acts on Itself Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Then $\\struct {G, \\circ}$ acts on itself by the rule: :$\\forall g, h \\in G: g * h = g \\circ h$"}
{"_id": "10140", "text": "Characteristic Function of Union/Variant 1 :$\\chi_{A \\mathop \\cup B} = \\min \\set {\\chi_A + \\chi_B, 1}$"}
{"_id": "14265", "text": "Pandigital Sum whose Components are Multiples The following sums use each of the digits $1$ to $9$ exactly once each, while one summand equals $2$ times the other summand: {{begin-eqn}} {{eqn | l = 192 + 384       | r = 576       | c = where $2 \\times 192 = 384$ and so $3 \\times 192 = 576$ }} {{eqn | l = 219 + 438       | r = 657       | c = where $2 \\times 219 = 438$ and so $3 \\times 219 = 657$ }} {{eqn | l = 273 + 546       | r = 819       | c = where $2 \\times 273 = 546$ and so $3 \\times 273 = 819$ }} {{eqn | l = 327 + 654       | r = 981       | c = where $2 \\times 327 = 654$ and so $3 \\times 327 = 981$ }} {{end-eqn}} {{OEIS|A039667}} It is interesting to note how the $4$ sums can be divided into two pairs in which each element of the sum is an anagram of its corresponding element in the other."}
{"_id": "9442", "text": "Primitive of x cubed over Root of x squared plus a squared cubed :$\\displaystyle \\int \\frac {x^3 \\ \\mathrm d x} {\\left({\\sqrt {x^2 + a^2} }\\right)^3} = \\sqrt {x^2 + a^2} + \\frac {a^2} {\\sqrt {x^2 + a^2} } + C$"}
{"_id": "9889", "text": "Primitive of Power of x by Power of Logarithm of x :$\\displaystyle \\int x^m \\ln^n x \\ \\mathrm d x = \\frac {x^{m + 1} \\ln^n x} {m + 1} - \\frac n {m + 1} \\int x^m \\ln^{n - 1} x \\ \\mathrm d x + C$ where $m \\ne -1$."}
{"_id": "16221", "text": "Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 1 Let $\\struct {R, \\norm {\\,\\cdot\\,}}$ be a normed division ring. Let $\\sequence {x_n}$,  $\\sequence {y_n} $ be sequences in $R$. Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be convergent in the norm $\\norm {\\,\\cdot\\,}$ to the following limits: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} y_n = m$ Then: {{:Combination Theorem for Sequences/Normed Division Ring/Product Rule}}"}
{"_id": "9969", "text": "Primitive of Hyperbolic Sine of a x over Power of x :$\\displaystyle \\int \\frac {\\sinh a x \\ \\mathrm d x} {x^n} = \\frac {-\\sinh a x} {\\left({n - 1}\\right) x^{n - 1} } + \\frac a {n - 1} \\int \\frac {\\cosh a x \\ \\mathrm d x} {x^{n - 1} } + C$"}
{"_id": "4322", "text": "K-Connectivity Implies Lesser Connectivity If a graph $G$ is $k$-connected, then $G$ is $l$-connected for all $l \\in \\Z : 0 < l < k$."}
{"_id": "3552", "text": "Triangles with Proportional Sides are Similar Let two triangles have corresponding sides which are proportional. Then their corresponding angles are equal. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/5}}"}
{"_id": "17034", "text": "Sum of Two Fifth Powers :$x^5 + y^5 = \\paren {x + y} \\paren {x^4 - x^3 y + x^2 y^2 - x y^3 + y^4}$"}
{"_id": "4020", "text": "Second Derivative of Convex Real Function is Non-Negative Let $f$ be a real function which is twice differentiable on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Then $f$ is convex on $\\left({a \\,.\\,.\\, b}\\right)$ {{iff}} its second derivative $f'' \\ge 0$ on $\\left({a \\,.\\,.\\, b}\\right)$."}
{"_id": "19718", "text": "Partial Derivative wrt x of sin x y over cos (x + y) :$\\dfrac \\partial {\\partial x} \\dfrac {\\sin x y} {\\map \\cos {x + y} } = \\dfrac {y \\map \\cos {x + y} \\cos x y + \\map \\sin {x + y} \\sin x y} {\\map {\\cos^2} {x + y} }$"}
{"_id": "5258", "text": "Tartaglia's Formula Let $T$ be a tetrahedron with vertices $\\mathbf d_1, \\mathbf d_2, \\mathbf d_3$ and $\\mathbf d_4$. For all $i$ and $j$, let the distance between $\\mathbf d_i$ and $\\mathbf d_j$ be denoted $d_{ij}$. Then the volume $V_T$ of $T$ satisfies: :$V_T^2 = \\dfrac {1} {288} \\det \\ \\begin{vmatrix}   0 & 1 & 1 & 1 & 1\\\\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\\\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\\\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\\\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \\end{vmatrix}$"}
{"_id": "14055", "text": "Killing Form of Symplectic Lie Algebra Let $\\mathbb K \\in \\left\\{ {\\C, \\R}\\right\\}$. Let $n$ be a positive integer. Let $\\mathfrak{sp}_{2 n} \\left({\\mathbb K}\\right)$ be the Lie algebra of the symplectic group $\\operatorname{Sp} \\left({2 n, \\mathbb K}\\right)$. Then its Killing form is $B: \\left({X, Y}\\right) \\mapsto \\left({2 n + 2}\\right) \\operatorname{tr} \\left({X Y}\\right)$."}
{"_id": "8370", "text": "Group Action on Subgroup by Right Regular Representation is not Transitive Let $G$ be a group. Let $H$ be a proper subgroup of $G$. Let $*: H \\times G \\to G$ be the group action defined as: :$\\forall \\tuple {h, g} \\in H \\times G: h * g = \\map {\\rho_{h^{-1} } } g$ where $\\map {\\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$. Then $*$ is not transitive."}
{"_id": "16470", "text": "Intersection of Left Cosets of Subgroups is Left Coset of Intersection Let $G$ be a group. Let $H, K \\le G$ be subgroups of $G$. Let $a, b \\in G$. Let: :$a H \\cap b K \\ne \\O$ where $a H$ denotes the left coset of $H$ by $a$. Then $a H \\cap b K$ is a left coset of $H \\cap K$."}
{"_id": "11756", "text": "Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0 The special case of Bessel's equation: :$(1): \\quad x^2 y'' + x y' + \\left({x^2 - \\dfrac 1 4}\\right) y = 0$ has the general solution: :$y = C_1 \\dfrac {\\sin x} {\\sqrt x} + C_2 \\dfrac {\\cos x} {\\sqrt x}$"}
{"_id": "16789", "text": "Sequence of Powers of Number less than One/Rational Numbers Let $x \\in \\Q$. Let $\\sequence {x_n}$ be the sequence in $\\Q$ defined as $x_n = x^n$. Then: :$\\size x < 1$ {{iff}} $\\sequence {x_n}$ is a null sequence."}
{"_id": "17948", "text": "Primitive of Cosine Integral Function :$\\displaystyle \\int \\map \\Ci x \\rd x = x \\map \\Ci x + \\sin x + C$"}
{"_id": "18768", "text": "Sum from -m to m of Sine of n + alpha of theta over n + alpha For $0 < \\theta < 2 \\pi$: :$\\displaystyle \\sum_{n \\mathop = -m}^m \\dfrac {\\sin \\paren {n + \\alpha} \\theta} {n + \\alpha} = \\int_0^\\theta \\map \\cos {\\alpha \\theta} \\dfrac {\\sin \\paren {m + \\frac 1 2} \\theta \\rd \\theta} {\\sin \\frac 1 2 \\theta}$"}
{"_id": "4026", "text": "Classical Probability is Probability Measure The classical probability model is a probability measure."}
{"_id": "16633", "text": "Subgroup Generated by Subgroup Let $G$ be a group. Let $H \\le G$ be a subgroup of $G$. Then: :$H = \\gen H$ where $\\gen H$ denotes the subgroup generated by $H$."}
{"_id": "7712", "text": "Sufficient Condition for Vector Equals Inverse iff Zero Let $\\left({\\mathbf V, +, \\circ}\\right)_{\\mathbb F}$ be a vector space over $\\mathbb F$, as defined by the vector space axioms. Let $\\mathbb F$ be infinite. Then: :$\\forall \\mathbf v, -\\mathbf v \\in \\mathbf V: \\mathbf v = - \\mathbf v \\iff \\mathbf v = \\mathbf 0$"}
{"_id": "12571", "text": "Sum over k of m Choose k by k minus m over 2 :$\\displaystyle \\sum_{k \\mathop = 0}^n \\binom m k \\left({k - \\dfrac m 2}\\right) = -\\dfrac m 2 \\binom {m - 1} n$ where $\\dbinom m k$ etc. are binomial coefficients."}
{"_id": "4536", "text": "Equivalence of Definitions of Continuous Mapping between Topological Spaces/Everywhere Let $T_1 = \\struct {S_1, \\tau_1}$ and $T_2 = \\struct {S_2, \\tau_2}$ be topological spaces. Let $f: S_1 \\to S_2$ be a mapping from $S_1$ to $S_2$. {{TFAE|def = Everywhere Continuous Mapping Between Topological Spaces|view = everywhere continuous mapping between topological spaces}} === Definition by Pointwise Continuity === {{Definition:Continuous Mapping (Topology)/Everywhere/Pointwise}} === Definition by Open Sets === {{Definition:Continuous Mapping (Topology)/Everywhere/Open Sets}}"}
{"_id": "9647", "text": "Primitive of x cubed by Sine of a x :$\\displaystyle \\int x^3 \\sin a x \\rd x = \\paren {\\frac {3 x^2} {a^2} - \\frac 6 {a^4} } \\sin a x + \\paren {\\frac {6 x} {a^3} - \\frac {x^3} a} \\cos a x + C$"}
{"_id": "6631", "text": "Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Proof 1 :$p \\lor \\left ({p \\land q}\\right) \\dashv \\vdash p$"}
{"_id": "13683", "text": "Sum of Reciprocals in Base 10 with Zeroes Removed The infinite series :$\\displaystyle \\sum_{\\map P n} \\frac 1 n$ where $\\map P n$ is the propositional function: :$\\forall n \\in \\Z_{>0}: \\map P n \\iff$ the decimal representation of $n$ contains no instances of the digit $0$ converges to the approximate limit $23 \\cdotp 10345 \\ldots$"}
{"_id": "4155", "text": "Diagonal Relation is Smallest Equivalence Relation The diagonal relation $\\Delta_S$ on $S$ is the smallest equivalence in $S$, in the sense that: :$\\forall \\mathcal E \\subseteq S \\times S: \\Delta_S \\subseteq \\mathcal E$ where $\\mathcal E$ denotes a general equivalence relation."}
{"_id": "1042", "text": "Number of Matrix Equivalence Classes Let $K$ be a field. Let $\\mathcal M_K \\left({m, n}\\right)$ be the $m \\times n$ matrix space over $K$. Let $\\mathbf A$ be an $m \\times n$ matrix of rank $r$ over $K$. Then: :$\\mathbf A \\equiv \\begin{cases} \\left[{0_K}\\right]_{m n} & : r = 0 \\\\ & \\\\ \\begin{bmatrix}   \\mathbf I_r & \\mathbf 0 \\\\   \\mathbf 0 & \\mathbf 0  \\end{bmatrix} & : 0 < r < \\min \\left\\{{n, m}\\right\\} \\\\ & \\\\ \\begin{bmatrix}   \\mathbf I_r & \\mathbf 0  \\end{bmatrix} & : r = m < n \\\\ & \\\\ \\begin{bmatrix}   \\mathbf I_r \\\\   \\mathbf 0  \\end{bmatrix} & : r = n < m \\\\ & \\\\ \\mathbf I_r & : r = m = n \\end{cases}$ Thus there are exactly $\\min \\left\\{{m, n}\\right\\} + 1$ equivalence classes for the relation of equivalence on $\\mathcal M_K \\left({m, n}\\right)$, one of which contains only the zero matrix."}
{"_id": "6797", "text": "Complement of Top/Bounded Lattice Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded lattice. Then the top $\\top$ has a unique complement, namely $\\bot$, bottom."}
{"_id": "12437", "text": "Summation of Product of Differences :$\\displaystyle \\sum_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\left({u_j - u_k}\\right) \\left({v_j - v_k}\\right) = n \\sum_{j \\mathop = 1}^n u_j v_j - \\sum_{j \\mathop = 1}^n u_j \\sum_{j \\mathop = 1}^n v_j$"}
{"_id": "3011", "text": "Boundary of Subset of Discrete Space is Null Let $A \\subseteq S$. Then: :$\\partial A = \\O$ where: :$\\partial A$ is the boundary of $A$ in $T$."}
{"_id": "3040", "text": "Discrete Space is Complete Metric Space Let $T = \\struct {S, \\tau}$ be a discrete topological space. Then $T$ is a complete metric space."}
{"_id": "4561", "text": "Angle Between Vectors in Terms of Dot Product The angle between two non-zero  vectors in $\\R^n$ can be calculated by: {{begin-eqn}} {{eqn | l = \\theta       | r = \\arccos \\frac {\\mathbf v \\cdot \\mathbf w} {\\norm {\\mathbf v} \\norm {\\mathbf w} } }} {{end-eqn}} where: :$\\mathbf v \\cdot \\mathbf w$ represents the dot product of $\\mathbf v$ and $\\mathbf w$  :$\\norm {\\, \\cdot \\,}$ represents vector length. :$\\arccos$ represents arccosine"}
{"_id": "13612", "text": "Triangular Number cannot be Cube Let $T_n$ be the $n$th triangular number such that $n > 1$. Then $T_n$ cannot be a cube."}
{"_id": "16173", "text": "Triple Angle Formulas/Cosine/2 cos 3 theta + 1 :$2 \\cos 3 \\theta + 1 = \\paren {\\cos \\theta - \\cos \\dfrac {2 \\pi} 9} \\paren {\\cos \\theta - \\cos \\dfrac {4 \\pi} 9} \\paren {\\cos \\theta - \\cos \\dfrac {8 \\pi} 9}$"}
{"_id": "1859", "text": "Matrix Space is Module Let $\\struct {R, +, \\circ}$ be a ring. Let $\\mathbf A = \\sqbrk a_{m n}$ be an $m \\times n$ matrix over $\\struct {R, +, \\circ}$. Then the matrix space $\\map {\\MM_R} {m, n}$ of all $m \\times n$ matrices over $R$ is a module."}
{"_id": "17428", "text": "Kernel of Group Homomorphism is not Empty Let $G$ and $H$ be groups whose identity elements are $e_G$ and $e_H$ respectively. Let $\\phi: G \\to H$ be a homomorphism from $G$ to $H$. Let $\\map \\ker \\phi$ denote the kernel of $\\phi$. Then: :$\\map \\ker \\phi \\ne \\O$ where $\\O$ denotes the empty set."}
{"_id": "11776", "text": "Linear Second Order ODE/2 y'' + 2 y' + 3 y = 0 The second order ODE: :$(1): \\quad 2 y'' + 2 y' + 3 y = 0$ has the general solution: :$y = e^{-x/2} \\paren {C_1 \\cos \\dfrac {\\sqrt 5} 2 x + C_2 \\sin \\dfrac {\\sqrt 5} 2 x}$"}
{"_id": "16605", "text": "Topological Properties of Non-Archimedean Division Rings/Intersection of Closed Balls :$\\map { {B_r}^-} x \\cap \\map { {B_s}^-} y \\ne \\O \\iff \\map { {B_r}^-} x \\subseteq \\map { {B_s}^-} y$ or $\\map { {B_s}^-} y \\subseteq \\map { {B_r}^-} x$"}
{"_id": "8286", "text": "Set of Rational Numbers whose Numerator Divisible by p is Closed under Addition Let $p$ be a prime number. Let $A_p$ be the set of all rational numbers which, when expressed in canonical form has a numerator which is divisible by $p$. Then $A_p$ is closed under rational addition."}
{"_id": "5762", "text": "Metacategory Induces Morphisms-Only Metacategory Let $\\mathbf C$ be a metacategory. Then the collection of morphisms $\\mathbf C_1$ of $\\mathbf C$ is a morphisms-only metacategory."}
{"_id": "17924", "text": "Definite Integral from 0 to 2 Pi of Reciprocal of Square of a plus b Sine x :$\\displaystyle \\int_0^{2 \\pi} \\frac {\\d x} {\\paren {a + b \\sin x}^2} = \\frac {2 \\pi a} {\\paren {a^2 - b^2}^{3/2} }$"}
{"_id": "2035", "text": "Binomial Coefficient of Prime Plus One Modulo Prime Let $p$ be a prime number. Then: :$2 \\le k \\le p - 1 \\implies \\dbinom {p + 1} k \\equiv 0 \\pmod p$ where $\\dbinom {p + 1} k$ denotes a binomial coefficient."}
{"_id": "16543", "text": "Normalizer of Subgroup of Symmetric Group that Fixes n Let $S_n$ denote the symmetric group on $n$ letters. Let $H$ denote the subgroup of $S_n$ which consists of all $\\pi \\in S_n$ such that: :$\\map \\pi n = n$ The normalizer of $H$ is given by: :$\\map {N_{S_n} } H = \\map {N_{S_n} } {S_{n - 1} } = S_{n - 1}$"}
{"_id": "10001", "text": "Primitive of Cube of Hyperbolic Cotangent of a x :$\\ds \\int \\coth^3 a x \\rd x = \\frac {\\ln \\size {\\sinh a x} } a - \\frac {\\coth^2 a x} {2 a} + C$"}
{"_id": "17257", "text": "Quotient of Ring of Polynomials in Ring Element on Integral Domain by that Polynomial is that Domain Let $\\struct {D, +, \\times}$ be an integral domain. Let $X \\in R$ be transcencental over $D$. Let $D \\sqbrk X$ be the ring of polynomials in $X$ over $D$. Let $D \\sqbrk X / \\ideal X$ denote the quotient ring of $D \\sqbrk X$ by the ideal of $D$ generated by $X$. Then: :$D \\sqbrk X / \\ideal X \\cong D$"}
{"_id": "10219", "text": "Elements of Geometric Sequence from One where First Element is not Power of Number Let $G_n = \\sequence {a_n}_{0 \\mathop \\le i \\mathop \\le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Let $k \\in \\Z_{> 1}$. Let $a_1$ not be a power of $k$. Then $a_m$ is not a power of $k$ except for: :$\\forall m, k \\in \\set {1, 2, \\ldots, n}: k \\divides m$ where $\\divides$ denotes divisibility. {{:Euclid:Proposition/IX/10}}"}
{"_id": "9970", "text": "Primitive of Hyperbolic Cosine of a x over Power of x :$\\ds \\int \\frac {\\cosh a x \\rd x} {x^n} = \\frac {-\\cosh a x} {\\paren {n - 1} x^{n - 1} } + \\frac a {n - 1} \\int \\frac {\\sinh a x \\rd x} {x^{n - 1} } + C$"}
{"_id": "16430", "text": "Hilbert Proof System Instance 2 Independence Results/RST4 is Derivable Rule of inference $RST \\, 4$ is derivable from $RST \\, 1, RST \\, 2, RST \\, 3$ and the axioms $(A1)$ through $(A4)$."}
{"_id": "18738", "text": "Fourier Series/Sawtooth Wave 600pxthumbrightSawtooth Wave and $6$th Approximation Let $\\map S x$ be the sawtooth wave defined on the real numbers $\\R$ as: :$\\forall x \\in \\R: \\map S x = \\begin {cases} x & : x \\in \\openint {-l} l \\\\ \\map S {x + 2 l} & : x < -l \\\\ \\map S {x - 2 l} & : x > +l \\end {cases}$ where $l$ is a given real constant. Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \\map S x       | o = \\sim       | r = \\frac {2 l} \\pi \\sum_{n \\mathop = 1}^\\infty \\frac {\\paren {-1}^{n + 1} } n \\sin \\dfrac {n \\pi x} l       | c =  }} {{eqn | r = \\frac {2 l} \\pi \\paren {\\sin \\dfrac {\\pi x} l - \\frac 1 2 \\sin \\dfrac {2 \\pi x} l + \\frac 1 3 \\sin \\dfrac {3 \\pi x} l + \\dotsb}       | c =  }} {{end-eqn}}"}
{"_id": "3078", "text": "Open Set in Partition Topology is Component Let $T = \\struct {S, \\tau}$ be a partition topological space. Then each of its open sets are components of $T$."}
{"_id": "963", "text": "Basis for Finite Submodule of Function Space Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $A$ be a set. For each $a \\in A$, let $f_a: A \\to R$ be defined as: :$\\forall x \\in A: \\map {f_a} x = \\begin{cases} 1 & : x = a \\\\ 0 & : x \\ne a \\end{cases}$ Then $B = \\set {f_a: a \\in A}$ is a basis of the Finite Submodule of Function Space $R^{\\paren A}$."}
{"_id": "16527", "text": "Order of Additive Group of Integers Modulo m Let $\\struct {\\Z_m, +_m}$ denote the additive group of integers modulo $m$. The order of $\\struct {\\Z_m, +_m}$ is $m$."}
{"_id": "7722", "text": "Continuum Property implies Well-Ordering Principle The Continuum Property of the positive real numbers $\\R_{\\ge 0}$ implies the Well-Ordering Principle of the natural numbers $\\N$."}
{"_id": "295", "text": "Strictly Monotone Mapping is Monotone A mapping that is strictly monotone is a monotone mapping."}
{"_id": "17385", "text": "Complex Number is Algebraic over Real Numbers Let $z \\in \\C$ be a complex number. Then $z$ is algebraic over $\\R$."}
{"_id": "11349", "text": "Point is Isolated iff belongs to Set less Derivative Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $H \\subseteq S$. Let $x \\in S$. Then: :$x$ is an isolated point in $H$ {{iff}}: :$x \\in H \\setminus H'$ where :$H'$ denotes the derivative of $H$."}
{"_id": "17074", "text": "Sequence of Imaginary Reciprocals/Countability The set $S$ is countably infinite."}
{"_id": "609", "text": "Positive Integer is Well-Defined \"Positive\" as applied to an integer is well-defined."}
{"_id": "11490", "text": "Orthogonal Trajectories/Rectangular Hyperbolas Consider the one-parameter family of curves of rectangular hyperbolas: :$(1): \\quad x y = c$ Its family of orthogonal trajectories is given by the equation: :$x^2 - y^2 = c$ :600px"}
{"_id": "12924", "text": "Dominated Strategy may be Optimal A dominated strategy of a game may be the optimal strategy for a player of that game."}
{"_id": "12803", "text": "Termial on Real Numbers is Extension of Integers The termial function as defined on the real numbers is an extension of its definition on the integers $\\Z$."}
{"_id": "2329", "text": "Solution to Legendre's Differential Equation The solution of Legendre's differential equation: {{:Definition:Legendre's Differential Equation}} can be obtained by Power Series Solution Method. {{refactor|Include the actual solution here in the Theorem section and then proceed to derive each instance of that solution in the Proof section.  If necessary, split it up into bits.  At the moment it is too amorphous to be able to be followed easily.  My eyes are too sore to do a good job on this tonight so I won't.}}"}
{"_id": "19590", "text": "Leigh.Samphier/Sandbox/Independent Superset of Dependent Set Minus Singleton Doesn't Contain Singleton Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $C$ be a dependent subset of $M$. Let $x \\in S$. Let $X$ be an independent subset of $M$ such that: :$C \\setminus \\set x \\subseteq X$. Then: :$x \\notin X$"}
{"_id": "4370", "text": "Eigenvalues of Normal Operator have Orthogonal Eigenspaces Let $H$ be a Hilbert space. Let $A \\in B \\left({H}\\right)$ be a normal operator. Let $\\lambda, \\mu$ be distinct eigenvalues of $A$. Then: :$\\ker \\left({A - \\lambda}\\right) \\perp \\ker \\left({A - \\mu}\\right)$ where: : $\\ker$ denotes kernel : $\\perp$ denotes orthogonality."}
{"_id": "4032", "text": "Preimage of Set Difference under Mapping/Corollary 1 Let $f: S \\to T$ be a mapping. Let $T_1 \\subseteq T_2 \\subseteq T$. Then: :$\\complement_{f^{-1} \\left[{T_2}\\right]} \\left({f^{-1} \\left[{T_1}\\right]}\\right) = f^{-1} \\left[{\\complement_{T_2} \\left({T_1}\\right)}\\right]$ where: : $\\complement$ (in this context) denotes relative complement : $f^{-1} \\left[{T_1}\\right]$ denotes preimage."}
{"_id": "14781", "text": "Square of Repunit times Sum of Digits The following pattern emerges: {{begin-eqn}} {{eqn | l = 121 \\times \\paren {1 + 2 + 1}       | r = 22^2 }} {{eqn | l = 12 \\, 321 \\times \\paren {1 + 2 + 3 + 2 + 1}       | r = 333^2 }} {{eqn | l = 1 \\, 234 \\, 321 \\times \\paren {1 + 2 + 3 + 4 + 3 + 2 + 1}       | r = 4444^2 }} {{end-eqn}} and so on, up until $999 \\, 999 \\, 999^2$ after which the pattern breaks down."}
{"_id": "15323", "text": "Countable Open Ordinal Space is Metrizable Let $\\Omega$ denote the first uncountable ordinal. Let $\\Gamma$ be a limit ordinal which strictly precedes $\\Omega$. Let $\\hointr 0 \\Gamma$ denote the open ordinal space on $\\Gamma$. Then $\\hointr 0 \\Gamma$ is a metrizable space."}
{"_id": "1534", "text": "Composite Number has Prime Factor not Greater Than its Square Root Let $n \\in \\N$ and $n = p_1 \\times p_2 \\times \\cdots \\times p_j$, $j \\ge 2$, where $p_1, \\ldots, p_j \\in \\Bbb P$ are prime factors of $n$. Then $\\exists p_i \\in \\Bbb P$ such that $p_i \\le \\sqrt n$. That is, if $n \\in \\N$ is composite, then $n$ has a prime factor $p \\le \\sqrt n$."}
{"_id": "8863", "text": "Satisfiable Set minus Formula is Satisfiable Let $\\mathcal L$ be a logical language. Let $\\mathscr M$ be a formal semantics for $\\mathcal L$. Let $\\mathcal F$ be an $\\mathscr M$-satisfiable set of formulas from $\\mathcal L$. Let $\\phi \\in \\mathcal F$. Then $\\mathcal F \\setminus \\left\\{{\\phi}\\right\\}$ is also $\\mathscr M$-satisfiable."}
{"_id": "10026", "text": "Primitive of Inverse Hyperbolic Cosecant of x over a :$\\displaystyle \\int \\operatorname{csch}^{-1} \\frac x a \\ \\mathrm d x = \\begin{cases} x \\operatorname{csch}^{-1} \\dfrac x a + a \\sinh^{-1} \\dfrac x a + C & : x > 0 \\\\ x \\operatorname{csch}^{-1} \\dfrac x a - a \\sinh^{-1} \\dfrac x a + C & : x < 0 \\end{cases}$"}
{"_id": "6087", "text": "Fort Space is T0 Let $T = \\left({S, \\tau_p}\\right)$ be a Fort space on an infinite set $S$. Then $T$ is a $T_0$ (Kolmogorov) space."}
{"_id": "17943", "text": "Power Series Expansion for Cosine Integral Function plus Logarithm :$\\displaystyle \\map \\Ci x = -\\gamma - \\ln x + \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n + 1} \\frac {x^{2 n} } {\\paren {2 n} \\times \\paren {2 n}!}$"}
{"_id": "6088", "text": "Pullback Functor is Functor Let $\\mathbf C$ be a metacategory having all pullbacks. Let $f: C \\to D$ be a morphism of $\\mathbf C$. Let $\\mathbf C \\mathbin / C$ and $\\mathbf C \\mathbin / D$ be the slice categories over $C$ and $D$, respectively. Let $f^* : \\mathbf C \\mathbin / D \\to \\mathbf C \\mathbin / C$ be the pullback functor defined by $f$. Then $f^*$ is a functor."}
{"_id": "9912", "text": "Primitive of Hyperbolic Cosine of a x over x squared :$\\displaystyle \\int \\frac {\\cosh a x \\ \\mathrm d x} {x^2} = -\\frac {\\cosh a x} x + a \\int \\frac {\\sinh a x \\ \\mathrm d x} x$"}
{"_id": "19524", "text": "Subspace of Normed Vector Space with Induced Norm forms Normed Vector Space Let $\\struct {X, \\norm {\\, \\cdot \\,}_X}$ be a normed vector space. Let $Y \\subseteq X$ be a vector subspace. Let $\\norm {\\, \\cdot \\,}_Y$ be the induced norm on $Y$. Then $\\struct {Y, \\norm {\\, \\cdot \\,}_Y}$ is a normed vector space."}
{"_id": "5113", "text": "Image of Union under Mapping/General Result Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $\\powerset S$ be the power set of $S$. Let $\\mathbb S \\subseteq \\powerset S$. Then: :$\\displaystyle f \\sqbrk {\\bigcup \\mathbb S} = \\bigcup_{X \\mathop \\in \\mathbb S} f \\sqbrk X$"}
{"_id": "16556", "text": "Stabilizer of Subgroup Action is Identity Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $\\struct {H, \\circ}$ be a subgroup of $G$. Let $*: H \\times G \\to G$ be the subgroup action defined for all $h \\in H, g \\in G$ as: :$\\forall h \\in H, g \\in G: h * g := h \\circ g$ The stabilizer of $x \\in G$ is $\\set e$: :$\\Stab x = \\set e$"}
{"_id": "4801", "text": "Integral with respect to Dirac Measure Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $x \\in X$, and let $\\delta_x$ be the Dirac measure at $x$. Let $f \\in \\mathcal M _{\\overline \\R}, f: X \\to \\overline \\R$ be a measurable function. Then: :$\\displaystyle \\int f \\, \\mathrm d \\delta_x = f \\left({x}\\right)$ where the integral sign denotes the $\\delta_x$-integral."}
{"_id": "16649", "text": "Equivalence of Definitions of Generated Subgroup {{TFAE|def = Generated Subgroup}} Let $G$ be a group. Let $S \\subset G$ be a subset."}
{"_id": "1864", "text": "Graph Components are Equivalence Classes The components of a graph are equivalence classes under the relation '''is connected to''' on the set of vertices."}
{"_id": "12164", "text": "Complex Numbers cannot be Extended to Algebra in Three Dimensions with Real Scalars It is not possible to extend the complex numbers to an algebra of $3$ dimensions with real scalars."}
{"_id": "15048", "text": "Smallest Titanic Palindromic Prime The smallest titanic prime that is also palindromic is: :$10^{1000} + 81 \\, 918 \\times 10^{498} + 1$ which can be written as: :$1 \\underbrace {000 \\ldots 000}_{497} 81918 \\underbrace {000 \\ldots 000}_{497} 1$"}
{"_id": "806", "text": "Parity Group is Group The parity group is in fact a group."}
{"_id": "4001", "text": "At Most Two Horizontal Asymptotes The graph of a real function has at most two horizontal asymptotes."}
{"_id": "17421", "text": "P-adic Valuation Extends to P-adic Numbers Let $p$ be a prime number. Let $\\nu_p^\\Q: \\Q \\to \\Z \\cup \\set {+\\infty}$ be the $p$-adic valuation on the set of  rational numbers. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\nu_p: \\Q_p \\to \\Z \\cup \\set {+\\infty}$ be defined by: :$\\forall x \\in \\Q_p : \\map {\\nu_p} x = \\begin {cases} -\\log_p \\norm {x}_p : x \\ne 0 \\\\ +\\infty : x = 0 \\end{cases}$ Then $\\nu_p: \\Q_p \\to \\Z \\cup \\set {+\\infty}$ is a valuation that extends $\\nu_p^\\Q$ from $\\Q$ to $\\Q_p$."}
{"_id": "14573", "text": "Odd Numbers not Sum of Prime and Power The sequence of odd numbers which cannot be expressed as the sum of a perfect power and a prime number begins: :$1, 5, 1549, 1 \\, 771 \\, 561, \\ldots$ {{OEIS|A119747}} It is not known if there are any more terms."}
{"_id": "12506", "text": "Modulo Operation/Examples/-2 mod 3 :$-2 \\bmod 3 = 1$"}
{"_id": "18865", "text": "Geometrical Interpretation of Complex Modulus Let $z \\in \\C$ be a complex number expressed in the complex plane. Then the modulus of $z$ can be interpreted as the distance of $z$ from the origin."}
{"_id": "11618", "text": "Linear First Order ODE/y' + y cot x = 2 x cosec x The linear first order ODE: :$(1): \\quad y' + y \\cot x = 2 x \\csc x$ has the general solution: :$y = x^2 \\csc x + C \\csc x$"}
{"_id": "16683", "text": "Power of Group Element in Kernel of Homomorphism iff Power of Image is Identity Let $G$ be a group whose identity is $e_G$. Let $H$ be a group whose identity is $e_H$. Let $\\phi: G \\to H$ be a (group) homomorphism. Let $x^n \\in \\map \\ker \\phi$ for some integer $n$. Then: :$\\paren {\\map \\phi x}^n = e_H$"}
{"_id": "8325", "text": "Non-Zero Natural Numbers under Multiplication form Commutative Monoid Let $\\N_{>0}$ be the set of natural numbers without zero, i.e. $\\N_{>0} = \\N \\setminus \\set 0$. The structure $\\struct{\\N_{>0}, \\times}$ forms a commutative monoid."}
{"_id": "8557", "text": "Rational Number Space is Topological Space Let $\\struct {\\Q, \\tau_d}$ be the rational number space formed by the rational numbers $\\Q$ under the usual (Euclidean) topology $\\tau_d$. Then $\\tau_d$ forms a topology."}
{"_id": "8784", "text": "Inverse Hyperbolic Cosine of Imaginary Number :$\\cosh^{-1} x = \\pm \\, i \\cos^{-1} x$"}
{"_id": "16536", "text": "Finite Group whose Subsets form Nest is Cyclic P-Group Let $G$ be a group. Let $G$ be such that its subgroups form a nest. Then $G$ is a cyclic $p$-group."}
{"_id": "9775", "text": "Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Sine :$\\displaystyle \\int \\frac {\\d x} {\\sin^m a x \\cos^n a x} = \\frac {-1} {a \\paren {n - 1} \\sin^{m - 1} a x \\cos^{n - 1} a x} + \\frac {m + n - 2} {m - 1} \\int \\frac {\\d x} {\\sin^{m - 2} a x \\cos^n a x} + C$"}
{"_id": "16570", "text": "Floor Function/Examples/Floor of Root 5 :$\\floor {\\sqrt 5} = 2$"}
{"_id": "4095", "text": "Function is Odd Iff Inverse is Odd Let $f$ be an odd real function with an inverse $f^{-1}$. Then $f^{-1}$ is also odd."}
{"_id": "14501", "text": "Mapping at Element is Supremum of Compact Elements implies Mapping is Increasing Let $\\left({S, \\vee_1, \\wedge_1, \\preceq_1}\\right)$ be a lattice. Let $\\left({T, \\vee_2, \\wedge_2, \\preceq_2}\\right)$ be a complete lattice. Let $f: S \\to T$ be a mapping such that :$\\forall x \\in S: f\\left({x}\\right) = \\sup \\left\\{ {f\\left({w}\\right): w \\in S \\land w \\preceq_1 x \\land w}\\right.$ is compact$\\left.{}\\right\\}$ Then $f$ is increasing."}
{"_id": "524", "text": "Quotient Group of Ideal is Coset Space Let $\\struct {R, +, \\circ}$ be a ring. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +}$ be the quotient group of $\\struct {R, +}$ by $\\struct {J, +}$. Then each element of $\\struct {R / J, +}$ is a coset of $J$ in $R$, that is, is of the form $x + J = \\set {x + j: j \\in J}$ for some $x \\in R$. The rule of addition of these cosets is: $\\paren {x + J} + \\paren {y + J} = \\paren {x + y} + J$. The identity of $\\struct {R / J, +}$ is $J$ and for each $x \\in R$, the inverse of $x + J$ is $\\paren {-x} + J$."}
{"_id": "8587", "text": "Rational Number Space is Second-Countable Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is second-countable."}
{"_id": "10784", "text": "Existence of Non-Square Residue Let $m \\in \\Z$ be an integer such that $m > 2$. Let $\\Z_m$ be the set of integers modulo $m$: :$\\Z_m = \\left\\{{\\left[\\!\\left[{0}\\right]\\!\\right]_m, \\left[\\!\\left[{1}\\right]\\!\\right]_m, \\ldots, \\left[\\!\\left[{m - 1}\\right]\\!\\right]_m}\\right\\}$ Then there exists at least one residue in $\\Z_m$ which is not the product modulo $m$ of a residue with itself: :$\\exists p \\in \\Z_M: \\forall x \\in \\Z_m: x \\cdot_m x \\ne p$"}
{"_id": "10529", "text": "Positive Multiple of Metric is Metric Let $M = \\left({A, d}\\right)$ be a metric space. Let $k \\in \\R_{>0}$ be a (strictly) positive real number. Let $d_k: A \\times A \\to \\R$ be the function defined as: :$\\forall \\left({x, y}\\right) \\in A: d_k \\left({x, y}\\right) = k \\cdot d \\left({x, y}\\right)$ Then $M_k = \\left({A, d_k}\\right)$ is a metric space."}
{"_id": "16770", "text": "Construction of Smith Number from Prime Repunit Let $R_n$ be a repunit which is prime where $n \\ge 3$. Then $3304 \\times R_n$ is a Smith number. $3304$ is not the only number this works for, but it is the smallest."}
{"_id": "16614", "text": "Zero Element of Multiplication on Numbers On all the number systems: * natural numbers $\\N$ * integers $\\Z$ * rational numbers $\\Q$ * real numbers $\\R$ * complex numbers $\\C$ the zero element of multiplication is zero ($0$)."}
{"_id": "14168", "text": "Odd Integers whose Smaller Odd Coprimes are Prime Let $n \\in \\Z_{>0}$ be an odd positive integer such that all smaller odd integers greater than $1$ which are coprime to it are prime. The complete list of such $n$ is as follows: :$1, 3, 5, 7, 9, 15, 21, 45, 105$ {{OEIS|A327823}}"}
{"_id": "6262", "text": "Reparameterization of Contour is Contour Let $\\closedint a b$ and $\\closedint c d$ be closed real intervals. Let $\\gamma: \\closedint a b \\to \\C$ be a contour in $\\C$. That is, there exists a subdivision $a_0, a_1 , \\ldots, a_n$ of $\\closedint a b$ such that $\\gamma \\restriction_{I_i}$ is a smooth path for all $i \\in \\set {1, \\ldots, n}$, where $I_i = \\closedint {a_{i - 1} } {a_i}$. Here, $\\gamma \\restriction_{I_i}$ denotes the restriction of $\\gamma$ to $I_i$. Let $\\phi: \\closedint c d \\to \\closedint a b$  be a differentiable real function with $\\map \\phi c = a$, and $\\map \\phi d = b$. Suppose that $\\phi$ is either bijective or strictly increasing. Define $\\sigma: \\closedint c d \\to \\C$ as a composite function by $\\sigma = \\gamma \\circ \\phi$. Then $\\sigma$ is a contour. For all $i \\in \\set {1, \\ldots, n}$, let $J_i := \\closedint {\\map {\\phi^{-1} } {a_{i - 1} } } {\\map {\\phi^{-1} } {a_i} }$. Then: :$\\map {\\sigma' \\restriction_{J_i} } t = \\map {\\gamma' \\restriction_{I_i} } {\\map \\phi t} \\map {\\phi'} t$"}
{"_id": "17963", "text": "Power Series Expansion for Fresnel Sine Integral Function :$\\displaystyle \\map {\\operatorname S} x = \\sqrt {\\frac 2 \\pi} \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{4 n + 3} } {\\paren {4 n + 3} \\paren {2 n + 1}!}$"}
{"_id": "14842", "text": "Algebra Defined by Ring Homomorphism on Ring with Unity is Unitary Let $R$ be a commutative ring. Let $\\struct {S, +, *}$ be a ring with unity. Let $f: R \\to S$ be a ring homomorphism. Let $\\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$. Then $\\struct {S_R, *}$ is a unitary algebra."}
{"_id": "11060", "text": "Sine in terms of Tangent {{begin-eqn}} {{eqn | l = \\sin x       | r = +\\frac {\\tan x} {\\sqrt {1 + \\tan^2 x} }       | c = if there exists an integer $n$ such that $\\paren {2 n - \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\sin x       | r = -\\frac {\\tan x} {\\sqrt {1 + \\tan^2 x} }       | c = if there exists an integer $n$ such that $\\paren {2 n + \\dfrac 1 2} \\pi < x < \\paren {2 n + \\dfrac 3 2} \\pi$ }} {{end-eqn}}"}
{"_id": "16833", "text": "Symmetric Group on 3 Letters is Isomorphic to Dihedral Group D3 Let $S_3$ denote the Symmetric Group on 3 Letters. Let $D_3$ denote the dihedral group $D_3$. Then $S_3$ is isomorphic to $D_3$."}
{"_id": "17111", "text": "Squares of Diagonals of Parallelogram Let $ABCD$ be a parallelogram. :400px Then: :$AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + DA^2$"}
{"_id": "4076", "text": "Quaternion Group not Dihedral Group Let $Q$ be the quaternion group. Then $Q$ is not isomorphic to the dihedral group $D_4$."}
{"_id": "4199", "text": "Mappings Partially Ordered by Extension Let $S$ and $T$ be sets. Let $F$ be the set of all mappings from $S$ to $T$. Let $\\mathcal R \\subseteq F \\times F$ be the relation defined as: :$\\left({f, g}\\right) \\in \\mathcal R \\iff \\operatorname{Dom} \\left({f}\\right) \\subseteq \\operatorname{Dom} \\left({g}\\right) \\land \\forall x \\in \\operatorname{Dom} \\left({f}\\right): f \\left({x}\\right) = g \\left({x}\\right)$ That is, $f \\mathrel{\\mathcal R} g$ {{iff}} $g$ is an extension of $f$. Then $\\mathcal R$ is an ordering on $F$."}
{"_id": "18885", "text": "Positive Real Axis forms Subgroup of Complex Numbers under Multiplication Let $S$ be the subset of the set of complex numbers $\\C$ defined as: :$S = \\set {z \\in \\C: z = x + 0 i, x > 0}$ That is, let $S$ be the positive real axis of the complex plane. Then the algebraic structure $\\struct {S, \\times}$ is a subgroup of the multiplicative group of complex numbers $\\struct {\\C_{\\ne 0}, \\times}$."}
{"_id": "4541", "text": "P-adic Valuation of Rational Number is Well Defined The $p$-adic valuation: : $\\nu_p: \\Q \\to \\Z \\cup \\left\\{{+\\infty}\\right\\}$ is well defined."}
{"_id": "8474", "text": "Limit of Sets Exists iff Limit Inferior contains Limit Superior Let $\\Bbb S = \\set {E_n : n \\in \\N}$ be a sequence of sets. Then $\\Bbb S$ converges to a limit {{iff}}: :$\\ds \\limsup_{n \\mathop \\to \\infty} E_n \\subseteq \\liminf_{n \\mathop \\to \\infty}E_n$"}
{"_id": "12479", "text": "Ceiling Function/Examples/Ceiling of Root 2 :$\\ceiling {\\sqrt 2} = 2$"}
{"_id": "18743", "text": "Inverse Image Mapping Induced by Projection Let $\\family {S_i}_{i \\mathop \\in I}$ be a family of sets. Let $\\displaystyle S = \\prod_{i \\mathop \\in I} S_i$ be the Cartesian product of $\\family {S_i}_{i \\mathop \\in I}$. For each $j \\in I$, let $\\pr_j: S \\to S_j$ denote the $j$-th projection. For each $j \\in I$ let $\\pr_j^\\gets: \\powerset {S_i} \\to \\powerset S$ denote the inverse image mapping induced by $\\pr_j$ Then for all $j \\in I$, $\\pr_j^\\gets$ is the mapping defined by: :$\\forall T \\subseteq S_i: \\map {\\pr_j^\\gets} T = \\displaystyle \\prod_{i \\mathop \\in I} T_i$ where: :$T_i = \\begin {cases} T & i = j \\\\ S_i & i \\ne j \\end {cases}$"}
{"_id": "10270", "text": "Condition for Commensurability of Roots of Quadratic Equation Consider the quadratic equation: :$(1): \\quad a x - x^2 = \\dfrac {b^2} 4$ Then $x$ and $a - x$ are commensurable {{iff}} $\\sqrt{a^2 - b^2}$ and $a$ are commensurable. {{:Euclid:Proposition/X/17}}"}
{"_id": "4751", "text": "Measurable Functions Determine Measurable Sets Let $\\struct {X, \\Sigma}$ be a measurable space. Let $f, g: X \\to \\overline \\R$ be $\\Sigma$-measurable functions. Then the following sets are measurable: :$\\set {f < g}$ :$\\set {f \\le g}$ :$\\set {f = g}$ :$\\set {f \\ne g}$ where, for example, $\\set {f < g}$ is short for $\\set {x \\in X: \\map f x < \\map g x}$."}
{"_id": "16675", "text": "C6 is not Isomorphic to S3 Let $C_6$ denote the cyclic group of order $6$. Let $S_3$ denote the symmetric group on $3$ letters. Then $C_6$ and $S_3$ are not isomorphic."}
{"_id": "9463", "text": "Primitive of Root of x squared minus a squared over x squared :$\\displaystyle \\int \\frac {\\sqrt {x^2 - a^2} } {x^2} \\rd x = \\frac {-\\sqrt {x^2 - a^2} } x + \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$"}
{"_id": "16596", "text": "Set of Even Integers is Equivalent to Set of Integers Let $\\Z$ denote the set of integers. Let $2 \\Z$ denote the set of even integers. Then: :$2 \\Z \\sim \\Z$ where $\\sim$ denotes set equivalence."}
{"_id": "9840", "text": "Primitive of Arctangent of x over a over x squared :$\\displaystyle \\int \\frac {\\arctan \\frac x a \\ \\mathrm d x} {x^2} = \\frac {-\\arctan \\frac x a} x - \\frac 1 {2 a} \\ln \\left({\\frac {x^2 + a^2} {x^2} }\\right) + C$"}
{"_id": "12656", "text": "Row in Pascal's Triangle forms Palindromic Sequence Each of the rows of Pascal's triangle forms a palindromic sequence."}
{"_id": "7531", "text": "Negative Binomial Distribution as Generalized Geometric Distribution/First Form The first form of the negative binomial distribution is a generalization of the geometric distribution: Let $\\left \\langle{X_i}\\right \\rangle$ be a Bernoulli process with parameter $p$. Let $\\mathcal E$ be the experiment which consists of: : Perform the Bernoulli trial $X_i$ until $n$ failures occur, and then stop. Let $k$ be the number of successes before before $n$ failures have been encountered. Let $\\mathcal E'$ be the experiment which consists of: : Perform the Bernoulli trial $X_i$ until '''one''' failure occurs, and then stop. Then $k$ is modelled by the experiment: : Perform experiment $\\mathcal E'$ until $n$ failures occur, and then stop."}
{"_id": "11728", "text": "Solution of Linear 2nd Order ODE Tangent to X-Axis Let $\\map {y_p} x$ be a particular solution to the homogeneous linear second order ODE: :$(1): \\quad \\dfrac {\\d^2 y} {\\d x^2} + \\map P x \\dfrac {\\d y} {\\d x} + \\map Q x y = 0$ on a closed interval $\\closedint a b$. Let there exist $\\xi \\in \\closedint a b$ such that the curve in the cartesian plane described by $y = \\map {y_p} x$ is tangent to the $x$-axis at $\\xi$. Then $\\map {y_p} x$ is the zero constant function: :$\\forall x \\in \\closedint a b: \\map {y_p} x = 0$"}
{"_id": "19025", "text": "Differentiable Function with Bounded Derivative is of Bounded Variation Let $a, b$ be real numbers with $a < b$. Let $f : \\closedint a b \\to \\R$ be a continuous function. Let $f$ be differentiable on $\\openint a b$, with bounded derivative. Then $f$ is of bounded variation."}
{"_id": "12239", "text": "Set Intersection is not Cancellable Set intersection is not a cancellable operation. That is, for a given $A, B, C \\subseteq S$ for some $S$, it is not always the case that: :$A \\cap B = A \\cap C \\implies B = C$"}
{"_id": "15537", "text": "Sum of Series of Product of Power and Cosine Let $r \\in \\R$. Let $x \\in \\R$ such that $x \\ne 2 m \\pi$ for any $m \\in \\Z$. Then: {{begin-eqn}} {{eqn | l = \\sum_{k \\mathop = 0}^n r^k \\map \\cos {k x}       | r = 1 + r \\cos x + r^2 \\cos 2 x + r^3 \\cos 3 x + \\cdots + r^n \\cos n x       | c =  }} {{eqn | r = \\dfrac {r^{n + 2} \\cos n x - r^{n + 1} \\map \\cos {n + 1} x - r \\cos x + 1} {1 - 2 r \\cos x + r^2}       | c =  }} {{end-eqn}}"}
{"_id": "7112", "text": "Transitive Relation whose Symmetric Closure is not Transitive Let $S = \\set {p, q}$, where $p$ and $q$ are distinct elements. Let $\\RR = \\set {\\tuple {p, q} }$. Then $\\RR$ is transitive but its symmetric closure is not."}
{"_id": "15035", "text": "Equivalence of Definitions of Minimal Polynomial {{TFAE|def = Minimal Polynomial}} Let $L / K$ be a field extension. Let $\\alpha \\in L$ be algebraic over $K$."}
{"_id": "2218", "text": "Image of Successor Mapping forms Peano Structure Let $\\left({P, s, 0}\\right)$ be a Peano structure. Let $P'$ be the set $s \\left[{P}\\right]$, that is: :$P' = \\left\\{{s \\left({n}\\right): n \\in P}\\right\\}$ Let $s'$ be the restriction of $s$ to $P'$. Then $\\left({P', s', s \\left({0}\\right)}\\right)$ is also a Peano structure."}
{"_id": "19262", "text": "Finite Dimensional Real Vector Space with Euclidean Norm form Normed Vector Space Let $\\R^n$ be an n-dimensional real vector space. Let $\\norm {\\, \\cdot \\,}_2$ be the Euclidean norm. Then $\\struct {\\R^n, \\norm {\\, \\cdot \\,}_2}$ is a normed vector space."}
{"_id": "1113", "text": "Laplace's Expansion Theorem Let $D$ be the determinant of order $n$. Let $r_1, r_2, \\ldots, r_k$ be integers such that: :$1 \\le k < n$ :$1 \\le r_1 < r_2 < \\cdots < r_k \\le n$ Let $\\map D {r_1, r_2, \\ldots, r_k \\mid u_1, u_2, \\ldots, u_k}$ be an order-$k$ minor of $D$. Let $\\map {\\tilde D} {r_1, r_2, \\ldots, r_k \\mid u_1, u_2, \\ldots, u_k}$ be the cofactor of $\\map D {r_1, r_2, \\ldots, r_k \\mid u_1, u_2, \\ldots, u_k}$. Then: :$\\displaystyle D = \\sum_{1 \\mathop \\le u_1 \\mathop < \\cdots \\mathop < u_k \\mathop \\le n} \\map D {r_1, r_2, \\ldots, r_k \\mid u_1, u_2, \\ldots, u_k} \\, \\map {\\tilde D} {r_1, r_2, \\ldots, r_k \\mid u_1, u_2, \\ldots, u_k}$ A similar result applies for columns."}
{"_id": "8872", "text": "Semantic Consequence as Tautological Conditional Let $\\mathcal F$ be a finite set of WFFs of propositional logic. Let $\\mathbf A$ be another WFF. Then the following are equivalent: {{begin-eqn}} {{eqn|l = \\mathcal F      |o = \\models_{\\mathrm{BI} }      |r = \\mathbf A }} {{eqn|o = \\models_{\\mathrm{BI} }       |r = \\bigwedge \\mathcal F \\implies \\mathbf A }} {{end-eqn}} that is, $\\mathbf A$ is a semantic consequence of $\\mathcal F$ iff $\\displaystyle \\bigwedge \\mathcal F \\implies \\mathbf A$ is a tautology. Here, $\\displaystyle \\bigwedge \\mathcal F$ is the conjunction of $\\mathcal F$."}
{"_id": "6083", "text": "Half-Open Real Interval is Closed in some Open Intervals Let $\\R$ be the real number line considered as an Euclidean space. Let $\\left[{a \\,.\\,.\\, b}\\right) \\subset \\R$ be a half-open interval of $\\R$. Let $c < a$. Then $\\left[{a \\,.\\,.\\, b}\\right)$ is a closed set of $\\left({c \\,.\\,.\\, b}\\right)$. Similarly, let $d > b$. Then the half-open interval $\\left({a \\,.\\,.\\, b}\\right]$ is a closed set of $\\left({a \\,.\\,.\\, d}\\right)$."}
{"_id": "17154", "text": "Unity of Subring is not necessarily Unity of Ring Let $\\struct {S, +, \\circ}$ be a ring with unity whose unity is $1_S$. Let $\\struct {T, + \\circ}$ be a subring of $\\struct {S, + \\circ}$ whose unity is $1_T$. Then it is not necessarily the case that $1_T = 1_S$."}
{"_id": "12246", "text": "Image of Real Square Function The image of the real square function is the entire set of positive real numbers $\\R_{\\ge 0}$."}
{"_id": "12602", "text": "Stirling Number of the Second Kind of Number with Greater Let $\\displaystyle {n \\brace k}$ denote a Stirling number of the second kind. Then: :$\\displaystyle {n \\brace k} = 0$"}
{"_id": "8507", "text": "Relation is Reflexive Symmetric and Antisymmetric iff Diagonal Relation Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a relation in $S$. Then: :$\\RR$ is reflexive, symmetric and antisymmetric {{iff}}: :$\\RR$ is the diagonal relation $\\Delta_S$."}
{"_id": "14938", "text": "Upper Bound for Abscissa of Absolute Convergence of Product of Dirichlet Series Let $f, g: \\N \\to \\C$ be arithmetic functions with Dirichlet convolution $h = f * g$. Let $F, G, H$ be their Dirichlet series. Let $\\sigma_f, \\sigma_g, \\sigma_h$ be their abscissae of absolute convergence. Then: :$\\sigma_h \\le \\max \\set {\\sigma_f, \\sigma_g}$"}
{"_id": "1885", "text": "Number of Components after Removal of Bridge Let $G = \\left({V, E}\\right)$ be a graph. Let $e \\in E$ be a bridge. Let $m$ be the number of components of $G$. Then when $e$ is removed from $G$, the number of components in the remaining graph is $m + 1$."}
{"_id": "17115", "text": "Real Part of Complex Product Let $z_1$ and $z_2$ be complex numbers. Then: :$\\map \\Re {z_1 z_2} = \\map \\Re {z_1} \\, \\map \\Re {z_2} - \\map \\Im {z_1} \\, \\map \\Im {z_2}$"}
{"_id": "14040", "text": "Dirichlet Convolution is Commutative : $f * g = g * f$"}
{"_id": "16342", "text": "Equivalence Relation on Integers Modulo 5 induced by Squaring/Multiplication Modulo Beta is Well-Defined Let the $\\times_\\beta$ operator (\"multiplication\") on the $\\beta$-equivalence classes be defined as: :$\\eqclass a \\beta \\times_\\beta \\eqclass b \\beta := \\eqclass {a \\times b} \\beta$ Then such an operation is well-defined."}
{"_id": "277", "text": "Set is Equivalent to Proper Subset of Power Set Every set is equivalent to a proper subset of its power set: : $\\forall S: \\exists T \\subset \\mathcal P \\left({S}\\right): S \\sim T$"}
{"_id": "4640", "text": "Projection is Surjection/General Version For all non-empty sets $S_1, S_2, \\ldots, S_j, \\ldots, S_n$, the $j$th projection $\\operatorname{pr}_j$ on $\\displaystyle \\prod_{i \\mathop = 1}^n S_i$ is a surjection."}
{"_id": "16506", "text": "Bijection between R x (S x T) and (R x S) x T Let $R$, $S$ and $T$ be sets. Let $S \\times T$ be the Cartesian product of $S$ and $T$. Then there exists a bijection from $R \\times \\paren {S \\times T}$ to $\\paren {R \\times S} \\times T$. Hence: :$\\card {R \\times \\paren {S \\times T} } = \\card {\\paren {R \\times S} \\times T}$"}
{"_id": "14096", "text": "Necessary and Sufficient Condition for First Order System to be Field for Functional Let $\\mathbf y$ be an N-dimensional vector. Let $J$ be a (real) functional such that: :$\\ds J \\sqbrk {\\mathbf y} = \\int_a^b \\map F {x, \\mathbf y, \\mathbf y'} \\rd x$ Let the corresponding momenta and Hamiltonian be: {{begin-eqn}} {{eqn | l = \\map {\\mathbf p} {x, \\mathbf y, \\mathbf y'}       | r = \\dfrac {\\partial \\map F {x, \\mathbf y, \\mathbf y'} } {\\partial \\mathbf y'} }} {{eqn | l = \\map H {x, \\mathbf y, \\mathbf y'}       | r = -\\map F {x, \\mathbf y, \\mathbf y'} + \\mathbf p \\mathbf y' }} {{end-eqn}} Let the following be a family of boundary conditions: :$(1): \\quad \\map {\\mathbf y'} x = \\map {\\boldsymbol \\psi} {x, \\mathbf y}$ Then a family of boundary conditions is a field for the functional $J$ {{iff}} $\\forall x \\in \\closedint a b$ the following self-adjointness and consistency relations hold: {{begin-eqn}} {{eqn | l = \\dfrac {\\partial \\map {p_i} {x, \\mathbf y, \\map {\\boldsymbol \\psi} {x, \\mathbf y} } } {\\partial y_k}       | r = \\dfrac {\\partial \\map {p_k} {x, \\mathbf y, \\map {\\boldsymbol \\psi} {x, \\mathbf y} } } {\\partial y_i} }} {{eqn | l = \\dfrac {\\partial \\map {\\mathbf p} {x, \\mathbf y, \\map {\\boldsymbol \\psi} {x, \\mathbf y} } } {\\partial x}       | r = -\\dfrac {\\partial \\map H {x, \\mathbf y, \\map {\\boldsymbol \\psi} {x, \\mathbf y} } } {\\partial \\mathbf y} }} {{end-eqn}} {{explain|Why \"consistency\"?}} {{explain|Meaning of square brackets, above and throughout}}"}
{"_id": "15279", "text": "Fréchet Space (Functional Analysis) is Complete Metric Space Let $\\struct {\\R^\\omega, d}$ be the '''Fréchet space on $\\R^\\omega$'''. Then $\\struct {\\R^\\omega, d}$ is a complete metric space."}
{"_id": "17699", "text": "Sum of Sequence of k x k! Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = \\sum_{j \\mathop = 1}^n j \\times j!       | r = 1 \\times 1! + 2 \\times 2! + 3 \\times 3! + \\dotsb + n \\times n!       | c =  }} {{eqn | r = \\paren {n + 1}! - 1       | c =  }} {{end-eqn}}"}
{"_id": "10217", "text": "Elements of Geometric Sequence from One which are Powers of Number Let $G_n = \\sequence {a_n}_{0 \\mathop \\le i \\mathop \\le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Then: :$\\forall m, k \\in \\set {1, \\ldots, n}: k \\divides m \\implies a_m$ is a power of $k$ where $\\divides$ denotes divisibility. {{:Euclid:Proposition/IX/8}}"}
{"_id": "12995", "text": "Best Rational Approximations to Root 2 generate Pythagorean Triples Consider the Sequence of Best Rational Approximations to Square Root of 2: :$\\sequence S := \\dfrac 1 1, \\dfrac 3 2, \\dfrac 7 5, \\dfrac {17} {12}, \\dfrac {41} {29}, \\dfrac {99} {70}, \\dfrac {239} {169}, \\dfrac {577} {408}, \\ldots$ Every other term of $\\sequence S$ can be expressed as: :$\\dfrac {2 a + 1} b$ such that: :$a^2 + \\left({a + 1}\\right)^2 = b^2$ :$b$ is odd."}
{"_id": "15926", "text": "Line Parallel to Base of Trapezoid which Bisects Leg also Bisects Other Leg Let $ABCD$ be a trapezoid whose bases are $AB$ and $CD$. Let $EF$ be a straight line parallel to $AB$ and $CD$ which intersects its legs $AD$ and $BC$. Let $EF$ bisect $AD$. Then $EF$ also bisects $BC$."}
{"_id": "8198", "text": "Bijection iff exists Mapping which is Left and Right Inverse Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Then $f$ is a bijection {{iff}}: :there exists a mapping $g: T \\to S$ such that: ::$g \\circ f = I_S$ ::$f \\circ g = I_T$ :where $I_S$ and $I_T$ are the identity mappings on $S$ and $T$ respectively."}
{"_id": "13692", "text": "Numbers Divisible by Sum and Product of Digits The sequence of positive integers which are divisible by both the sum and product of its digits begins: :$1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 111, 112, 132, 135, \\ldots$ {{OEIS|A038186}}"}
{"_id": "3213", "text": "Excluded Point Space is not Locally Arc-Connected Let $T = \\struct {S, \\tau_{\\bar p} }$ be an excluded point space. Then $T$ is not locally arc-connected."}
{"_id": "19610", "text": "Characterization of Stationary Gaussian Process Let $S$ be a Gaussian stochastic process giving rise to a time series $T$. Let the the mean of $S$ be fixed. Let the autocovariance matrix of $S$ also be fixed. Then $S$ is stationary."}
{"_id": "5636", "text": "Unique Constant in Category of Monoids Let $\\mathbf{Mon}$ be the category of monoids. Then every object $M$ of $\\mathbf{Mon}$ has precisely one constant."}
{"_id": "16616", "text": "Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y') Let $F$ and its partial derivatives $F_y, F_{y'}$ be real functions, defined on the closed interval $I = \\closedint a b$. Let $F, F_y, F_{y'} $ be continuous at every point $\\tuple {x, y}$ for all finite $y'$. Suppose there exists a constant $k > 0$ such that: :$\\map {F_y} {x, y, y'} > k$ Suppose there exist real functions $\\alpha = \\map \\alpha {x, y} \\ge 0$, $\\beta = \\map \\beta {x, y}\\ge 0$ bounded in every bounded region of the plane such that: :$\\size {\\map F {x, y, y'} } \\le \\alpha y'^2 + \\beta$ Then one and only one integral curve of equation $y'' = \\map F {x, y, y'}$ passes through any two points $\\tuple {a, A}$ and $\\tuple {b, B}$ such that $a \\ne b$."}
{"_id": "13161", "text": "Fibonacci Number of Index 3n as Sum of Cubes of Fibonacci Numbers Let $F_n$ denote the $n$th Fibonacci number. Then: :$F_{3 n} = {F_{n + 1} }^3 + {F_n}^3 - {F_{n - 1} }^3$"}
{"_id": "18274", "text": "Cycle Matroid is Matroid Let $G = \\struct{V,E}$ be a graph. Let $\\struct{E, \\mathscr I}$ be the cycle matroid of $G$. Then $\\struct{E, \\mathscr I}$ is a matroid."}
{"_id": "14281", "text": "Smallest Magic Constant of Order 3 Multiplicative Magic Square The magic constant of the smallest possible multiplicative magic square with the smallest magic constant is as follows. {{:Multiplicative Magic Square/Examples/Order 3/Smallest}} </onlyinclude>"}
{"_id": "19458", "text": "Derivative of Cotangent of Function :$\\map {\\dfrac \\d {\\d x} } {\\cot u} = -\\csc^2 u \\dfrac {\\d u} {\\d x}$"}
{"_id": "18651", "text": "Successor Mapping is Inflationary Let $\\omega$ denote the set of natural numbers as defined by the von Neumann construction. Let $s: \\omega \\to \\omega$ denote the successor mapping on $\\omega$. Then $s$ is an inflationary mapping."}
{"_id": "4557", "text": "Measure Space Sigma-Finite iff Cover by Sets of Finite Measure Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Then $\\left({X, \\Sigma, \\mu}\\right)$ is $\\sigma$-finite iff there exists a sequence $\\left({E_n}\\right)_{n \\in \\N}$ in $\\Sigma$ such that: :$(1):\\quad \\displaystyle \\bigcup_{n \\mathop \\in \\N} E_n = X$ :$(2):\\quad \\forall n \\in \\N: \\mu \\left({E_n}\\right) < +\\infty$"}
{"_id": "13115", "text": "Taylor Series reaches closest Singularity Let the singularities of a function be the points at which the function is not analytic. Let $F$ be a complex function. Let $F$ be analytic everywhere except at a finite number of singularities. Let $x_0$ be a point in $\\R$ where $F$ is analytic. Let $R \\in \\R_{>0}$ be the distance from $x_0$ to the closest singularity of $F$. Let $f = F {\\restriction_\\R}$ be a real function. Then: :the Taylor series of $f$ about $x_0$ converges to $f$ at every point $x \\in \\R$ satisfying $\\size {x - x_0} < R$"}
{"_id": "6569", "text": "Odd Power Function is Strictly Increasing/Real Numbers Let $n \\in \\Z_{> 0}$ be an odd positive integer. Let $f_n: \\R \\to \\R$ be the real function defined as: :$\\map {f_n} x = x^n$ Then $f_n$ is strictly increasing."}
{"_id": "6008", "text": "Infinite Intersection of Open Sets of Metric Space may not be Open Let $M = \\struct {A, d}$ be a metric space. Let $U_1, U_2, U_3, \\ldots$ be an infinite set of open sets of $M$. Then it is not necessarily the case that $\\displaystyle \\bigcap_{n \\mathop \\in \\N} U_n$ is itself an open set of $M$."}
{"_id": "17586", "text": "Unbounded Monotone Sequence Diverges to Infinity/Increasing Let $\\sequence {x_n}$ be increasing and unbounded above. Then $x_n \\to +\\infty$ as $n \\to \\infty$."}
{"_id": "15594", "text": "Power Series Expansion for Logarithm of 1 + x over 1 + x {{begin-eqn}} {{eqn | l = \\frac {\\map \\ln {1 + x} } {1 + x}       | r = \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n + 1} H_n x^n       | c =  }} {{eqn | r = x - H_2 x^2 + H_3 x^3 - H_4 x^4 + \\cdots       | c =  }} {{end-eqn}} where $H_n$ denotes the $n$th harmonic number: :$H_n = \\ds \\sum_{r \\mathop = 1}^n \\dfrac 1 r = 1 + \\dfrac 1 2 + \\dfrac 1 3 \\cdots + \\dfrac 1 r$ valid for all $x \\in \\R$ such that $\\size x < 1$."}
{"_id": "13232", "text": "Regular Heptagon is Smallest with no Compass and Straightedge Construction The regular heptagon is the smallest regular polygon (smallest in the sense of having fewest sides) that cannot be constructed using a compass and straightedge construction."}
{"_id": "13490", "text": "Infimum Precedes Coarser Infimum Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $X, Y$ be subsets of $S$ such that :$Y$ is coarser than $X$. Then $\\inf X \\preceq \\inf Y$ where $\\inf X$ denotes the infimum of $X$."}
{"_id": "8485", "text": "Complement of Limit Superior is Limit Inferior of Complements Let $\\left\\{{E_n : n \\in \\N}\\right\\}$ be a sequence of sets. Then: :$\\displaystyle \\complement \\left({\\limsup_{n \\to \\infty} \\ E_n}\\right) = \\liminf_{n \\to \\infty} \\ \\complement \\left({E_n}\\right)$ where $\\limsup$ and $\\liminf$ denote the limit superior and limit inferior, respectively."}
{"_id": "18325", "text": "Area inside Cardioid Consider the cardioid $C$ embedded in a polar plane given by its polar equation: :$r = 2 a \\paren {1 + \\cos \\theta}$ The area inside $C$ is $6 \\pi a^2$."}
{"_id": "5125", "text": "Image of Intersection under Injection/General Result Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $\\mathcal P \\left({S}\\right)$ be the power set of $S$. Then: :$\\displaystyle \\forall \\mathbb S \\subseteq \\mathcal P \\left({S}\\right): f \\left[{\\bigcap \\mathbb S}\\right] = \\bigcap_{X \\mathop \\in \\mathbb S} f \\left[{X}\\right]$  {{iff}} $f$ is an injection."}
{"_id": "10618", "text": "Point in Metric Space has Countable Neighborhood Basis Let $M = \\left({A, d}\\right)$ be a metric space. Let $a \\in A$. Then there exists a basis for the neighborhood system of $a$ which is countable."}
{"_id": "11780", "text": "Linear Second Order ODE/x^2 y'' + 3 x y' + 10 y = 0 The second order ODE: :$(1): \\quad x^2 y'' + 3 x y' + 10 y = 0$ has the general solution: :$y = \\dfrac 1 x \\paren {C_1 \\, \\map \\cos {\\ln x^3} + C_2 \\, \\map \\sin {\\ln x^3} }$"}
{"_id": "172", "text": "Preimage of Set Difference under Relation Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $C$ and $D$ be subsets of $T$. Then: :$\\mathcal R^{-1} \\sqbrk C \\setminus \\mathcal R^{-1} \\sqbrk D \\subseteq \\mathcal R^{-1} \\sqbrk {C \\setminus D}$ where: :$\\setminus$ denotes set difference :$\\mathcal R^{-1} \\sqbrk C$ denotes the preimage of $C$ under $\\mathcal R$."}
{"_id": "12148", "text": "Evolute of Parabola The evolute of the parabola $y = x^2$ is the curve: :$27 X^2 = 16 \\paren {Y - \\dfrac 1 2}^3$"}
{"_id": "10678", "text": "Set of Isolated Points of Metric Space is Disjoint from Limit Points Let $M = \\left({A, d}\\right)$ be a metric space. Let $H \\subseteq A$ be a subset of $A$. Let $H'$ be the set of limit points of $H$. Let $H^i$ be the set of isolated points of $H$. Then: :$H' \\cap H^i = \\varnothing$"}
{"_id": "13604", "text": "Product of Consecutive Triangular Numbers :$2 T_n T_{n - 1} = T_{n^2 - 1}$ where $T_n$ denotes the $n$th triangular number."}
{"_id": "11896", "text": "Eccentricity of Orbit indicates its Total Energy Consider a planet $p$ of mass $m$ orbiting a star $S$ of mass $M$ under the influence of the gravitational field which the two bodies give rise to. Then the total energy of the system determines the eccentricity of the orbit of $p$ around $S$."}
{"_id": "641", "text": "Existence of Greatest Common Divisor Let $a, b \\in \\Z$ be integers such that $a \\ne 0$ or $b \\ne 0$. Then the greatest common divisor of $a$ and $b$ exists."}
{"_id": "16790", "text": "Composite of Injection on Surjection is not necessarily Either Let $f$ be an injection. Let $g$ be a surjection. Let $f \\circ g$ denote the composition of $f$ with $g$. Then it is not necessarily the case that $f \\circ g$ is either a surjection or an injection."}
{"_id": "3186", "text": "Directed Hamilton Cycle Problem is NP-complete Both versions of the Directed Hamilton Cycle problem are NP-complete."}
{"_id": "9272", "text": "Primitive of x cubed over a x + b cubed :$\\displaystyle \\int \\frac {x^3 \\rd x} {\\paren {a x + b}^3} = \\frac x {a^3} - \\frac {3 b^2} {a^4 \\paren {a x + b} } + \\frac {b^3} {2 a^4 \\paren {a x + b}^2} - \\frac {3 b} {a^4} \\ln \\size {a x + b} + C$"}
{"_id": "13909", "text": "Smallest Prime Number not Difference between Power of 2 and Power of 3 $41$ is the smallest prime number which is not the difference between a power of $2$ and a power of $3$."}
{"_id": "3284", "text": "Ramanujan Sum is Multiplicative Let $q \\in \\N_{>0}$, $n \\in \\N$. Let $\\map {c_q} n$ be the Ramanujan sum. Then $\\map {c_q} n$ is multiplicative in $q$."}
{"_id": "15895", "text": "Sum over k of n Choose k by x to the k by kth Harmonic Number/x = -1 While for $x \\in \\R_{> 0}$ be a real number: :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\binom n k x^k H_k = \\left({x + 1}\\right)^n \\left({H_n - \\ln \\left({1 + \\frac 1 x}\\right)}\\right) + \\epsilon$ when $x = -1$ we have: :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\binom n k x^k H_k = \\dfrac {-1} n$ where: : $\\dbinom n k$ denotes a binomial coefficient : $H_k$ denotes the $k$th harmonic number."}
{"_id": "14755", "text": "Numbers whose Cube equals Sum of Sequence of that many Squares The integers $m$ in the following sequence all have the property that $m^3$ is equal to the sum of $m$ consecutive squares: :$m^3 = \\displaystyle \\sum_{k \\mathop = 1}^m \\left({n + k}\\right)^2$ for some $n \\in \\Z_{\\ge 0}$: :$0, 1, 47, 2161, 99 \\, 359, 4 \\, 568 \\, 353, \\ldots$ {{OEIS|A189173}}"}
{"_id": "12011", "text": "Derivative at Point of Inflection Let $f$ be a real function which is twice differentiable on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Let $f$ have a point of inflection at $\\xi \\in \\left({a \\,.\\,.\\, b}\\right)$. Then: :$f'' \\left({\\xi}\\right) = 0$ where $f'' \\left({\\xi}\\right)$ denotes the second derivative of $f$ at $\\xi$."}
{"_id": "3713", "text": "Field Homomorphism Preserves Product Inverses Let $\\phi: \\struct {F_1, +_1, \\times_1} \\to \\struct {F_2, +_2, \\times_2}$ be a field homomorphism. Then: :$\\forall x \\in F_1^*: \\map \\phi {x^{-1} } = \\map \\phi x^{-1}$"}
{"_id": "14892", "text": "Palindromic Cube with Non-Palindromic Root The only known palindromic cube with a root that is not itself palindromic is $10 \\, 662 \\, 526 \\, 601$."}
{"_id": "7481", "text": "Restriction to Subset of Strict Total Ordering is Strict Total Ordering Let $S$ be a set or class. Let $\\prec$ be a strict total ordering on $A$. Let $T$ be a subset or subclass of $A$. Then the restriction of $\\prec$ to $B$ is a strict total ordering of $B$."}
{"_id": "8243", "text": "Division Theorem/Positive Divisor/Existence/Proof 1 For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \\le r < b$: :$\\forall a, b \\in \\Z, b > 0: \\exists q, r \\in \\Z: a = q b + r, 0 \\le r < b$"}
{"_id": "9508", "text": "Primitive of Power of x over a x squared plus b x plus c Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {x^m \\rd x} {a x^2 + b x + c} = \\frac {x^{m - 1} } {\\paren {m - 1} a} - \\frac b a \\int \\frac {x^{m - 1} \\rd x} {a x^2 + b x + c} - \\frac c a \\int \\frac {x^{m - 2} \\rd x} {a x^2 + b x + c}$"}
{"_id": "10197", "text": "Number does not divide Number iff Square does not divide Square Let $a, b \\in \\Z$ be integers. Then: :$a \\nmid b \\iff a^2 \\nmid b^2$ where $a \\nmid b$ denotes that $a$ is not a divisor of $b$. {{:Euclid:Proposition/VIII/16}}"}
{"_id": "17138", "text": "Quaternion Group has Normal Subgroup without Complement Let $Q$ denote the quaternion group. There exists a normal subgroup of $Q$ which has no complement."}
{"_id": "2553", "text": "Sum of All Ring Products is Associative Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\left({S, +}\\right), \\left({T, +}\\right), \\left({U, +}\\right)$ be additive subgroups of $\\left({R, +, \\circ}\\right)$. Let $S T$ be defined as: :$\\displaystyle S T = \\left\\{{\\sum_{i \\mathop = 1}^n s_i \\circ t_i: s_1 \\in S, t_i \\in T, i \\in \\left[{1 \\,.\\,.\\,n}\\right]}\\right\\}$ Then: :$\\left({S T}\\right) U = S \\left({T U}\\right)$"}
{"_id": "1118", "text": "Row Equivalence is Equivalence Relation Row equivalence is an equivalence relation."}
{"_id": "5173", "text": "Mappings in Product of Sets are Surjections/Family of Sets Let $\\family {S_i}_{i \\mathop \\in I}$ be an indexed family of sets. Let $\\struct {P, \\family {\\phi_i} _{i \\mathop \\in I} }$ be a product of $S$ and $T$. Then for all $i \\in I$, $\\phi_i$ is a surjection."}
{"_id": "2620", "text": "Completion of Valued Field Let $\\struct {k, \\norm {\\,\\cdot\\,} }$ be a valued field. Then there exists a completion $\\struct {k', \\norm {\\,\\cdot\\,}'}$ of $\\struct {k, \\norm {\\,\\cdot\\,} }$ such that $\\struct {k', \\norm {\\,\\cdot\\,}'}$ is a valued field. Furthermore, every completion of $\\struct{k, \\norm {\\,\\cdot\\,} }$ is isometrically isomorphic to $\\struct {k', \\norm {\\,\\cdot\\,}'}$."}
{"_id": "2661", "text": "Absolute Value on Ordered Integral Domain is Strictly Positive except when Zero Let $\\struct {D, +, \\times, \\le}$ be an ordered integral domain. For all $a \\in D$, let $\\size a$ denote the absolute value of $a$. Then $\\size a$ is (strictly) positive except when $a = 0$."}
{"_id": "15493", "text": "Definite Integral from 0 to Pi of Sine of m x by Cosine of n x Let $m, n \\in \\Z$ be integers. Then: :$\\displaystyle \\int_0^\\pi \\sin m x \\cos n x \\rd x = \\begin{cases} 0 & : m + n \\text { even} \\\\ \\dfrac {2 m} {m^2 - n^2} & : m + n \\text { odd} \\end{cases}$"}
{"_id": "4596", "text": "Decomposition of Probability Measures Let $\\struct {\\Omega, \\Sigma, P}$ be a probability space. Suppose that for every $\\omega \\in \\Omega$, it holds that: :$\\set \\omega \\in \\Sigma$ that is, that $\\Sigma$ contains all singletons. Then there exist a unique diffuse measure $\\mu$ and a unique discrete measure $\\nu$ such that: :$P = \\mu + \\nu$"}
{"_id": "17780", "text": "Equivalence of Definitions of Weakly Locally Connected at Point Let $T = \\struct {S, \\tau}$ be a topological space. Let $x \\in S$. {{TFAE|def = Weakly Locally Connected at Point}}"}
{"_id": "13036", "text": "Equivalence of Definitions of Abundant Number The following definitions of a abundant number are equivalent:"}
{"_id": "7296", "text": "Closure is Closed Let $\\struct {S, \\preceq}$ be an ordered set. Let $\\cl: S \\to S$ be a closure operator. Let $x \\in S$. Then $\\map \\cl x$ is a closed element of $S$ with respect to $\\cl$."}
{"_id": "3333", "text": "Compact Complement Topology is Topology Let $T = \\struct {\\R, \\tau}$ be the compact complement space. Then $\\tau$ is a topology on $T$."}
{"_id": "2760", "text": "Characterisation of Local Rings Let $R$ be a ring. Let $J \\lhd R$ be a maximal ideal. :$(1): \\quad$ If the set $R \\setminus J$ is precisely the group of units $R^\\times$ of $R$, then $\\tuple {R, J}$ is a local ring. :$(2): \\quad$ If $1 + x$ is a unit in $R$ for all $x \\in J$ then $\\tuple {R, J}$ is local. {{explain|The specific interpretation of the notation $\\tuple {R, J}$ is unclear. The usual notation for a ring is something like $\\struct {R, +, \\times}$.}}"}
{"_id": "9685", "text": "Primitive of Cosecant of a x/Cosecant minus Cotangent Form :$\\ds \\int \\csc a x \\rd x = \\frac 1 a \\ln \\size {\\csc a x - \\cot a x} + C$ where $\\csc a x - \\cot a x \\ne 0$."}
{"_id": "18469", "text": "Rational Cut has Smallest Upper Number Let $r \\in \\Q$ be rational. Let $\\alpha$ be the rational cut consisting of all rational numbers $p$ such that $p < r$. Then $\\alpha$ is indeed a cut, and has a smallest upper number that is $r$."}
{"_id": "8321", "text": "Complex Numbers under Addition form Monoid The set of complex numbers under addition $\\left({\\C, +}\\right)$ forms a monoid."}
{"_id": "2525", "text": "Integration on Polynomials is Linear Operator Let $P \\left({\\R}\\right)$ be the vector space of all polynomial functions on the real number line $\\R$. Let $S$ be the mapping defined as: :$\\displaystyle \\forall p \\in P \\left({\\R}\\right): \\forall x \\in \\R: S \\left({p \\left({x}\\right)}\\right) = \\int_0^x p \\left({t}\\right) \\mathrm d t$ Then $S$ is a linear operator on $P \\left({\\R}\\right)$."}
{"_id": "10241", "text": "Number whose Half is Odd is Even-Times Odd Let $a \\in \\Z$ be an integer such that $\\dfrac a 2$ is an odd integer. Then $a$ is even-times odd. {{:Euclid:Proposition/IX/33}}"}
{"_id": "9768", "text": "Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Sine :$\\displaystyle \\int \\frac {\\sin^m a x} {\\cos^n a x} \\rd x = \\frac {-\\sin^{m - 1} a x} {a \\paren {m - n} \\cos^{n - 1} a x} + \\frac {m - 1} {m - n} \\int \\frac {\\sin^{m - 2} a x} {\\cos^n a x} \\rd x + C$"}
{"_id": "11594", "text": "First Order ODE/x dy - y dx = (1 + y^2) dy The first order ODE: :$(1): \\quad x \\rd y - y \\rd x = \\paren {1 + y^2} \\rd y$ has the general solution: :$\\dfrac x y = \\dfrac 1 y - y + C$"}
{"_id": "4120", "text": "Linear Subspace is Convex Set Let $V$ be a vector space over $\\R$ or $\\C$, and let $L$ be a linear subspace of $V$. Then $L$ is a convex set."}
{"_id": "13766", "text": "Squares of form 2 n^2 - 1 The sequence of integers $\\left\\langle{n}\\right\\rangle$ such that $2 n^2 - 1$ is square begins: :$1, 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, \\ldots$ {{OEIS|A001653}}"}
{"_id": "5911", "text": "Real Number Line with Point Removed is Not Path-Connected Let $\\R$ be the real number line considered as an Euclidean space. Let $x \\in \\R$ be a real number. Then $\\R \\setminus \\set x$, where $\\setminus$ denotes set difference, is not path-connected."}
{"_id": "12482", "text": "Floor Function/Examples/Floor of Minus One Half :$\\floor {-\\dfrac 1 2} = -1$"}
{"_id": "19684", "text": "Ring of Endomorphisms is not necessarily Commutative Ring Let $\\struct {G, \\oplus}$ be an abelian group. Let $\\mathbb G$ be the set of all group endomorphisms of $\\struct {G, \\oplus}$. Let $\\struct {\\mathbb G, \\oplus, *}$ denote the '''ring of endomorphisms''' on $\\struct {G, \\oplus}$. Then $\\struct {\\mathbb G, \\oplus, *}$ is not necessarily a commutative ring with unity."}
{"_id": "3046", "text": "Discrete Uniformity generates Discrete Topology Let $S$ be a set. Let $\\UU$ be the discrete uniformity on $S$. Then the topology generated by $\\UU$ is the discrete topology on $S$. The diagonal relation $\\Delta_S$ generates the basis for this discrete topology."}
{"_id": "9271", "text": "Primitive of x squared over a x + b cubed :$\\ds \\int \\frac {x^2 \\rd x} {\\paren {a x + b}^3} = \\frac {2 b} {a^3 \\paren {a x + b} } - \\frac {b^2} {2 a^3 \\paren {a x + b}^2} + \\frac 1 {a^3} \\ln \\size {a x + b} + C$"}
{"_id": "1868", "text": "Odd Order Complete Graph is Eulerian Let $K_n$ be the complete graph of $n$ vertices. Then $K_n$ is Eulerian {{iff}} $n$ is odd. If $n$ is even, then $K_n$ is traversable iff $n = 2$."}
{"_id": "12754", "text": "Conditions for Floor of Log base b of x to equal Floor of Log base b of Floor of x Let $b \\in \\R$ be a real number. :$\\forall x \\in \\R_{\\ge 1}: \\left\\lfloor{\\log_b x}\\right\\rfloor = \\left\\lfloor{\\log_b \\left\\lfloor{x}\\right\\rfloor}\\right\\rfloor \\iff b \\in \\Z_{> 1}$ where $\\left\\lfloor{x}\\right\\rfloor$ denotes the floor of $x$."}
{"_id": "14367", "text": "Completely Irreducible Element equals Infimum of Subset implies Element Belongs to Subset Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a complete lattice. Let $X \\subseteq S$, $p \\in S$ such that :$p$ is completely irreducible and $p = \\inf X$ Then $p \\in X$"}
{"_id": "17413", "text": "Isomorphism between Additive Group Modulo 16 and Multiplicative Group Modulo 17 Let $\\struct {\\Z_{16}, +}$ denote the additive group of integers modulo $16$. Let $\\struct {\\Z'_{17}, \\times}$ denote the multiplicative group of reduced residues modulo $17$. Let $\\phi: \\struct {\\Z_{16}, +} \\to \\struct {\\Z'_{17}, \\times}$ be the mapping defined as: :$\\forall \\eqclass k {16} \\in \\struct {\\Z_{16}, +}: \\map \\phi {\\eqclass k {16} } = \\eqclass {3^k} {17}$ Then $\\phi$ is a group isomorphism."}
{"_id": "2790", "text": "Completely Normal Space is Preserved under Homeomorphism If $T_A$ is a completely normal space, then so is $T_B$."}
{"_id": "2297", "text": "Open Set is G-Delta Set Let $T = \\struct {S, \\tau}$ be a topological space. Let $U$ be an open set of $T$. Then $U$ is a $G_\\delta$ set of $T$."}
{"_id": "2071", "text": "Pairwise Independence does not imply Independence Just because all the events in a family of events in a probability space are pairwise independent, it does not mean that the family is independent."}
{"_id": "4207", "text": "Intersection of Subset with Upper Bounds Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$. Let $T^*$ be the set of all upper bounds of $T$ in $S$. Then $T^* \\cap T \\ne \\varnothing$ {{iff}}: :$T$ has a greatest element $M$ and :$T^* \\cap T$ is a singleton such that $T^* \\cap T = \\left\\{{M}\\right\\}$"}
{"_id": "11247", "text": "Power Set less Empty Set has no Smallest Element iff not Singleton Let $S$ be a set which is non-empty. Let $\\mathcal C = \\mathcal P \\left({S}\\right) \\setminus \\varnothing$, that is, the power set of $S$ without the empty set. Then the ordered structure $\\left({\\mathcal C, \\subseteq}\\right)$ has no smallest element {{iff}} $S$ is not a singleton."}
{"_id": "19197", "text": "Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms Let $S$ be a finite set. Let $\\rho : \\powerset S \\to \\Z$ be a mapping from the power set of $S$ to the integers. Then $\\rho$ is the rank function of a matroid on $S$ {{iff}} $\\rho$ satisfies the rank axioms: {{:Definition:Rank Axioms (Matroid)/Definition 1}}"}
{"_id": "14353", "text": "Discrepancy between Gregorian Year and Tropical Year The Gregorian year and the tropical year differ such that the Gregorian calendar becomes $1$ day further out approximately every $3318$ years."}
{"_id": "5998", "text": "Combination Theorem for Sequences/Real/Quotient Rule/Corollary Let $\\sequence {x_n}$ be convergent to the following limit: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\frac 1 {x_n} = \\frac 1 l$ provided that $l \\ne 0$."}
{"_id": "7535", "text": "Equivalence of Definitions of Variance of Discrete Random Variable Let $X$ be a discrete random variable. Let $\\mu = \\expect X$ be the expectation of $X$. {{TFAE|def = Variance of Discrete Random Variable}}"}
{"_id": "14770", "text": "333,667 is Only Prime whose Reciprocal is of Period 9 The only prime number whose reciprocal has a period of $9$ is $333 \\, 667$: :$\\dfrac 1 {333 \\, 667} = 0 \\cdotp \\dot 00000 \\, 299 \\dot 7$"}
{"_id": "13193", "text": "Set of 3 Integers each Divisor of Sum of Other Two There exists exactly one set of distinct coprime positive integers such that each is a divisor of the sum of the other two: :$\\set {1, 2, 3}$"}
{"_id": "15386", "text": "Kernel of Composition of Ring Homomorphisms Let $f : A \\to B$ and $g : B \\to C$ be ring homomorphisms. Let $g \\circ f$ be their composition. Then the kernel of $g \\circ f$ is the preimage under $f$ of the kernel of $g$: :$\\map \\ker {g \\circ f} = f^{-1} \\sqbrk {\\ker g}$"}
{"_id": "12245", "text": "Domain of Real Square Function The domain of the real square function is the entire set of real numbers $\\R$."}
{"_id": "13593", "text": "Element is Meet Irreducible iff Complement of Element is Irreducible Let $T = \\struct {S, \\tau}$ be a topological space. Let $P = \\struct {\\tau, \\preceq}$ be an ordered set where $\\mathord \\preceq = \\mathord \\subseteq \\cap \\paren {\\tau \\times \\tau}$ Let $A \\in \\tau$ such that :$A \\ne \\top_P$ where $\\top_P$ denotes the greatest element in $P$. Then $A$ is meet irreducible in $P$ {{iff}} $\\complement_S\\left({A}\\right)$ is irreducible where $\\complement_S\\left({A}\\right)$ denotes the relative complement of $A$ relative to $S$."}
{"_id": "14208", "text": "121 is Square Number in All Bases greater than 2 Let $b \\in \\Z$ be an integer such that $b \\ge 3$. Let $n$ be a positive integer which can be expressed in base $b$ as $121_b$. Then $n$ is a square number."}
{"_id": "4545", "text": "Null Measure is Measure Let $\\struct {X, \\Sigma}$ be a measurable space. Then the null measure $\\mu$ on $\\struct {X, \\Sigma}$ is a measure."}
{"_id": "12899", "text": "Isomorphism between Group Generated by Reciprocal of z and 1 minus z and Symmetric Group on 3 Letters Let $S_3$ denote the symmetric group on $3$ letters. Let $G$ be the group generated by $1 / z$ and $1 - z$. Then $S_3$ and $G$ are isomorphic algebraic structures."}
{"_id": "15140", "text": "Product of Paracompact Spaces is not always Paracompact Let $I$ be an indexing set. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be a paracompact space. Then it is not necessarily the case that $\\struct {S, \\tau}$ is also paracompact space."}
{"_id": "3646", "text": "Multiplicative Inverse in Nicely Normed Algebra Let $A = \\left({A_F, \\oplus}\\right)$ be a nicely normed $*$-algebra whose conjugation is denoted $*$. Let $a \\in A$. Then the multiplicative inverse of $a$ is given by: :$a^{-1} = \\dfrac {a^*} {\\left \\Vert {a}\\right \\Vert^2}$ where: : $a^*$ is the conjugate of $a$ : $\\left \\Vert {a}\\right \\Vert$ is the norm of $a$."}
{"_id": "15453", "text": "Ideal is Contained in Contraction of Extension Let $A$ and $B$ be commutative ring with unity. Let $f : A \\to B$ be a ring homomorphism. Let $\\mathfrak a \\subseteq A$ be an ideal. Then $\\mathfrak a$ is contained in the contraction of its extension by $f$: :$\\mathfrak a \\subseteq \\mathfrak a^{ec}$"}
{"_id": "15555", "text": "Power Series Expansion of Reciprocal of Cube Root of 1 + x Let $x \\in \\R$ such that $-1 < x \\le 1$. Then: :$\\dfrac 1 {\\sqrt [3] {1 + x} } = 1 - \\dfrac 1 3 x + \\dfrac {1 \\times 4} {3 \\times 6} x^2 - \\dfrac {1 \\times 4 \\times 7} {3 \\times 6 \\times 9} x^3 + \\cdots$"}
{"_id": "11760", "text": "Second Order ODE/y'' - x f(x) y' + f(x) y = 0 The second order ODE: :$(1): \\quad y'' - x \\, \\map f x y' + \\map f x y = 0$ has the general solution: :$\\displaystyle y = C_1 x + C_2 x \\int x^{-2} e^{\\int x \\, \\map f x \\rd x} \\rd x$"}
{"_id": "18946", "text": "Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring/Corollary Let $\\struct {R, \\norm {\\, \\cdot \\,} }$ be a complete normed division ring. Let $\\struct {S, \\norm {\\, \\cdot \\,}}$ be a dense normed division subring of $\\struct {R, \\norm {\\, \\cdot \\,}}$. Then for all $x \\in R$, there exists a sequence $\\sequence{x_n}$ in $S$: :$x = \\displaystyle \\lim_{n \\mathop \\to \\infty} x_n$"}
{"_id": "16613", "text": "Subtraction has no Identity Element The operation of subtraction on numbers of any kind has no identity."}
{"_id": "3337", "text": "Compact Complement Space is not T2, T3, T4 or T5 Let $T = \\struct {\\R, \\tau}$ be the compact complement topology on $\\R$. Then $T$ is not a $T_2$ (Hausdorff) space, $T_{2 \\frac 1 2}$ (completely Hausdorff) space, $T_3$ space, $T_4$ space or $T_5$ space."}
{"_id": "2393", "text": "Count of Commutative Binary Operations with Identity Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different commutative binary operations that can be applied to $S$ which have an identity element is given by: :$N = n^{\\frac {n \\left({n-1}\\right)}2 + 1}$"}
{"_id": "4962", "text": "Scalar Product with Inverse Unity :$\\paren {-1_R} \\circ x = - x$"}
{"_id": "18262", "text": "Arctangent Function in terms of Gaussian Hypergeometric Function :$\\displaystyle \\arctan x = x \\, {}_2 \\map {F_1} {\\frac 1 2, 1; \\frac 3 2; -x^2}$"}
{"_id": "14403", "text": "Field is Galois over Fixed Field of Automorphism Group Let $E/F$ be a finite field extension. Let $K = \\operatorname{Fix}_E(\\operatorname{Aut}(E/F))$ be the fixed field of the automorphism group of $E/F$. Then $E/K$ is Galois."}
{"_id": "8357", "text": "Permutation Representation defines Group Action Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $\\map \\Gamma X$ be the symmetric group of $X$. Let $\\rho: G \\to \\map \\Gamma X$ be a permutation representation, that is, a homomorphism. The mapping $\\phi: G \\times X \\to X$ associated to $\\rho$, defined by: :$\\map \\phi {g, x} = \\map {\\map \\rho g} x$ is a group action."}
{"_id": "16771", "text": "Arithmetic Sequence of 16 Primes The $16$ integers in arithmetic sequence defined as: :$2\\,236\\,133\\,941 + 223\\,092\\,870 n$ are prime for $n = 0, 1, \\ldots, 15$."}
{"_id": "3278", "text": "Relation of Boubaker Polynomials to Dickson Polynomials The Boubaker polynomials $B_n$ are linked to the  Dickson polynomials by the relations: : $B_{n+1} \\left({x}\\right) B_{n+j} \\left({x}\\right) - B_{n+j+1} \\left({x}\\right) B_n \\left({x}\\right) = \\left({3 x^2 + 4}\\right) D_{n+1} \\left({x, \\dfrac 1 4}\\right)$ : $B_n \\left({x}\\right) = D_n \\left({2x, \\dfrac 1 4}\\right) + 4 D_{n-1} \\left({2 x, \\dfrac 1 4}\\right)$"}
{"_id": "19071", "text": "Cauchy Sequences in Vector Spaces with Equivalent Norms Coincide Let $M_a = \\struct {X, \\norm {\\, \\cdot \\, }_a}$ and $M_b = \\struct {X, \\norm {\\, \\cdot \\,}_b}$ be normed vector spaces. Let $\\sequence {x_n}_{n \\mathop \\in \\N}$ be a Cauchy sequence in $M_a$. Suppose, $\\norm {\\, \\cdot \\, }_a$ and $\\norm {\\, \\cdot \\,}_b$ are equivalent norms, i.e. $\\norm {\\, \\cdot \\, }_a \\sim \\norm {\\, \\cdot \\,}_b$. Then $\\sequence {x_n}_{n \\mathop \\in \\N}$ is also a Cauchy sequence in $M_b$."}
{"_id": "7213", "text": "Order Topology equals Dual Order Topology Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Let $\\tau$ be the $\\preceq$-order topology on $S$. Let $\\tau'$ be the $\\succeq$-order topology on $S$, where $\\succeq$ is the dual ordering of $\\preceq$. Then $\\tau' = \\tau$."}
{"_id": "18801", "text": "Max is Half of Sum Plus Absolute Difference For all numbers $a, b$ where $a, b$ in $\\N, \\Z, \\Q$ or $\\R$: :$\\max \\set {a, b} = \\frac 1 2 \\paren {a + b + \\size {a - b} }$"}
{"_id": "12765", "text": "Summation over k of Floor of k over 2 :$\\displaystyle \\sum_{k \\mathop = 1}^n \\floor {\\dfrac k 2} = \\floor {\\dfrac {n^2} 4}$"}
{"_id": "13654", "text": "Prime to Own Power minus 1 over Prime minus 1 being Prime Let $n \\in \\Z_{>1}$ be an integer greater than $1$. Then $\\dfrac {n^n - 1} {n - 1}$ is a prime for $n$ equal to: :$2, 3, 19, 31$ {{OEIS|A088790}}"}
{"_id": "3448", "text": "Singleton Set is not Dense-in-itself Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$. Then the singleton set $\\left\\{{x}\\right\\}$ is not dense-in-itself."}
{"_id": "17885", "text": "Limit to Infinity of Complementary Error Function :$\\displaystyle \\lim_{x \\mathop \\to \\infty} \\map \\erfc x = 0$"}
{"_id": "16184", "text": "Factorisation of z^(2n+1)+1 in Real Domain Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$z^{2 n + 1} + 1 = \\paren {z + 1} \\displaystyle \\prod_{k \\mathop = 0}^{n - 1} \\paren {z^2 - 2 z \\cos \\dfrac {\\paren {2 k + 1} \\pi} {2 n + 1} + 1}$"}
{"_id": "17378", "text": "Dimension of Vector Space of Polynomial Functions Let $\\struct {F, +, \\times}$ be a field whose unity is $1_F$. Let $F_n \\sqbrk X$ be the ring of polynomials over $F$ whose degree is less than $n$. Then the dimension of the vector space $F_n \\sqbrk X$ is $n$."}
{"_id": "17496", "text": "Laplace Transform of t^2 by Cosine a t Let $\\sin$ denote the real sine function. Let $\\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\\laptrans {t^2 \\cos a t} = \\dfrac {2 s^3 - 6 a^2 s} {\\paren {s^2 + a^2}^3}$"}
{"_id": "19382", "text": "Area of Integer Heronian Triangle is Multiple of 6 Let $\\triangle {ABC}$ be an integer Heronian triangle. Then the area of $\\triangle {ABC}$ is a multiple of $6$."}
{"_id": "4346", "text": "Expansion of Included Set Topology Let $S$ be a set. Let $A_1 \\subseteq S$ and $A_2 \\subseteq S$. Let $T_1 = \\struct {S, \\tau_{A_1} }$ and $T_2 = \\struct {S, \\tau_{A_2} }$ be included set spaces on $S$. Then: :$(1): \\quad T_1 \\ge T_2 \\iff A_1 \\subseteq A_2$ :$(2): \\quad T_1 > T_2 \\iff A_1 \\subsetneq A_2$ where: :$T_1 \\ge T_2$ denotes that $T_1$ is finer than $T_2$ :$T_1 > T_2$ denotes that $T_1$ is strictly finer than $T_2$."}
{"_id": "18960", "text": "Semilattice Homomorphism is Order-Preserving Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be semilattices. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a semilattice homomorphism. Let $\\preceq_1$ be the ordering on $S$ defined by: :$a \\preceq_1 b \\iff \\paren {a \\circ b} = b$ Let $\\preceq_2$ be the ordering on $T$ defined by: :$x \\preceq_2 y \\iff \\paren {x * y} = y$ Then: :$\\phi: \\struct {S, \\preceq_1} \\to \\struct {T, \\preceq_2}$ is order-preserving"}
{"_id": "18812", "text": "D'Alembert's Formula {{MissingLinks|throughout}} Let $u: \\R^2 \\to \\R$ be a twice-differentiable function in two variables.  Let $\\phi: \\R \\to \\R$ be a differentiable function in $x$.  Let $\\psi: \\R \\to \\R$ be an integrable function in $x$. {{explain|What does \"in $x$\" mean? $x$ has not been defined yet.}} Let $c \\in \\R_{> 0}$ be a constant. Then the solution to the partial differential equation: :$u_{tt} = c^2 u_{xx}$ {{explain|The condensed form of the partial differential equation is not explicit enough, as neither $x$ nor $t$ have been defined. I refer you to Definition:Mapping/Notation/Warning where it is explained that, say, you can't say \"Let $\\map u {x, t}$ be a function\". Recommend that something like $\\dfrac {\\partial^2} {\\partial x^2} \\map u {x, t} {{=}} c^2 \\dfrac {\\partial^2} {\\partial t^2} \\map u {x, t}$ be used.}} with initial conditions {{begin-eqn}} {{eqn | l = \\map u {x, 0}       | r = \\map \\phi x }} {{eqn | l = \\map {u_t} {x, 0}       | r = \\map \\psi x }} {{end-eqn}} is given by: :$\\displaystyle \\map u {x, t} = \\dfrac 1 2 \\paren {\\map \\phi {x + c t} + \\map \\phi {x - c t} } + \\dfrac 1 {2 c} \\int_{x - c t}^{x + c t} \\map \\psi s \\rd s$ The above solution formula is called d'Alembert's Formula."}
{"_id": "7822", "text": "Equivalence of Definitions of Cosine of Angle Let $\\theta$ be an angle. {{TFAE|def = Cosine of Angle|view = cosine}}"}
{"_id": "9617", "text": "Primitive of Reciprocal of x fourth minus a fourth :$\\displaystyle \\int \\frac {\\d x} {x^4 - a^4} = \\frac 1 {4 a^3} \\ln \\size {\\frac {x - a} {x + a} } - \\frac 1 {2 a^3} \\arctan \\frac x a$"}
{"_id": "14126", "text": "Product of Number of Edges, Edges per Face and Faces of Regular Dodecahedron The product of the number of edges, edges per face and faces of a regular dodecahedron is $1800$."}
{"_id": "14005", "text": "Sum of 2 Squares in 2 Distinct Ways/Examples/65 $65$ can be expressed as the sum of two square numbers in two distinct ways: {{begin-eqn}} {{eqn | l = 65       | r = 8^2 + 1^2 }} {{eqn | r = 7^2 + 4^2 }} {{end-eqn}}"}
{"_id": "13774", "text": "4 Consecutive Integers cannot be Square-Free Let $n, n + 1, n + 2, n + 3$ be four consecutive positive integers. At least one of these is not square-free."}
{"_id": "18228", "text": "Numbers Equal to Sum of Squares of Digits There are exactly $2$ integers which are equal to the sum of the squares of their digits when expressed in base $10$: :$0 = 0^2$ :$1 = 1^2$"}
{"_id": "8649", "text": "Absorption Laws (Set Theory)/Corollary :$S \\cup \\paren {S \\cap T} = S \\cap \\paren {S \\cup T}$"}
{"_id": "12922", "text": "Card Game with Bluffing is Completely Mixed Game The game of cards with bluffing is a completely mixed game."}
{"_id": "19688", "text": "Rational Numbers whose Denominators are not Divisible by 4 do not form Ring Let $S$ be the set defined as: :$S = \\set {\\dfrac m n : m, n \\in \\Z, m \\perp n, 4 \\nmid n}$ That is, $S$ is defined as the set of rational numbers such that, when expressed in canonical form, their denominators are not divisible by $4$. Then the algebraic structure $\\struct {S, +, \\times}$ is not a ring."}
{"_id": "8319", "text": "Real Numbers under Addition form Monoid The set of real numbers under addition $\\struct {\\R, +}$ forms a monoid."}
{"_id": "10206", "text": "If Ratio of Cube to Number is as between Two Cubes then Number is Cube Let $a, b, c, d \\in \\Z$ be integers such that: :$\\dfrac a b = \\dfrac {c^3} {d^3}$ Let $a$ be a cube number. Then $b$ is also a cube number. {{:Euclid:Proposition/VIII/25}}"}
{"_id": "16646", "text": "Condition for Elements of Group to be in Subgroup Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $x, y \\in G$ be such that $2$ elements of $\\set {x, y, x y}$ are elements of $h$. Then ''all'' the elements of $\\set {x, y, x y}$ are in $H$."}
{"_id": "8642", "text": "Equivalence of Definitions of Semantic Equivalence for Boolean Interpretations {{TFAE|def = Semantic Equivalence for Boolean Interpretations}} Let $\\mathbf A, \\mathbf B$ be WFFs of propositional logic."}
{"_id": "3426", "text": "Real Number Line is Separable Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ is separable."}
{"_id": "18923", "text": "Loop-Digraph as a Relation A loop-digraph is the same thing as a relational structure."}
{"_id": "13598", "text": "Sum of Sequence of Cubes divides 3 times Sum of Sequence of Fifth Powers :$\\displaystyle \\sum_{i \\mathop = 1}^n i^3 \\divides 3 \\sum_{i \\mathop = 1}^n i^5$ where $\\divides$ denotes divisibility."}
{"_id": "15349", "text": "Parseval's Theorem/Formulation 2 Let $f$ be a real function which is square-integrable over the interval $\\openint {-\\pi} \\pi$. Let $f$ be expressed by the Fourier series: :$\\map f x = \\displaystyle \\sum_{n \\mathop = -\\infty}^\\infty c_n e^{i n x}$ where: :$c_n = \\displaystyle \\frac 1 {2 \\pi} \\int_{-\\pi}^\\pi \\map f t e^{-i n t} \\rd t$ Then: :$\\displaystyle \\frac 1 {2 \\pi} \\int_{-\\pi}^\\pi \\size {\\map f x}^2 \\rd x = \\sum_{n \\mathop = -\\infty}^\\infty \\size {c_n}^2$"}
{"_id": "11391", "text": "Euler Phi Function of 1 :$\\map \\phi 1 = 1$"}
{"_id": "11504", "text": "Terminal Velocity of Body under Fall Retarded Proportional to Velocity Let $B$ be a body falling in a gravitational field. Let $B$ be falling through a medium which exerts a resisting force $k \\mathbf v$ upon $B$ which is proportional to the velocity of $B$ relative to the medium. Then the terminal velocity of $B$ is given by: :$v = \\dfrac {g m} k$"}
{"_id": "4031", "text": "Image of Set Difference under Mapping/Corollary 2 Let $f: S \\to T$ be a mapping. Let $X$ be a subset of $S$. Then: :$\\relcomp {\\Img f} {f \\sqbrk X} \\subseteq f \\sqbrk {\\relcomp S X}$ where: :$\\Img f$ denotes the image of $f$ :$\\complement_{\\Img f}$ denotes the complement relative to $\\Img f$. This can be expressed in the language and notation of direct image mappings as: :$\\forall X \\in \\powerset S: \\relcomp {\\Img f} {\\map {f^\\to} X} \\subseteq \\map {f^\\to} {\\relcomp S X}$ That is: :$\\forall X \\in \\powerset S: \\map {\\paren {\\complement_{\\Img f} \\circ f^\\to} } X \\subseteq \\map {\\paren {f^\\to \\circ \\complement_S} } X$ where $\\circ$ denotes composition of mappings."}
{"_id": "5457", "text": "Category of Finite Sets is Category Let $\\mathbf{Set}$ be the category of finite sets. Then $\\mathbf{Set}$ is a metacategory."}
{"_id": "4194", "text": "Triangle Inequality for Generalized Sums Let $V$ be a Banach space, and let $\\left\\Vert{\\cdot}\\right\\Vert$ denote its norm. Let $\\left({v_i}\\right)_{i\\in I}$ be an indexed subset of $V$. Let the generalized sum $\\displaystyle \\sum \\left\\{{v_i: i \\in I}\\right\\}$ converge absolutely. Then $\\displaystyle \\left\\Vert{\\sum \\left\\{{v_i: i \\in I}\\right\\}}\\right\\Vert \\le \\sum \\left\\{{\\left\\Vert{v_i}\\right\\Vert: i \\in I}\\right\\}$."}
{"_id": "17895", "text": "Power Series Expansion for Exponential Integral Function plus Logarithm :$\\displaystyle \\map \\Ei x = -\\gamma - \\ln x + \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n + 1} \\frac {x^n} {n \\times n!}$"}
{"_id": "1488", "text": "Parallel Law for Extremal Length Let $X$ be a Riemann surface. Let $\\Gamma_1, \\Gamma_2$ be families of rectifiable curves (or, more generally, families of disjoint unions of rectifiable curves) on $X$. Let $\\Gamma_1$ and $\\Gamma_2$ be disjoint, in the sense that: :there exist disjoint Borel subsets $A_1, A_2 \\subseteq X$ such that: ::for any $\\gamma_1 \\in \\Gamma_1$ and $\\gamma_2 \\in \\Gamma_2$, we have $\\gamma_1 \\subseteq A_1$ and $\\gamma_2 \\subseteq A_2$. Let $\\Gamma$ be a third curve family, with the property that every element $\\Gamma_1$ and every element of $\\Gamma_2$ contains some element of $\\Gamma$. Then the extremal length of $\\Gamma$ satisfies: :$\\dfrac 1 {\\lambda \\left({\\Gamma}\\right)} \\ge \\dfrac 1 {\\lambda \\left({\\Gamma_1}\\right)} + \\dfrac 1 {\\lambda \\left({\\Gamma_2}\\right)}$"}
{"_id": "15418", "text": "Product of Coprime Ideals equals Intersection Let $A$ be a commutative ring with unity. Let $\\mathfrak a, \\mathfrak b \\subseteq A$ be coprime ideals. Then their product equals their intersection: :$\\mathfrak a \\mathfrak b = \\mathfrak a \\cap \\mathfrak b$"}
{"_id": "14972", "text": "Union of Blocks is Set of Points Let $\\struct {X, \\mathcal B}$ be a pairwise balanced design. That is, let $\\struct {X, \\mathcal B}$ be a design, with $\\size X \\ge 2$, and the number of occurrences of each pair of distinct points in $\\mathcal B$ be $\\lambda$ for some $\\lambda > 0$ constant. Then the set union of all the subset elements in $\\mathcal B$ is precisely $X$."}
{"_id": "4975", "text": "Linear Combination of Sequence is Linear Combination of Set Let $G$ be an $R$-module. Let $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ be a sequence of elements of $G$. Let $b$ be an element of $G$. Then: :$b$ is a linear combination of the sequence $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ {{iff}}: :$b$ is a linear combination of the set $\\set {a_k: 1 \\mathop \\le k \\mathop \\le n}$"}
{"_id": "972", "text": "Linear Transformations form Abelian Group Let $\\struct {G, +_G}$ and $\\struct {H, +_H}$ be groups. Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G, \\circ}_R$ and $\\struct {H, +_H, \\circ}_R$ be $R$-modules. Let $\\map {\\mathcal L_R} {G, H}$ be the set of all linear transformations from $G$ to $H$. Let $\\oplus_H$ be the operation on $H^G$ as defined in Addition of Linear Transformations. Then $\\struct {\\map {\\LL_R} {G, H}, \\oplus_H}$ is an abelian group."}
{"_id": "15194", "text": "Finite Product Space is Connected iff Factors are Connected/General Case Let $I$ be an indexing set. Let $\\family {T_\\alpha}_{\\alpha \\mathop \\in I}$ be an indexed family of topological spaces. Let $T = \\displaystyle \\prod_{\\alpha \\mathop \\in I} T_\\alpha$ be the Cartesian space of $\\family {T_\\alpha}_{\\alpha \\mathop \\in I}$. Let $T = \\displaystyle \\overline {\\bigcup_{\\alpha \\mathop \\in I} S_\\alpha}$. {{explain|Notation in the above -- explain what the overline means in this context.}} Let $\\tau$ be a topology on $T$ such that the subsets ${S'}_\\alpha \\subseteq \\displaystyle \\prod T_\\alpha$ where ${S'}_\\alpha = \\set {\\family {y_\\beta} \\in T: y_\\beta = x \\beta \\text { for all } \\beta \\ge \\alpha}$ is homeomorphic to $S_{\\alpha - 1} \\times T_\\alpha$. Then $T$ is connected {{iff}} each of $T_\\alpha: \\alpha \\in I$ are connected."}
{"_id": "11555", "text": "First Order ODE/y' = (x + y)^2 The first order ODE: :$\\dfrac {\\d y} {\\d x} = \\paren {x + y}^2$ has the general solution: :$x + y = \\map \\tan {x + C}$"}
{"_id": "16480", "text": "Intersection of Coprime Cyclic Subgroups is Trivial Let $G$ be a group whose identity is $e$. Let $x, y \\in G$ such that: :$\\order x \\perp \\order y$ where: :$\\order x, \\order y$ denotes the orders of $x$ and $y$ in $G$ respectively :$\\perp$ denotes the coprimality relation. Then: :$\\gen x \\cap \\gen y = \\set e$ where $\\gen x, \\gen y$ denotes the subgroup of $G$ generated by $x$ and $y$ in $G$ respectively."}
{"_id": "11653", "text": "Current in Electric Circuit/L, R in Series/Exponentially Decaying EMF at t = 0 Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let an EMF $E$ be imposed upon $K$ at time $t = 0$ defined by the equation: :$E = E_0 e^{-k t}$ The electric current $I$ in $K$ is given by the equation: :$I = \\dfrac {E_0} {R - k L} e^{-k t} + \\left({I_0 - \\dfrac {E_0} {R - k L} }\\right) e^{-R t / L}$"}
{"_id": "13426", "text": "11 is Only Palindromic Prime with Even Number of Digits $11$ is the only palindromic prime with an even number of digits when expressed in decimal notation."}
{"_id": "18714", "text": "Fourier's Theorem/Integral Form/Continuous Point Let $f$ be continuous at $t \\in \\R$. Then: :$\\displaystyle \\map f t = \\int_{-\\infty}^\\infty e^{2 \\pi i t s} \\paren {\\int_{-\\infty}^\\infty e^{-2 \\pi i d t} \\map f t \\rd t} \\rd s$"}
{"_id": "19107", "text": "Product of Absolutely Continuous Functions is Absolutely Continuous Let $a, b$ be real numbers with $a < b$. Let $f, g : \\closedint a b \\to \\R$ be absolutely continuous functions. Then $f \\times g$ is absolutely continuous."}
{"_id": "5944", "text": "Cartesian Product with Complement Let $S$ and $T$ be sets. Let $A \\subseteq S$ and $B \\subseteq T$ be subsets of $S$ and $T$, respectively. Let $\\relcomp S A$ denote the relative complement of $A$ in $S$. Then: {{begin-eqn}} {{eqn | l = \\relcomp S A \\times T        | r = \\relcomp {S \\times T} {A \\times T} }} {{eqn | l = S \\times \\relcomp T B       | r = \\relcomp {S \\times T} {S \\times B} }} {{end-eqn}}"}
{"_id": "8646", "text": "Existence of Prime between Prime and Factorial Let $p$ be a prime number. Then there exists a prime number $q$ such that: :$p < q \\le p! + 1$ where $p!$ denotes the factorial of $p$."}
{"_id": "5626", "text": "Rank of Ordinal Let $x$ be an ordinal. Let $\\map {\\operatorname {rank} } x$ denote the rank of $x$. Then: :$\\map {\\operatorname {rank} } x = x$"}
{"_id": "3398", "text": "Arens-Fort Space is Paracompact Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is a paracompact space."}
{"_id": "5128", "text": "Set Products on Same Set are Equivalent Let $S$ and $T$ be sets. Let $\\struct {P, \\phi_1, \\phi_2}$ and $\\struct {Q, \\psi_1, \\psi_2}$ be products of $S$ and $T$. Then there exists a unique bijection $\\chi: Q \\to P$ such that: :$\\phi_1 \\circ \\chi = \\psi_1$ :$\\phi_2 \\circ \\chi = \\psi_2$"}
{"_id": "3648", "text": "Nicely Normed Alternative Algebra is Normed Division Algebra $A = \\left({A_F, \\oplus}\\right)$ be a nicely normed $*$-algebra which is also an alternative algebra. Then $A$ is a normed division algebra."}
{"_id": "13780", "text": "Hensel's Lemma for Composite Numbers Let $b \\in \\Z \\setminus \\set {-1, 0, 1}$ be an integer. Let $k > 0$ be a positive integer. Let $\\map f X \\in \\Z \\sqbrk X$ be a polynomial. Let $x_k \\in \\Z$ such that: :$\\map f {x_k} \\equiv 0 \\pmod {b^k}$ :$\\gcd \\set {\\map {f'} {x_k}, b} = 1$ Then for every integer $l \\ge 0$, there exists an integer $x_{k + l}$ such that: :$\\map  f {x_{k + l} } \\equiv 0 \\pmod {b^{k + l} }$ :$x_{k + l} \\equiv x_k \\pmod {b^k}$ and any two integers satisfying these congruences are congruent modulo $b^{k + l}$. Moreover, for all $l \\ge 0$ and any solutions $x_{k + l}$ and $x_{k + l + 1}$: :$x_{k + l + 1} \\equiv x_{k + l} - \\dfrac {\\map f {x_{k + l} } } {\\map {f'} {x_{k + l} } } \\pmod {b^{k + l + 1} }$ :$x_{k + l + 1} \\equiv x_{k + l} \\pmod {b^{k + l} }$"}
{"_id": "2637", "text": "Product with Inverse equals Identity iff Equality Let $\\struct {G, \\circ}$ be a group whose identity element is $e$. Then: :$\\forall a, b \\in G: a \\circ b^{-1} = e \\iff a = b$"}
{"_id": "17860", "text": "Definite Integral from 0 to Pi of x by Logarithm of Sine x :$\\displaystyle \\int_0^\\pi x \\, \\map \\ln {\\sin x} \\rd x = -\\frac {\\pi^2} 2 \\ln 2$"}
{"_id": "19129", "text": "Uncountable Closed Ordinal Space is Sigma-Compact Let $\\Omega$ denote the first uncountable ordinal. Let $\\closedint 0 \\Omega$ denote the closed ordinal space on $\\Omega$. Then $\\closedint 0 \\Omega$ is a $\\sigma$-compact space."}
{"_id": "11399", "text": "Full Rook Matrix is Invertible A full rook matrix is invertible."}
{"_id": "11244", "text": "Equivalence of Definitions of Maximal Element Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $T \\subseteq S$ be a subset of $S$. {{TFAE|def = Maximal Element}}"}
{"_id": "1684", "text": "Set of Natural Numbers is Primitive Recursive The set of natural numbers $\\N$ is primitive recursive."}
{"_id": "18541", "text": "Doubleton of Sets can be Derived using Comprehension Principle Let $a$ and $b$ be sets. By application of the comprehension principle, the set $\\set {a, b}$ can be formed. Hence the '''doubleton''' $\\set {a, b}$ can be derived as a valid object in Frege set theory."}
{"_id": "18236", "text": "Complex Sine Function is Unbounded Let $\\sin: \\C \\to \\C$ be the complex sine function.  Then $\\sin$ is unbounded."}
{"_id": "677", "text": "Condition for Divisibility of Powers of Prime Let $p$ be a prime. Let $k, l \\in \\Z_{>0}$. Then: :$p^k \\divides p^l \\iff k \\le l$"}
{"_id": "18438", "text": "Linear Second Order ODE/y'' + 2 y' + 5 y = x sin x The second order ODE: :$(1): \\quad y'' + 2 y' + 5 y = x \\sin x$ has the general solution: :$y = e^{-x} \\paren {C_1 \\cos 2 x + C_2 \\sin 2 x} + \\ldots$"}
{"_id": "16317", "text": "Combination Theorem for Cauchy Sequences/Combined Sum Rule :$\\sequence {a x_n + b y_n }$ is a Cauchy sequence."}
{"_id": "15100", "text": "Separation Properties Preserved in Subspace/Corollary If $T$ has one of the following properties then $T_H$ has the same property: :Regular Property :Tychonoff (Completely Regular) Property :Completely Normal Property That is, the above properties are all hereditary."}
{"_id": "11507", "text": "Boundary of Union of Separated Sets equals Union of Boundaries Let $T$ be a topological space. Let $A, B$ be subsets of $T$. Let $A$ and $B$ are separated. Then: :$\\partial \\left({ A \\cup B }\\right) = \\partial A \\cup \\partial B$ where : $\\partial A$ denotes the boundary of $A$, : $A \\cup B$ denotes the union of $A$ and $B$."}
{"_id": "14123", "text": "Product of Number of Edges, Edges per Face and Faces of Cube The product of the number of edges, edges per face and faces of a cube is $288$."}
{"_id": "5735", "text": "Definition:Smith-Volterra-Cantor Set Let $G$ be a Cantor collection. Let $g_0 \\in G$ such that $\\map \\mu {g_0} = b$. Let $p$ be a natural number. Then there are two nonempty disjoint sets $N_p$ and $P$ such that: :$g_0 = N_p \\bigcup P$ where $N_p$ is nowhere dense in the relative topology on $g_0$ and $\\map \\mu P \\le b / p$."}
{"_id": "15115", "text": "Existence of Strongly Locally Compact Space which is not Weakly Sigma-Locally Compact There exists at least one example of a strongly locally compact topological space which is not also a weakly $\\sigma$-locally compact space."}
{"_id": "2689", "text": "Characteristic of Finite Ring is Non-Zero Let $\\struct {R, +, \\circ}$ be a finite ring with unity. Then the characteristic of $R$ is not zero."}
{"_id": "11624", "text": "First Order ODE/y - x y' = y' y^2 exp y The first order ODE: :$(1): \\quad y - x y' = y' y^2 e^y$ has the general solution: :$x y^2 = e^y + C$"}
{"_id": "19500", "text": "Space of Bounded Sequences with Supremum Norm forms Banach Space Let $\\ell^\\infty$ be a space of bounded sequences. Let $\\norm {\\, \\cdot \\,}_\\infty$ be a supremum norm. Then $\\struct {\\ell^\\infty, \\norm {\\, \\cdot \\,}_\\infty}$ is a Banach space."}
{"_id": "296", "text": "Order Embedding into Image is Isomorphism Let $\\struct {S, \\preceq_1}$ and $\\struct {T, \\preceq_2}$ be ordered sets. Let $S'$ be the image of a mapping $\\phi: \\struct {S, \\preceq_1} \\to \\struct {T, \\preceq_2}$. Then: :$\\phi$ is an order embedding from $\\struct {S, \\preceq_1}$ into $\\struct {T, \\preceq_2}$ {{iff}}: :$\\phi$ is an order isomorphism from $\\struct {S, \\preceq_1}$ into $\\struct {S', \\preceq_2 \\restriction_{S'} }$."}
{"_id": "14467", "text": "Carmichael Number has 3 Odd Prime Factors Let $n$ be a Carmichael number. Then $n$ has at least $3$ distinct odd prime factors."}
{"_id": "9820", "text": "Primitive of Arcsine of x over a over x squared :$\\displaystyle \\int \\frac {\\arcsin \\frac x a \\ \\mathrm d x} {x^2} = \\frac {-\\arcsin \\frac x a} x - \\frac 1 a \\ln \\left({\\frac {a + \\sqrt {a^2 - x^2} } x}\\right) + C$"}
{"_id": "11522", "text": "Discrete Space is Separable iff Countable Let $T = \\left({S, \\tau}\\right)$ be a discrete topological space. Then: :$T$ is separable {{iff}} $S$ is countable."}
{"_id": "11531", "text": "T1/2 Space is T0 Space Let $T = \\struct {S, \\tau}$ be a $T_{\\frac 1 2}$ topological space. Then $T$ is $T_0$ space."}
{"_id": "13110", "text": "Pi as Sum of Sequence of Reciprocal of Product of Three Consecutive Integers :$\\dfrac {\\pi - 3} 4 = \\dfrac 1 {2 \\times 3 \\times 4} - \\dfrac 1 {4 \\times 5 \\times 6} + \\dfrac 1 {6 \\times 7 \\times 8} \\cdots$"}
{"_id": "4827", "text": "Sum of Binomial Coefficients over Lower Index/Corollary :$\\displaystyle \\forall n \\in \\Z_{\\ge 0}: \\sum_{i \\mathop \\in \\Z} \\binom n i = 2^n$"}
{"_id": "10756", "text": "Chebyshev Distance is Limit of P-Product Metric Let $M_{1'} = \\left({A_{1'}, d_{1'}}\\right)$ and $M_{2'} = \\left({A_{2'}, d_{2'}}\\right)$ be metric spaces. Let $\\mathcal A = A_{1'} \\times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$. Let $p \\in \\R_{\\ge 1}$. Let $d_p: \\mathcal A \\times \\mathcal A \\to \\R$ be the $p$-product metric on $\\mathcal A$ : $d_p \\left({x, y}\\right) := \\left({\\left({d_{1'} \\left({x_1, y_1}\\right)}\\right)^p + \\left({d_{2'} \\left({x_2, y_2}\\right)}\\right)^p}\\right)^{1/p}$ and let $d_\\infty: \\mathcal A \\times \\mathcal A \\to \\R$ be the Chebyshev distance on $\\mathcal A$: : $\\displaystyle d_\\infty \\left({x, y}\\right) := \\max \\left\\{ {d_{1'} \\left({x_1, y_1}\\right), d_{2'} \\left({x_2, y_2}\\right)}\\right\\}$ where $x = \\left({x_1, x_2}\\right), y = \\left({y_1, y_2}\\right) \\in \\mathcal A$. Then: :$\\displaystyle d_\\infty = \\lim_{p \\mathop \\to \\infty} d_p$ in the sense that: :$\\displaystyle \\max \\left\\{ {d_{1'} \\left({x_1, y_1}\\right), d_{2'} \\left({x_2, y_2}\\right)}\\right\\} = \\lim_{p \\mathop \\to \\infty} \\left({\\left({d_{1'} \\left({x_1, y_1}\\right)}\\right)^p + \\left({d_{2'} \\left({x_2, y_2}\\right)}\\right)^p}\\right)^{1/p}$"}
{"_id": "19108", "text": "Accumulation Point of Sequence is not necessarily Limit Point Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\sequence {a_n}$ be a sequence in $T$. Let $q \\in S$ be an accumulation point of $\\sequence {a_n}$. Then it is not necessarily the case that $q$ is also a limit point of $\\sequence {a_n}$."}
{"_id": "19461", "text": "Derivative of General Logarithm of Function :$\\map {\\dfrac \\d {\\d x} } {\\log_a u} = \\dfrac {\\log_a e} u \\dfrac {\\d u} {\\d x}$"}
{"_id": "14244", "text": "Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares The largest odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers with greatest common divisor $1$ is $157$. The full sequence of such odd positive integers which cannot be so expressed is: :$1, \\ldots, 37, 41, 43, 45, 47, 49, 53, 55, 59, 61, 67, 69, 73, 77, 83, 89, 97, 101, 103, 115, 157$ where the sequence contains all odd integers between $1$ and $37$. This sequence appears not to be documented on the {{OEISLink}}."}
{"_id": "13678", "text": "23 is Largest Integer not Sum of Distinct Perfect Powers The largest integer which cannot be expressed as the sum of distinct perfect powers is $23$."}
{"_id": "5799", "text": "Topology as Magma of Sets The concept of a topology is an instance of a magma of sets."}
{"_id": "2559", "text": "Fourth Isomorphism Theorem Let $\\phi: R \\to S$ be a ring homomorphism. Let $K = \\map \\ker \\phi$ be the kernel of $\\phi$. Let $\\mathbb K$ be the set of all subrings of $R$ which contain $K$ as a subset. Let $\\mathbb S$ be the set of all subrings of $\\Img \\phi$. Let $\\phi^\\to: \\powerset R \\to \\powerset S$ be the direct image mapping of $\\phi$. Then its restriction $\\phi^\\to: \\mathbb K \\to \\mathbb S$ is a bijection. Also: :$(1): \\quad \\phi^\\to$ and its inverse both preserve subsets. :$(2): \\quad \\phi^\\to$ and its inverse both preserve ideals: :::: If $J$ is an ideal of $R$, then $\\map {\\phi^\\to} J$ is an ideal of $S$ :::: If $J'$ is an ideal of $S$, then $\\map {\\paren {\\phi^\\to}^{-1} } {J'}$ is an ideal of $R$"}
{"_id": "15330", "text": "Integral Operator is Linear/Corollary 1 :$\\forall \\alpha, \\beta \\in \\R: \\map T {f + g} = \\map T f + \\map T g$"}
{"_id": "16082", "text": "Product of Negative Real Numbers is Positive Let $a, b \\in \\R_{\\le 0}$ be negative real numbers. Then: :$a \\times b \\in \\R_{\\ge 0}$ That is, their product $a \\times b$ is a positive real number."}
{"_id": "6225", "text": "Recursive Construction of Transitive Closure Given the relation $\\mathcal R$, its transitive closure $\\mathcal R^+$ can be constructed as follows: Let: :$\\mathcal R_n := \\begin{cases} \\mathcal R & : n = 0 \\\\ \\mathcal R_{n-1} \\cup \\left\\{{\\left({x_1, x_3}\\right): \\exists x_2: \\left({x_1, x_2}\\right) \\in \\mathcal R_{n-1} \\land \\left({x_2, x_3}\\right) \\in \\mathcal R_{n-1}}\\right\\} & : n > 0 \\end{cases}$ Finally, let: :$\\displaystyle \\mathcal R^+ = \\bigcup_{i \\in \\N} \\mathcal R_i$. Then $R^+$ is the transitive closure of $R$."}
{"_id": "18868", "text": "Canonical Injection of Real Number Line into Complex Plane Let $\\struct {\\C, +}$ be the additive group of complex numbers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $f: \\R \\to \\C$ be the mapping from the real numbers to the complex numbers defined as: :$\\forall x \\in \\R: \\map f z = x + 0 y$ Then $f: \\struct {\\R, +} \\to \\struct {\\C, +}$ is a monomorphism."}
{"_id": "12644", "text": "Sum over k to n of Unsigned Stirling Number of the First Kind of k with m by n factorial over k factorial Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_{k \\mathop \\le n} {k \\brack m} \\frac {n!} {k!} = {n + 1 \\brack m + 1}$ where: :$\\displaystyle {k \\brack m}$ denotes an unsigned Stirling number of the first kind :$ n!$ denotes a factorial."}
{"_id": "18445", "text": "Complex Numbers under Addition form Group Let $\\C$ be the set of complex numbers. The structure $\\struct {\\C, +}$ is a group."}
{"_id": "18407", "text": "Linear Second Order ODE/y'' - 3 y' + 2 y = 5 exp 3 x The second order ODE: :$(1): \\quad y'' - 3 y' + 2 y = 5 e^{3 x}$ has the general solution: :$y = C_1 e^x + C_2 e^{2 x} + \\dfrac {5 e^{3 x} } 2$"}
{"_id": "9536", "text": "Sum of Infinite Arithmetic-Geometric Sequence Let $\\sequence {a_k}$ be an arithmetic-geometric sequence defined as: :$a_k = \\paren {a + k d} r^k$ for $n = 0, 1, 2, \\ldots$ Let: :$\\size r < 1$ where $\\size r$ denotes the absolute value of $r$. Then: :$\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\paren {a + k d} r^k = \\frac a {1 - r} + \\frac {r d} {\\paren {1 - r}^2}$"}
{"_id": "767", "text": "Odd Order Group Element is Square Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $x \\in G$. Then: :$\\exists y \\in G: y^2 = x$ {{iff}}: :the order  $\\order x$ is odd"}
{"_id": "16598", "text": "Composition of Permutations is not Commutative Let $S$ be a set. Let $\\map \\Gamma S$ denote the set of permutations on $S$. Let $\\pi, \\rho$ be elements of $\\map \\Gamma S$ Then it is not necessarily the case that: :$\\pi \\circ \\rho = \\rho \\circ \\pi$ where $\\circ$ denotes composition."}
{"_id": "15402", "text": "Series Expansion for Pi over 8 Root 2 :$\\displaystyle \\frac \\pi {8 \\sqrt 2} = \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n - 1} \\frac {2 n - 1} {\\paren {4 n - 1} \\paren {4 n - 3} }$"}
{"_id": "17691", "text": "Sum of Powers of 2 Let $n \\in \\N_{>0}$ be a (strictly positive) natural number. Then: {{begin-eqn}} {{eqn | n = 1       | l = 2^n - 1       | r = \\sum_{j \\mathop = 0}^{n - 1} 2^j       | c =  }} {{eqn | r = 1 + 2 + 2^2 + 2^3 + \\dotsb + 2^{n - 1}       | c =  }} {{end-eqn}}"}
{"_id": "1150", "text": "Divergent Sequence may be Bounded While every Convergent Sequence is Bounded, it does not follow that every bounded sequence is convergent. That is, there exist bounded sequences which are divergent."}
{"_id": "487", "text": "Smallest Field is Field The ring $\\struct {\\set {0_R, 1_R}, +, \\circ}$ is the smallest algebraic structure which is a field."}
{"_id": "19446", "text": "Derivative of Inverse Hyperbolic Secant Function :$\\map {\\dfrac \\d {\\d x} } {\\sech^{-1} u} = \\dfrac {-1} {u \\sqrt {1 - u^2} } \\dfrac {\\d u} {\\d x}$  where $0 < u < 1$"}
{"_id": "15626", "text": "Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors Let $\\mathbf a, \\mathbf b, \\mathbf c$ be vectors in a vector space of $3$ dimensions: Let $\\mathbf a \\cdot \\left({\\mathbf b \\times \\mathbf c}\\right)$ denote the scalar triple product of $\\mathbf a, \\mathbf b, \\mathbf c$. Then $\\left\\lvert{\\mathbf a \\cdot \\left({\\mathbf b \\times \\mathbf c}\\right)}\\right\\rvert$ equals the volume of the parallelepiped contained by $\\mathbf a, \\mathbf b, \\mathbf c$."}
{"_id": "13800", "text": "Pythagorean Triangles whose Area equal their Perimeter There exist exactly $2$ Pythagorean triples which define a Pythagorean triangle whose area equals its perimeter: :$(1): \\quad \\tuple {6, 8, 10}$, leading to an area and perimeter of $24$ :$(2): \\quad \\tuple {5, 12, 13}$, leading to an area and perimeter of $30$."}
{"_id": "7192", "text": "Condition for Well-Foundedness/Reverse Implication/Proof 2 Let $\\struct {S, \\preceq}$ be an ordered set. Suppose that there is no infinite sequence $\\sequence {a_n}$ of elements of $S$ such that: :$\\forall n \\in \\N: a_{n + 1} \\prec a_n$ Then $\\struct {S, \\preceq}$ is well-founded."}
{"_id": "13716", "text": "Integer as Sum of 27 Primes Every positive integer greater than $1$ can be expressed as the sum of no more than $27$ primes."}
{"_id": "1696", "text": "Factorial is Primitive Recursive The factorial function $\\operatorname{fac}: \\N \\to \\N$ defined as: :$\\operatorname{fac} \\left({n}\\right) = n!$ is primitive recursive."}
{"_id": "8365", "text": "Left Regular Representation is Transitive Group Action Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $*: G \\times G \\to G$ be the group action: :$\\forall g, h \\in G: g * h = \\map {\\lambda_g} h$ where $\\lambda_g$ is the left regular representation of $G$ with respect to $g$. Then $*$ is a transitive group action."}
{"_id": "12114", "text": "Arc of Cycloid is Concave Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. Then the locus of $P$ is concave."}
{"_id": "2529", "text": "Equivalence of Definitions of Infimum of Real-Valued Function Let $S \\subseteq \\R$ be a subset of the real numbers. Let $f: S \\to \\R$ be a real function on $S$. {{TFAE|def = Infimum of Real-Valued Function}}"}
{"_id": "14207", "text": "Multiply Perfect Number of Order 8 The number defined as: :$n = 2^{65} \\times 3^{23} \\times 5^9 \\times 7^{12} \\times 11^3 \\times 13^3 \\times 17^2 \\times 19^2 \\times 23 \\times 29^2 \\times 31^2$ ::$\\times 37 \\times 41 \\times 53 \\times 61 \\times 67^2 \\times 71^2 \\times 73 \\times 83 \\times 89 \\times 103 \\times 127 \\times 131$ ::$\\times 149 \\times 211 \\times 307 \\times 331 \\times 463 \\times 521 \\times 683 \\times 709 \\times 1279 \\times 2141 \\times 2557 \\times 5113$ ::$\\times 6481 \\times 10 \\, 429 \\times 20 \\, 857 \\times 110 \\, 563 \\times 599 \\, 479 \\times 16 \\, 148 \\, 168 \\, 401$ is multiply perfect of order $8$."}
{"_id": "16997", "text": "Wilson's Theorem/Corollary 1 Let $p$ be a prime number. Then $p$ is the smallest prime number which divides $\\paren {p - 1}! + 1$."}
{"_id": "17183", "text": "Moment Generating Function of Discrete Uniform Distribution Let $X$ be a discrete random variable with a discrete uniform distribution with parameter $n$ for some $n \\in \\N$. Then the moment generating function $M_X$ of $X$ is given by:  :$\\map {M_X} t = \\dfrac {e^t \\paren {1 - e^{n t} } } {n \\paren {1 - e^t} }$"}
{"_id": "5172", "text": "Order Topology is Normal Let $\\struct {S, \\preceq}$ be a toset. Let $\\tau$ be the order topology on $S$. Then $\\struct {S, \\tau}$ is normal."}
{"_id": "12572", "text": "Binomial Coefficient is instance of Gaussian Binomial Coefficient Let $\\dbinom r m_q$ denote the Gaussian binomial coefficient: Then: :$\\displaystyle \\lim_{q \\mathop \\to 1^-} \\dbinom r m_q = \\dbinom r m$ where $\\dbinom r m$ denotes the conventional binomial coefficient."}
{"_id": "16507", "text": "Bijection between Power Set of Disjoint Union and Cartesian Product of Power Sets Let $S$ and $T$ be disjoint sets. Let $\\powerset S$ denote the power set of $S$. Then there exists a bijection between $\\powerset {S \\cup T}$ and $\\paren {\\powerset S} \\times \\paren {\\powerset T}$. Hence: :$\\powerset {S \\cup T} \\sim \\paren {\\powerset S} \\times \\paren {\\powerset T}$ where $\\sim$ denotes set equivalence."}
{"_id": "6109", "text": "Category has Finite Limits iff Finite Products and Equalizers Let $\\mathbf C$ be a metacategory. Then: :$\\mathbf C$ has all finite limits {{iff}}: :$\\mathbf C$ has all finite products and equalizers."}
{"_id": "10050", "text": "Primitive of x squared by Inverse Hyperbolic Cotangent of x over a :$\\displaystyle \\int x^2 \\coth^{-1} \\frac x a \\ \\mathrm d x = \\frac {a x^2} 6 + \\frac {x^3} 3 \\coth^{-1} \\frac x a + \\frac {a^3} 6 \\ln \\left({x^2 - a^2}\\right) + C$"}
{"_id": "15850", "text": "Minimal Uncountable Well-Ordered Set Unique up to Isomorphism Let $\\Omega, \\Omega'$ be minimal uncountable well-ordered sets. Then $\\Omega$ is order isomorphic to $\\Omega'$. That is, the minimal uncountable well-ordered set is unique up to order isomorphism."}
{"_id": "19233", "text": "Scalar Multiplication Corresponds to Multiplication by 1x1 Matrix Let $\\map \\MM 1$ denote the matrix space of square matrices of order $1$. Let $\\map \\MM {1, n}$ denote the matrix space of order $1 \\times n$. Let $\\mathbf A = \\begin {pmatrix} a \\end {pmatrix} \\in \\map \\MM 1$ and $\\mathbf B = \\begin {pmatrix} b_1 & b_2 & \\cdots & b_n \\end{pmatrix} \\in \\map \\MM {1, n}$. Let $\\mathbf C = \\mathbf A \\mathbf B$ denote the (conventional) matrix product of $\\mathbf A$ with $\\mathbf B$. Let $\\mathbf D = a \\mathbf B$ denote the matrix scalar product of $a$ with $\\mathbf B$. Then $\\mathbf C = \\mathbf D$."}
{"_id": "13458", "text": "Superabundant Numbers are Infinite in Number There are infinitely many superabundant numbers."}
{"_id": "14698", "text": "Sequence of 9 Primes of form 4n+1 The following sequence of $9$ consecutive prime numbers are all of the form $4 n + 1$: :$11 \\, 593, 11 \\, 597, 11 \\, 617, 11 \\, 621, 11 \\, 633, 11 \\, 657, 11 \\, 677, 11 \\, 681, 11 \\, 689$"}
{"_id": "18466", "text": "Set of Rationals Less than Root 2 has no Greatest Element Let $A$ be the set of all positive rational numbers $p$ such that $p^2 < 2$. Then $A$ has no greatest element."}
{"_id": "16933", "text": "Different Representations to Number Base represent Different Integers Let $k \\in \\Z$ such that $k \\ge 2$. Let $a$ and $b$ be representations of integers in base $k$ notation: :$a = \\displaystyle \\sum_{j \\mathop = 0}^r a_j k^j$ :$b = \\displaystyle \\sum_{j \\mathop = 0}^s b_j k^j$ such that either: :$r \\ne s$ or: :$\\exists j \\in \\set {0, 1, \\ldots, r}: a_j \\ne b_j$ Then $a$ and $b$ represent different integers."}
{"_id": "10839", "text": "Subset not necessarily Submagma Let $\\struct {S, \\circ}$ be a magma. Let $T \\subseteq S$. Then it is not necessarily the case that: : $\\struct {T, \\circ} \\subseteq \\struct {S, \\circ}$ That is, it does not always follow that $\\struct {T, \\circ}$ is a submagma of $\\struct {S, \\circ}$."}
{"_id": "17740", "text": "Set of Common Divisors of Integers is not Empty Let $a, b \\in \\Z$ be integers. Let $S$ be the set of common divisors of $a$ and $b$. Then $S$ is not empty."}
{"_id": "12711", "text": "Torelli's Sum :$\\displaystyle \\left({x + y}\\right)^{\\overline n} = \\sum_k \\binom n k x \\left({x - k z + 1}\\right)^{\\overline {k - 1} } \\left({y + k z}\\right)^{\\overline {n - k} }$ where: :$\\dbinom n k$ denotes a binomial coefficient :$x^{\\overline k}$ denotes $x$ to the $k$ rising."}
{"_id": "18272", "text": "Free Matroid is Matroid Let $S$ be a finite set. Let $\\struct{S, \\powerset S}$ be the free matroid of $S$. Then $\\struct{S, \\powerset S}$ is a matroid."}
{"_id": "18597", "text": "Real Function with Strictly Negative Derivative is Strictly Decreasing If $\\forall x \\in \\openint a b: \\map {f'} x < 0$, then $f$ is strictly decreasing on $\\closedint a b$."}
{"_id": "15855", "text": "Addition Rule for Gaussian Binomial Coefficients/Formulation 2 :$\\dbinom n m_q = \\dbinom {n - 1} m_q q^m + \\dbinom {n - 1} {m - 1}_q$"}
{"_id": "14893", "text": "Open Set of Irreducible Space is Irreducible Let $T = \\struct {S, \\tau}$ be an irreducible topological space. Let $U$ be a non-empty open set of $T$. Then $U$ is irreducible in its induced subspace topology."}
{"_id": "18431", "text": "Linear Second Order ODE/y'' + 4 y' + 5 y = 2 exp -2 x The second order ODE: :$(1): \\quad y'' + 4 y' + 5 y = 2 e^{-2 x}$ has the general solution: :$y = e^{-2 x} \\paren {C_1 \\cos x + C_2 \\sin x + 2}$"}
{"_id": "6976", "text": "Order Isomorphism between Linearly Ordered Spaces is Homeomorphism Let $\\struct {S_1, \\le_1, \\tau_1}$ and $\\struct {S_2, \\le_2, \\tau_2}$ be linearly ordered spaces. Let $\\phi: S_1 \\to S_2$ be an order isomorphism from $\\struct {S_1, \\le_1}$ to $\\struct {S_2, \\le_2}$. Then $\\phi$ is a homeomorphism from $\\struct {S_1, \\tau_1}$ to $\\struct {S_2, \\tau_2}$."}
{"_id": "4417", "text": "Non-Zero Real Numbers Closed under Multiplication The set of non-zero real numbers is closed under multiplication: :$\\forall x, y \\in \\R_{\\ne 0}: x \\times y \\in \\R_{\\ne 0}$"}
{"_id": "9410", "text": "Completely Multiplicative Function of Quotient Let $f: \\R \\to \\R$ be a completely multiplicative function. Then: :$\\forall x, y \\in \\R, y \\ne 0: f \\left({\\dfrac x y}\\right) = \\dfrac {f \\left({x}\\right)} {f \\left({y}\\right)}$ whenever $f \\left({y}\\right) \\ne 0$."}
{"_id": "11398", "text": "Identity Matrix is Permutation Matrix An identity matrix is an example of a permutation matrix."}
{"_id": "4968", "text": "Trivial Module is Not Unitary Let $\\struct {G, +_G}$ be an abelian group whose identity is $e_G$. Let $\\struct {R, +_R, \\circ_R}$ be a ring. Let $\\struct {G, +_G, \\circ}_R$ be the trivial $R$-module, such that: :$\\forall \\lambda \\in R: \\forall x \\in G: \\lambda \\circ x = e_G$ Then unless $R$ is a ring with unity and $G$ contains only one element, this is ''not'' a unitary module."}
{"_id": "5006", "text": "Modulo Addition is Closed/Real Numbers Let $z \\in \\R$ be a real number. Then addition modulo $z$ on the set of residue classes modulo $z$ is closed: :$\\forall \\eqclass x z, \\eqclass y z \\in \\R_z: \\eqclass x z +_z \\eqclass y z \\in \\R_z$."}
{"_id": "10612", "text": "Open Rational-Number Balls form Neighborhood Basis in Real Number Line Let $\\R$ be the real number line with the usual (Euclidean) metric. Let $a \\in \\R$ be a point in $\\R$. Let $\\BB_a$ be defined as: :$\\BB_a := \\set {\\map {B_\\epsilon} a: \\epsilon \\in \\Q_{>0} }$ that is, the set of all open $\\epsilon$-balls of $a$ for rational $\\epsilon$. Then the $\\BB_a$ is a basis for the neighborhood system of $a$."}
{"_id": "14226", "text": "Numbers which are Sum of Increasing Powers of Digits The following integers are the sum of the increasing powers of their digits taken in order begin as follows: :$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2 \\, 646 \\, 798, 12 \\, 157 \\, 692 \\, 622 \\, 039 \\, 623 \\, 539$ {{OEIS|A032799}}"}
{"_id": "1763", "text": "Minimal Element of an Ordinal The minimal element of any nonempty ordinal is the empty set. That is, if $S$ is a nonempty ordinal, $\\bigcap S = \\O$"}
{"_id": "308", "text": "Left Operation is Anticommutative The left operation is anticommutative: :$\\forall x, y: x \\leftarrow y = y \\leftarrow x \\iff x = y$"}
{"_id": "17616", "text": "Farey Sequence has Convergent Subsequences for all x in Closed Unit Interval Consider the Farey sequence: :$\\sequence {a_n} = \\dfrac 1 2, \\dfrac 1 3, \\dfrac 2 3, \\dfrac 1 4, \\dfrac 2 4, \\dfrac 3 4, \\dfrac 1 5, \\dfrac 2 5, \\dfrac 3 5, \\dfrac 4 5, \\dfrac 1 6, \\ldots$ Every element of the closed real interval $\\closedint 0 1$ is the limit of a subsequence of $\\sequence {a_n}$."}
{"_id": "16037", "text": "Generating Function of Sequence by Index Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{a_n}\\right\\rangle$. Then: :$z G' \\left({z}\\right)$ is the generating function for the sequence $\\left\\langle{n a_n}\\right\\rangle$ where $G' \\left({z}\\right)$ is the derivative of $G \\left({z}\\right)$ {{WRT|Differentiation}} $z$."}
{"_id": "13701", "text": "Mapping Assigning to Element Its Lower Closure is Isomorphism Let $L = \\left({S, \\vee, \\preceq}\\right)$ be a bounded below join semilattice. Let $I = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an inclusion ordered set where :$\\mathit{Ids}\\left({L}\\right)$ denotes the set of all ideals in $L$, :$\\mathord\\precsim = \\mathord\\subseteq \\cap \\left({\\mathit{Ids}\\left({L}\\right) \\times \\mathit{Ids}\\left({L}\\right)}\\right)$ Let $P = \\left({K\\left({I}\\right), \\precsim'}\\right)$ be an ordered subset of $I$ where :$K\\left({I}\\right)$ denotes the compact subset of $I$. Let $f:S \\to K\\left({I}\\right)$ be a mapping such that :$\\forall x \\in S: f\\left({x}\\right) = x^\\preceq$ Then $f$ is an order isomorphism between $L$ and $P$."}
{"_id": "19196", "text": "Bounds for Rank of Subset Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $\\rho: \\powerset S \\to \\Z$ be the rank function of $M$. Let $A \\subseteq S$ be subset of $S$. Then: :$0 \\le \\map \\rho A \\le \\size A$"}
{"_id": "9914", "text": "Primitive of Reciprocal of Hyperbolic Cosine of a x :$\\ds \\int \\frac {\\d x} {\\cosh a x} = \\frac {2 \\map \\arctan {e^{a x} } } a + C$"}
{"_id": "16075", "text": "Doubly Sequenced Generating Function for Binomial Coefficients Let $\\sequence {a_{m n} }$ be the doubly subscripted sequence defined as: :$\\forall m, n \\in \\N_{\\ge 0}: a_{m n} = \\dbinom n m$ where $\\dbinom n m$ denotes a binomial coefficient. Then the generating function for $\\sequence {a_{m n} }$ is given as: :$\\map G {w, z} = \\dfrac 1 {1 - z - w z}$"}
{"_id": "16645", "text": "Non-Zero Integers under Multiplication are not Subgroup of Reals Let $\\struct {\\Z_{\\ne 0}, \\times}$ denote the algebraic structure formed by the set of non-zero integers under multiplication. Let $\\struct {\\R_{\\ne 0}, \\times}$ denote the algebraic structure formed by the set of non-zero real numbers under multiplication. Then, while $\\struct {\\Z_{\\ne 0}, \\times}$ is closed, it is not a subgroup of $\\struct {\\R_{\\ne 0}, \\times}$."}
{"_id": "10674", "text": "Closed Subset of Real Numbers with Lower Bound contains Infimum Consider the real number line as a metric space under the usual metric. Let $A \\subseteq \\R$ such that $A$ is closed in $\\R$ and $A \\neq \\varnothing$. Let $A$ be bounded below. Then $A$ contains its infimum."}
{"_id": "6142", "text": "Empty Set Satisfies Topology Axioms Let $T = \\struct {\\O, \\set \\O}$ where $\\O$ denotes the empty set. Then $T$ satisfies the open set axioms for a topological space."}
{"_id": "16068", "text": "Newton-Girard Formulas/Examples/Order 2 :$\\displaystyle \\sum_{a \\mathop \\le i \\mathop < j \\mathop \\le b} x_i x_j = \\dfrac 1 2 \\left({\\left({\\sum_{i \\mathop = a}^b x_i}\\right)^2 - \\left({\\sum_{i \\mathop = a}^b {x_i}^2}\\right)}\\right)$"}
{"_id": "2161", "text": "Floquet's Theorem Let $\\mathbf A \\left({t}\\right)$ be a continuous matrix function with period $T$. Let $\\Phi \\left({t}\\right)$ be a fundamental matrix of the Floquet system $\\mathbf x' = \\mathbf A \\left({t}\\right) \\mathbf x$. Then $\\Phi \\left({t + T}\\right)$ is also a fundamental matrix. Moreover, there exists: : A nonsingular, continuously differentiable matrix function $\\mathbf P \\left({t}\\right)$ with period $T$ : A constant (possibly complex) matrix $\\mathbf B$ such that: ::$\\Phi \\left({t}\\right) = \\mathbf P \\left({t}\\right) e^{\\mathbf Bt}$"}
{"_id": "9679", "text": "Pumping Lemma for Regular Languages Let $\\mathcal L_3$ be the set of regular languages. {{explain|Is it a set? Does this need to be proved? Intuition would suggest that it would be a class.}} Then the following holds: $\\forall L \\in \\mathcal L_3: \\exists n_0 \\in \\N_0: \\forall z \\in L: \\left|{z}\\right| > n_0 \\implies \\exists u, v, w$ such that: :$z = u \\cdot v \\cdot w$ :$\\left|{v}\\right| > 0$ :$\\left|{u v}\\right| < n_0$ :$\\forall i \\in \\N_0: u \\cdot v^i \\cdot w \\in L$"}
{"_id": "12020", "text": "Gelfond's Constant is Transcendental '''Gelfond's constant''': :$e^\\pi$ is transcendental."}
{"_id": "4040", "text": "Relative Frequency is Probability Measure The relative frequency model is a probability measure."}
{"_id": "5528", "text": "Free Monoid is Unique Let $S$ be a set. Let $\\struct {M, i}$ and $\\struct {N, j}$ be free monoids over $S$. Then there is a unique monoid isomorphism $f: M \\to N$ such that: :$\\size f \\circ i = j$ :$\\size {f^{-1} } \\circ j = i$ where $\\size {\\, \\cdot \\,}$ denotes the underlying set functor on $\\mathbf{Mon}$."}
{"_id": "16437", "text": "Graph of Nonlinear Additive Function is Dense in the Plane Let $f: \\R \\to \\R$ be an additive function which is not linear. Then the graph of $f$ is dense in the real number plane."}
{"_id": "17539", "text": "Convergence of P-Series/Absolute Convergence if Real Part of p Greater than 1 Let $\\map \\Re p > 1$. Then the $p$-series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty n^{-p}$ converges absolutely."}
{"_id": "17199", "text": "Idempotent Ring has Characteristic Two/Corollary :$\\forall x \\in R: -x = x$"}
{"_id": "7838", "text": "Rational Numbers with Denominators Coprime to Prime under Addition form Group Let $p$ be a prime number. Let $\\Q_p$ denote the set: :$\\set {\\dfrac r s : s \\perp p}$ where $s \\perp p$ denotes that $s$ is coprime to $p$. Then $\\struct {\\Q_p, +}$ is a group."}
{"_id": "3118", "text": "Countable Particular Point Space is Lindelöf Let $T = \\struct {S, \\tau_p}$ be a countable particular point space. Then $T$ is a Lindelöf apace."}
{"_id": "11563", "text": "First Order ODE/exp y dx + (x exp y + 2 y) dy = 0 is an exact differential equation with solution: :$x e^y + y^2 = C$"}
{"_id": "12776", "text": "Floor Function is Replicative The floor function is a replicative function in the sense that: :$\\displaystyle \\forall n \\in \\Z_{> 0}: \\sum_{k \\mathop = 0}^{n - 1} \\left \\lfloor{x + \\frac k n}\\right \\rfloor = \\left \\lfloor{n x}\\right \\rfloor$"}
{"_id": "13617", "text": "Integers with Prime Values of Sigma Function The sequence of integer whose $\\sigma$ value is prime begins: {{begin-eqn}} {{eqn | l = \\map \\sigma 2       | r = 3 }} {{eqn | l = \\map \\sigma 4       | r = 7 }} {{eqn | l = \\map \\sigma 6       | r = 13 }} {{eqn | l = \\map \\sigma {16}       | r = 31 }} {{eqn | l = \\map \\sigma {25}       | r = 31 }} {{eqn | l = \\map \\sigma {64}       | r = 127 }} {{eqn | l = \\map \\sigma {289}       | r = 307 }} {{end-eqn}} {{OEIS|A023194}}"}
{"_id": "7444", "text": "Equivalence of Definitions of Transitive Closure (Set Theory) Let $x$ and $y$ be sets. {{TFAE|def = Transitive Closure (Set Theory)|view = Transitive Closure|context = Set Theory}}"}
{"_id": "15628", "text": "Differential Entropy of Continuous Uniform Distribution Let $X \\sim \\ContinuousUniform a b$ for some $a, b \\in \\R$, $a \\ne b$, where $\\operatorname U$ is the continuous uniform distribution.  Then the differential entropy of $X$, $\\map h X$, is given by:  :$\\map h X = \\map \\ln {b - a}$"}
{"_id": "12806", "text": "Characteristics of Floor and Ceiling Function/Real Domain Let $f: \\R \\to \\Z$ be an integer-valued function which satisfies both of the following: :$(1): \\quad \\map f {x + 1} = \\map f x + 1$ :$(2): \\quad \\forall n \\in \\Z_{> 0}: \\map f x = \\map f {\\dfrac {\\map f {n x} } n}$ Then it is not necessarily the case that either: :$\\forall x \\in \\R: \\map f x = \\floor x$ or: :$\\forall x \\in \\R: \\map f x = \\ceiling x$"}
{"_id": "18330", "text": "Equation of Trochoid Consider a circle $C$ of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on on the line of a radius of $C$ at a distance $b$ from the center of $C$. Let $P$ be on the y-axis when the center of $C$ is also on the y-axis. Consider the trochoid traced out by the point $P$. Let $\\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \\tuple {x, y}$ is described by the equations: :$x = a \\theta  - b \\sin \\theta$ :$y = a - b \\cos \\theta$"}
{"_id": "15096", "text": "Existence of Completely Hausdorff Space which is not Regular There exists at least one example of a topological space which is a completely Hausdorff space, but is not also a regular space."}
{"_id": "13448", "text": "Prime Number is Deficient Let $p$ be a prime number. Then $p$ is deficient."}
{"_id": "15989", "text": "Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers Let $m \\in \\Z_{>0}$ be a (strictly) positive integer. Then: :$\\displaystyle \\sum_{k \\mathop \\in \\Z} \\dbinom m k_\\mathcal F \\left({-1}\\right)^{\\left\\lceil{\\left({m - k}\\right) / 2}\\right\\rceil} {F_{n + k} }^{m - 1} = 0$ where: :$\\dbinom m k_\\mathcal F$ denotes a Fibonomial coefficient :$F_{n + k}$ denotes the $n + k$th Fibonacci number :$\\left\\lceil{\\, \\cdot \\,}\\right\\rceil$ denotes the ceiling function"}
{"_id": "1943", "text": "Sum of Squares of Sum and Difference :$\\forall a, b \\in \\R: \\paren {a + b}^2 + \\paren {a - b}^2 = 2 \\paren {a^2 + b^2}$"}
{"_id": "18870", "text": "Multiplicative Group of Complex Numbers is not Direct Product of Reals with Reals Let $\\struct {\\C_{\\ne 0}, \\times}$ be the multiplicative group of complex numbers. Let $\\struct {\\R_{\\ne 0}, \\times}$ be the multiplicative group of real numbers. Then the direct product $\\struct {\\R_{\\ne 0}, \\times} \\times \\struct {\\R_{\\ne 0}, \\times}$ is not isomorphic with $\\struct {\\C_{\\ne 0}, \\times}$."}
{"_id": "13073", "text": "Circumcenter of Triangle is Orthocenter of Medial Let $\\triangle ABC$ be a triangle. Let $\\triangle DEF$ be the medial triangle of $\\triangle ABC$. Let $K$ be the circumcenter of $\\triangle ABC$. Then $K$ is the orthocenter of $\\triangle DEF$."}
{"_id": "8625", "text": "Interior may not equal Exterior of Exterior Let $T = \\struct {S, \\tau}$ be a topological space. Let $A \\subseteq S$ be a subset of the underlying set $S$ of $T$. Let $A^e$ be the  exterior of $A$. Let $A^\\circ$ be the  interior of $A$. Then it is not necessarily the case that: :$A^{ee} = A^\\circ$"}
{"_id": "2577", "text": "Inverses for Rational Addition Each element $x$ of the set of rational numbers $\\Q$ has an inverse element $-x$ under the operation of rational number addition: :$\\forall x \\in \\Q: \\exists -x \\in \\Q: x + \\paren {-x} = 0 = \\paren {-x} + x$"}
{"_id": "6543", "text": "Bottom is Unique Let $\\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a lattice. Then $S$ has at most one bottom."}
{"_id": "759", "text": "Recurrence Relation for Number of Derangements on Finite Set The number of derangements $D_n$ on a finite set $S$ of cardinality $n$ is: :$D_n = \\paren {n - 1} \\paren {D_{n - 1} + D_{n - 2} }$ where $D_1 = 0$, $D_2 = 1$."}
{"_id": "952", "text": "Elements of Module with Equal Images under Linear Transformations form Submodule Let $G$ and $H$ be $R$-modules. Let $\\phi$ and $\\psi$ be linear transformations from $G$ into $H$. Then the set $S = \\set {x \\in G: \\map \\phi x = \\map \\psi x}$ is a submodule of $G$."}
{"_id": "16286", "text": "Sum of Infinite Series of Product of nth Power of cos 2 theta by 2n+1th Multiple of Sine Let $\\theta \\in \\R$ such that $\\theta \\ne m \\pi$ for any $m \\in \\Z$. Then: {{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 0}^\\infty \\paren {\\cos 2 \\theta}^n \\sin \\paren {2 n + 1} \\theta       | r = \\sin \\theta + \\cos 2 \\theta \\sin 3 \\theta + \\paren {\\cos 2 \\theta}^2 \\sin 5 \\theta + \\paren {\\cos 2 \\theta}^3 \\sin 7 \\theta + \\cdots       | c =  }} {{eqn | r = \\dfrac {\\csc \\theta} 2       | c =  }} {{end-eqn}}"}
{"_id": "11254", "text": "Trisecting the Angle by Compass and Straightedge Construction is Impossible There is no compass and straightedge construction for the trisection of the general angle."}
{"_id": "1526", "text": "First Subsequence Rule Let $T = \\left({A, \\tau}\\right)$ be a Hausdorff space. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $T$. Suppose $\\left \\langle {x_n} \\right \\rangle$ has two convergent subsequences with different limit. Then $\\left \\langle {x_n} \\right \\rangle$ is divergent."}
{"_id": "4156", "text": "Trivial Relation is Largest Equivalence Relation The trivial relation $\\mathcal T$ on $S$ is the largest equivalence in $S$, in the sense that: :$\\forall \\mathcal E \\subseteq S \\times S: \\mathcal E \\subseteq \\mathcal T$ where $\\mathcal E$ denotes a general equivalence relation."}
{"_id": "2961", "text": "Scattered Space is T0 Let $T = \\struct {S, \\tau}$ be a scattered topological space. Then $T$ is also a $T_0$ (Kolmogorov) space."}
{"_id": "18638", "text": "Natural Number is not Subset of its Union Let $n \\in \\N$ be a natural number as defined by the von Neumann construction. Then, except in the degenerate case where $n = 0$, it is not the case that: :$n \\subseteq \\bigcup n$"}
{"_id": "15675", "text": "Beta Function expressed using Gamma Functions Let $\\map \\Beta {x, y}$ denote the Beta function. Then: :$\\map \\Beta {x, y} = \\dfrac {\\map \\Gamma x \\, \\map \\Gamma y} {\\map \\Gamma {x + y} }$ where $\\Gamma$ is the Gamma function:"}
{"_id": "9494", "text": "Primitive of Reciprocal of x cubed by Root of a squared minus x squared cubed :$\\displaystyle \\int \\frac {\\mathrm d x} {x^3 \\left({\\sqrt {a^2 - x^2} }\\right)^3} = \\frac {-1} {2 a^2 x^2 \\sqrt {a^2 - x^2} } + \\frac 3 {2 a^4 \\sqrt {a^2 - x^2} } - \\frac 3 {2 a^5} \\ln \\left({\\frac {a + \\sqrt {a^2 - x^2} } x}\\right) + C$"}
{"_id": "14158", "text": "Reciprocal of 99 The decimal expansion of the reciprocal of $99$ is: :$\\dfrac 1 {99} = 0 \\cdotp \\dot 0 \\dot 1$ {{OEIS|A000035}}"}
{"_id": "18598", "text": "Real Function with Negative Derivative is Decreasing If $\\forall x \\in \\openint a b: \\map {f'} x \\le 0$, then $f$ is decreasing on $\\closedint a b$."}
{"_id": "11820", "text": "Linear Second Order ODE/y'' - 3 y' + 2 y = 14 sine 2 x - 18 cosine 2 x The second order ODE: :$(1): \\quad y'' - 3 y' + 2 y = 14 \\sin 2 x - 18 \\cos 2 x$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x} - 4 x e^{-2 x}$"}
{"_id": "3422", "text": "Real Number Line is Non-Meager Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ is non-meager."}
{"_id": "5925", "text": "Leibniz's Rule/Real Valued Functions Let $f, g : \\R^n \\to \\R$ be real valued functions, $k$ times differentiable on some open set $U \\subseteq \\R^n$. Let $\\alpha = \\tuple {\\alpha_1, \\ldots, \\alpha_n}$ be a multiindex indexed by $\\set {1, \\ldots, n}$ with $\\size \\alpha \\le k$. For $i \\in \\set {1, \\ldots, n}$, let $\\partial_i$ denote the partial derivative: :$\\partial_i = \\dfrac {\\partial} {\\partial {x_i} }$ Let $\\partial^\\alpha$ denote the partial differential operator: :$\\partial^\\alpha = \\partial_1^{\\alpha_1} \\partial_2^{\\alpha_2} \\cdots \\partial_n^{\\alpha_n}$ Then as functions on $U$, we have: :$\\displaystyle \\map {\\partial^\\alpha} {f g} = \\sum_{\\beta \\mathop \\le \\alpha} \\binom \\alpha \\beta \\paren {\\partial^\\beta f} \\paren {\\partial^{\\alpha - \\beta} g}$"}
{"_id": "19642", "text": "Number of Permutations with Repetition Set $S$ be a set of $n$ elements. Let $\\sequence T_m$ be a sequence of $m$ terms of $S$. Then there are $n^m$ different instances of $\\sequence T_m$."}
{"_id": "9724", "text": "Primitive of Reciprocal of p squared plus square of q by Cosine of a x :$\\displaystyle \\int \\frac {\\d x} {p^2 + q^2 \\cos^2 a x} = \\frac 1 {a p \\sqrt{p^2 + q^2} } \\arctan \\frac {p \\tan a x} {\\sqrt {p^2 + q^2} } + C$"}
{"_id": "8277", "text": "Bottom-Up Form of Top-Down Grammar defines same Formal Language Let $\\mathcal L$ be a formal language. Let $\\mathcal T$ be a top-down grammar for $\\mathcal L$. Let $\\mathcal B$ be the bottom-up form of $\\mathcal T$. Then $\\mathcal B$ is also a formal grammar for $\\mathcal L$."}
{"_id": "1023", "text": "Lines are Subspaces of Plane The one-dimensional subspaces of $\\R^2$ are precisely the homogeneous lines of plane analytic geometry."}
{"_id": "3832", "text": "Gaussian Integer Units form Multiplicative Subgroup of Complex Numbers The group of Gaussian integer units under complex multiplication: :$\\struct {U_\\C, \\times} = \\struct {\\set {1, i, -1, -i}, \\times}$ forms a subgroup of the multiplicative group of complex numbers."}
{"_id": "3994", "text": "Lagrange Polynomial Approximation Let $f: D \\to \\R$ be $n+1$ times differentiable in an interval $I \\subset \\R$. Let $x_0, \\dotsc, x_n \\in I$ be pairwise distinct points. Let $P$ be the Lagrange Interpolation Formula of degree at most $n$ such that $P \\left({x_i}\\right) = f \\left({x_i}\\right)$ for all $i = 0, \\dotsc, n$. Let $R \\left({x}\\right) = f \\left({x}\\right) - P \\left({x}\\right)$. Then, for every $x \\in I$, there exists $\\xi$ in the interval spanned by $x$ and $x_i$, $i = 0, \\dotsc, n$, such that: :$R \\left({x}\\right) = \\dfrac{\\left({x - x_0}\\right) \\left({x - x_1}\\right) \\dotsm \\left({x - x_n}\\right) f^{\\left({n + 1}\\right)} \\left({\\xi}\\right)} {\\left({n + 1}\\right)!}$"}
{"_id": "5542", "text": "Inequality for Ordinal Exponentiation Let $x$ and $y$ be ordinals. Let $x$ be a limit ordinal and let $y > 0$. Let $\\sequence {a_i}$ be a sequence of ordinals that is strictly decreasing on $1 \\le i \\le n$. Let $\\sequence {b_i}$ be a sequence of natural numbers. Then: :$\\displaystyle \\paren {\\sum_{i \\mathop = 1}^n x^{a_i} \\times b_i}^y \\le x^{a_1 \\mathop \\times y} \\times \\paren {b_1 + 1}$"}
{"_id": "15399", "text": "Residue of Quotient Let $f$ and $g$ be functions holomorphic on some region containing $a$.  Let $g$ have a zero of multiplicity $1$ at $a$.  Then: :$\\Res {\\dfrac f g} a = \\dfrac {\\map f a} {\\map {g'} a}$ {{explain|definition of \"Res\", by a link to Residue}}"}
{"_id": "975", "text": "Linear Transformations of Commutative Scalar Ring Let $R$ be a commutative ring. Let $\\left({G, +_G, \\circ}\\right)_R$ and $\\left({H, +_H, \\circ}\\right)_R$ be $R$-modules. Let $\\mathcal L_R \\left({G, H}\\right)$ be the set of all linear transformations from $G$ to $H$. Then $\\mathcal L_R \\left({G, H}\\right)$ is a submodule of the $R$-module $H^G$. If $H$ is a unitary module, then so is $\\mathcal L_R \\left({G, H}\\right)$."}
{"_id": "1262", "text": "Relative Homotopy is Equivalence Relation Let $X$ and $Y$ be topological spaces. Let $K \\subseteq X$ be a (possibly empty) subset of $X$. Let $\\map \\CC {X, Y}$ be the set of all continuous mappings from $X$ to $Y$. Define a relation $\\sim$ on $\\map \\CC {X, Y}$ as: :$f \\sim g$ {{iff}} $f$ and $g$ are homotopic relative to $K$. Then $\\sim$ is an equivalence relation."}
{"_id": "1419", "text": "Basis for Topology on Cartesian Product Let $T_1 = \\struct{A_1, \\tau_1}$ and $T_2 = \\struct{A_2, \\tau_2}$ be topological spaces. Let $A_1 \\times A_2$ be the Cartesian product of $A_1$ and $A_2$. Let $\\PP = \\set{ U_1 \\times U_2 : U_1 \\in \\tau_1, U_2 \\in \\tau_2}$ Then $\\PP$ is a synthetic basis on $A_1 \\times A_2$."}
{"_id": "8807", "text": "Complex Modulus of Quotient of Complex Numbers Let $z_1, z_2 \\in \\C$ be complex numbers such that $z_2 \\ne 0$. Let $\\cmod z$ denote the modulus of $z$. Then: :$\\cmod {\\dfrac {z_1} {z_2} } = \\dfrac {\\cmod {z_1} } {\\cmod {z_2} }$"}
{"_id": "19019", "text": "Equivalence of Definitions of Boundary Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$. {{TFAE|def = Boundary (Topology)|view = boundary of $H$}}"}
{"_id": "3522", "text": "Inscribing Regular 15-gon in Circle In a given circle, it is possible to inscribe a regular 15-gon. {{:Euclid:Proposition/IV/16}}"}
{"_id": "15124", "text": "Abelianization of Free Group is Free Abelian Group Let $X$ be a set. Let $\\struct {F_X, \\iota}$ be a free group on $X$. Let $F_X^{\\mathrm {ab} }$ be its abelianization. Let $\\pi : F_X \\to F_X^{\\mathrm {ab} }$ be the quotient group epimorphism. Then $\\struct {F_X^{\\mathrm {ab} }, \\pi \\circ \\iota}$ is a free abelian group on $X$."}
{"_id": "2946", "text": "Totally Disconnected but Connected Set must be Singleton Let $T = \\struct {S, \\tau}$ be a topological space. Let $H \\subseteq S$ be both totally disconnected and connected. Then $H$ is a singleton."}
{"_id": "10951", "text": "Left Identity Element is Idempotent Let $\\struct {S, \\circ}$ be an algebraic structure. Let $e_L \\in S$ be a left identity with respect to $\\circ$. Then $e_L$ is idempotent under $\\circ$."}
{"_id": "8469", "text": "Element in Preimage of Image under Mapping Let $f: S \\to T$ be a mapping. Then: :$\\forall x \\in S: x \\in f^{-1} \\sqbrk {\\map f x}$"}
{"_id": "2937", "text": "Locally Euclidean Space is Locally Compact Let $M$ be a locally Euclidean space of some dimension $d$. Then $M$ is locally compact."}
{"_id": "3131", "text": "Equivalence of Definitions of Algebraically Closed Field Let $K$ be a field.  {{TFAE|def = Algebraically Closed Field}}"}
{"_id": "11388", "text": "Partition of Facets of Rubik's Cube Let $S$ denote the set of the facets of Rukik's cube. Then $S$ can be partitioned as follows: :$S = \\set {S_C \\mid S_E \\mid S_Z}$ where: :$S_C$ denotes the set of corner facets :$S_E$ denotes the set of edge facets :$S_Z$ denotes the set of center facets."}
{"_id": "12160", "text": "Field Norm of Quaternion is not Norm Let $\\mathbf x = a \\mathbf 1 + b \\mathbf i + c \\mathbf j + d \\mathbf k$ be a quaternion. Let $\\overline {\\mathbf x}$ be the conjugate of $\\mathbf x$. The field norm of $\\mathbf x$: :$\\map n {\\mathbf x} := \\size {\\mathbf x \\overline {\\mathbf x} }$ is not a norm in the abstract algebraic context of a division ring."}
{"_id": "10147", "text": "Multiplication of Real Numbers is Right Distributive over Subtraction {{:Euclid:Proposition/V/6}} That is, for any number $a$ and for any integers $m, n$: :$m a - n a = \\paren {m - n} a$"}
{"_id": "19615", "text": "Strict Ordering on Integers is Transitive Let $\\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Then: {{begin-eqn}} {{eqn | l = \\eqclass {a, b} {}       | o = <       | r = \\eqclass {c, d} {}       | c =  }} {{eqn | lo= \\land       | l = \\eqclass {c, d} {}       | o = <       | r = \\eqclass {e, f} {}       | c =  }} {{eqn | ll= \\implies       | l = \\eqclass {a, b} {}       | o = <       | r = \\eqclass {e, f} {}       | c =  }} {{end-eqn}} That is, strict ordering on the integers is transitive."}
{"_id": "11097", "text": "Real Power of Strictly Positive Real Number is Strictly Positive Let $x$ be a strictly positive real number. Let $y$ be a real number. Then: :$x^y > 0$ where $x^y$ denotes $x$ raised to the $y$th power."}
{"_id": "4666", "text": "Conditions for Internal Group Direct Product Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $H_1, H_2 \\le G$. Then $G$ is the internal group direct product of $H_1$ and $H_2$ {{iff}}: :$(1): \\quad \\forall h_1 \\in H_1, h_2 \\in H_2: h_1 \\circ h_2 = h_2 \\circ h_1$ :$(2): \\quad G = H_1 \\circ H_2$ :$(3): \\quad H_1 \\cap H_2 = \\set e$ Condition $(1)$ can also be stated as: :$(1): \\quad$ Either $H_1$ or $H_2$ is normal in $G$"}
{"_id": "5700", "text": "Identity Morphism of Product Let $\\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\\mathbf C$, and let $C \\times D$ be a binary product for $C$ and $D$. Then: :$\\operatorname{id}_{\\left({C \\mathop \\times D}\\right)} = \\operatorname{id}_C \\times \\operatorname{id}_D$ where $\\operatorname{id}$ denotes an identity morphism, and $\\times$ signifies a product of morphisms."}
{"_id": "16809", "text": "Conditions Satisfied by Linear Code Let $p$ be a prime number. Let $\\Z_p$ be the set of residue classes modulo $p$. Let $C := \\tuple {n, k}$ be a linear code of a master code $\\map V {n, p}$. Then $C$ satisfies the following conditions: :$(C \\, 1): \\quad \\forall \\mathbf x, \\mathbf y \\in C: \\mathbf x + \\paren {-\\mathbf y} \\in C$ :$(C \\, 2): \\quad \\forall \\mathbf x \\in C, m \\in \\Z_p: m \\times \\mathbf x \\in C$ where $+$ and $\\times$ are the operations of codeword addition and codeword multiplication respectively. {{expand|Add a page defining the difference between codewords.}}"}
{"_id": "11402", "text": "Equivalence of Definitions of Non-Invertible Matrix Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\mathbf A$ be an element of the ring of square matrices $\\struct {\\map {\\mathcal M_R} n, +, \\times}$. The following definitions for $\\mathbf A$ to be non-invertible are equivalent:"}
{"_id": "15561", "text": "Power Series Expansion for Logarithm of x/Formulation 2 {{begin-eqn}} {{eqn | l = \\ln x        | r = \\sum_{n \\mathop = 1}^\\infty \\paren {\\frac {x - 1} x}^n }} {{eqn | r = \\frac {x - 1} x + \\frac 1 2 \\paren {\\frac {x - 1} x}^2 + \\frac 1 3 \\paren {\\frac {x - 1} x}^3 + \\cdots }} {{end-eqn}} valid for all $x \\in \\R$ such that $x \\ge \\dfrac 1 2$."}
{"_id": "8186", "text": "Biconditional in terms of NAND :$p \\iff q \\dashv \\vdash \\paren {\\paren {p \\uparrow p} \\uparrow \\paren {q \\uparrow q} } \\uparrow \\paren {p \\uparrow q}$ where $\\iff$ denotes logical biconditional and $\\uparrow$ denotes logical NAND."}
{"_id": "10512", "text": "Images of Elements under Repeated Composition of Injection form Equivalence Classes Let $S$ be a set. Let $f: S \\to S$ be an injection. Let the sequence of mappings: :$f^0, f^1, f^2, \\ldots, f^n, \\ldots$ be defined as: :$\\forall n \\in \\N: \\map {f^n} x = \\begin {cases} x & : n = 0 \\\\ \\map f x & : n = 1 \\\\ \\map f {\\map {f^{n - 1} } x} & : n > 1 \\end{cases}$ Let $\\mathcal R \\subseteq S \\times S$ be the relation on $S$ defined as: :$\\mathcal R = \\set {\\tuple {a, b} \\in S \\times S: \\exists k \\in \\Z: b = \\map {f^k} a \\lor \\exists j \\in \\Z: a = \\map {f^j} b}$ Then $\\mathcal R$ is an equivalence relation."}
{"_id": "9854", "text": "Primitive of Power of x by Arcsine of x over a :$\\displaystyle \\int x^m \\arcsin \\frac x a \\rd x = \\frac {x^{m + 1} } {m + 1} \\arcsin \\frac x a - \\frac 1 {m + 1} \\int \\frac {x^{m + 1} \\rd x} {\\sqrt{a^2 - x^2} } + C$"}
{"_id": "12912", "text": "Equivalence of Definitions of Distance to Nearest Integer Function The following definitions of the distance to nearest integer function $\\norm \\cdot: \\R \\to \\closedint 0 {\\dfrac 1 2}$ are equivalent:"}
{"_id": "10176", "text": "LCM of Three Numbers Let $a, b, c \\in \\Z: a b c \\ne 0$. The lowest common multiple of $a, b, c$, denoted $\\lcm \\set {a, b, c}$, can always be found. {{:Euclid:Proposition/VII/36}}"}
{"_id": "11629", "text": "Second Order ODE/x y'' - y' = 3 x^2 The second order ODE: :$(1): \\quad x y'' - y' = 3 x^2$ has the general solution: :$y = x^3 + \\dfrac {C_1 x^2} 2 + C^2$"}
{"_id": "3917", "text": "Real Number Line is Locally Compact Hausdorff Space Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ is a locally compact Hausdorff space."}
{"_id": "7079", "text": "Equivalence of Definitions of Generalized Ordered Space Let $\\struct {S, \\preceq}$ be a totally ordered set. Let $\\tau$ be a topology for $S$. {{TFAE|def = Generalized Ordered Space}}"}
{"_id": "3692", "text": "Multiplicative Identity is Unique Let $\\struct {F, +, \\times}$ be a field. Then the multiplicative identity $1_F$ of $F$ is unique."}
{"_id": "9651", "text": "Primitive of Reciprocal of Sine of a x/Logarithm of Tangent Form :$\\ds \\int \\frac {\\d x} {\\sin a x} = \\frac 1 a \\ln \\size {\\tan \\frac {a x} 2} + C$"}
{"_id": "13694", "text": "Integers which are Sigma for 3 Integers The sequence of integers which are the $\\sigma$ value of $3$ different integers begins: :$24, 42, 48, 60, 84, 90, \\ldots$ {{OEIS|A007372}}"}
{"_id": "9794", "text": "Primitive of Reciprocal of Cotangent of a x :$\\displaystyle \\int \\frac {\\d x} {\\cot a x} = \\frac {-\\ln \\size {\\cos a x} } a + C$"}
{"_id": "19664", "text": "Shortest Distance between Two Points is Straight Line The shortest distance between $2$ points is a straight line."}
{"_id": "16111", "text": "Diagonals of Rhombus Intersect at Right Angles Let $ABCD$ be a rhombus. The diagonals $AC$ and $BD$ of $ABCD$ intersect each other at right angles."}
{"_id": "3254", "text": "Condition for Open Extension Space to be First-Countable Let $T = \\struct {S, \\tau}$ be a topological space. Let $T^*_{\\bar p} = \\struct {S^*_p, \\tau^*_{\\bar p} }$ be the open extension space of $T$. Then $T^*_{\\bar p}$ is a first-countable space {{iff}} $T$ is."}
{"_id": "2352", "text": "Lune of Hippocrates Take the circle whose center is $A$ and whose radius is $AB = AC = AD = AE$. Let $C$ be the center of a circle whose radius is $CD = CF = CE$. :400px Consider the lune $DFEB$. Its area is equal to that of the square $AEGC$."}
{"_id": "9706", "text": "Primitive of Cube of Cosine of a x :$\\displaystyle \\int \\cos^3 a x \\rd x = \\frac {\\sin a x} a - \\frac {\\sin^3 a x} {3 a} + C$"}
{"_id": "18180", "text": "Sine of Half Side for Spherical Triangles :$\\sin \\dfrac a 2 = \\sqrt {\\dfrac {-\\cos S \\, \\map \\cos {S - A} } {\\sin B \\sin C} }$ where $S = \\dfrac {A + B + C} 2$."}
{"_id": "5958", "text": "Equivalence of Definitions of Hereditarily Compact Let $T = \\left({S, \\tau}\\right)$ be a topological space. {{TFAE|def = Hereditarily Compact Space}}"}
{"_id": "13841", "text": "Triplets of Products of Two Distinct Primes The following triplets of consecutive positive integers are the smallest in which each number is the product of $2$ distinct prime numbers: :$33, 34, 35$ :$85, 86, 87$ :$93, 94, 95$ :$141, 142, 143$ :$201, 202, 203$ :$213, 214, 215$ :$217, 218, 219$"}
{"_id": "2950", "text": "Product Category is Category Let $\\mathbf C$ and $\\mathbf D$ be metacategories. Then the product category $\\mathbf C \\times \\mathbf D$ is a metacategory."}
{"_id": "6286", "text": "Composition of Ring Monomorphisms is Ring Monomorphism Let: * $\\left({R_1, +_1, \\circ_1}\\right)$ * $\\left({R_2, +_2, \\circ_2}\\right)$ * $\\left({R_3, +_3, \\circ_3}\\right)$ be rings. Let: * $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ * $\\psi: \\left({R_2, +_2, \\circ_2}\\right) \\to \\left({R_3, +_3, \\circ_3}\\right)$ be (ring) monomorphisms. Then the composite of $\\phi$ and $\\psi$ is also a (ring) monomorphism."}
{"_id": "19686", "text": "Set of Positive Integers does not form Ring Let $\\Z_{\\ge 0}$ denote the set of positive integers. Then the algebraic structure $\\struct {\\Z_{\\ge 0}, +, \\times}$ does not form a ring."}
{"_id": "10517", "text": "Equivalence Classes induced by Derivative Function on Set of Functions Let $X$ be the set of real functions $f: \\R \\to \\R$ which possess continuous derivatives. Let $\\mathcal R \\subseteq X \\times X$ be the equivalence relation on $X$ defined as: :$\\mathcal R = \\set {\\tuple {f, g} \\in X \\times X: D f = D g}$ where $D f$ denotes the first derivative of $f$. Then the equivalence classes of $\\mathcal R$ are defined as: :$\\map {\\eqclass f {\\mathcal R} } x = \\set {g \\in X: \\exists c \\in \\R: \\forall x \\in \\R: \\map g x = \\map f x + c}$ That is, it consists of the set of all real functions $f \\in X$ which differ by a real constant."}
{"_id": "309", "text": "Right Operation is Anticommutative The right operation is anticommutative: :$\\forall x, y: x \\rightarrow y = y \\rightarrow x \\iff x = y$"}
{"_id": "4866", "text": "Subring is not necessarily Ideal Let $\\left({R, +, \\circ}\\right)$ be a ring. Let $\\left({S, +_S, \\circ_S}\\right)$ be a subring of $R$. Then it is not necessarily the case that $S$ is also an ideal of $R$."}
{"_id": "2091", "text": "Coin-Tossing Modeled by Bernoulli Trial The act of coin-tossing can be modeled as a Bernoulli trial. This applies whether the coin is fair or biased."}
{"_id": "7246", "text": "Open Ray is Open in GO-Space/Definition 2 Let $\\struct {S, \\preceq, \\tau}$ be a generalized ordered space by Definition 2. That is: :Let $\\struct {S, \\preceq}$ be a totally ordered set. :Let $\\struct {S, \\tau}$ be a topological space. Let there be: :a linearly ordered space $\\struct {S', \\preceq', \\tau'}$ and: :a mapping $\\phi: S \\to S'$ which is both: ::a $\\preceq$-$\\preceq'$ order embedding :and: ::a $\\tau$-$\\tau'$ topological embedding. Let $p \\in S$. Then: :$p^\\prec$ and $p^\\succ$ are $\\tau$-open where: :$p^\\prec$ is the strict lower closure of $p$ :$p^\\succ$ is the strict upper closure of $p$."}
{"_id": "11353", "text": "Union of Set of Dense-in-itself Sets is Dense-in-itself Let $T$ be a topological space. Let $\\mathcal F \\subseteq \\mathcal P \\left({T}\\right)$ such that  :every element of $\\mathcal F$ is dense-in-itself. Then the union $\\bigcup \\mathcal F$ is also dense-in-itself."}
{"_id": "13714", "text": "Smallest Non-Palindromic Number with Palindromic Square $26$ is the smallest non-palindromic integer whose square is palindromic."}
{"_id": "10622", "text": "Positive Image of Point of Continuous Real Function implies Positive Closed Interval of Domain Let $f: \\R \\to \\R$ be a continuous real function. Let $a \\in \\R$ such that $f \\left({a}\\right) > 0$. Then: :$\\exists k \\in \\R_{>0}: \\exists \\delta \\in \\R_{>0}: \\forall x \\in \\left[{a - \\delta \\,.\\,.\\, a + \\delta}\\right]: f \\left({x}\\right) \\ge k$"}
{"_id": "8516", "text": "Equivalence of Definitions of Ordering/Proof 1 The following definitions of ordering are equivalent:"}
{"_id": "10041", "text": "Primitive of x by Inverse Hyperbolic Cosine of x over a :$\\displaystyle \\int x \\cosh^{-1} \\frac x a \\rd x = \\begin{cases} \\paren {\\dfrac {x^2} 2 - \\dfrac {a^2} 4} \\cosh^{-1} \\dfrac x a - \\dfrac {x \\sqrt {x^2 - a^2} } 4 + C & : \\cosh^{-1} \\dfrac x a > 0 \\\\ \\paren {\\dfrac {x^2} 2 - \\dfrac {a^2} 4} \\cosh^{-1} \\dfrac x a + \\dfrac {x \\sqrt {x^2 - a^2} } 4 + C & : \\cosh^{-1} \\dfrac x a < 0 \\end{cases}$"}
{"_id": "7038", "text": "Convex Set is Star Convex Set Let $V$ be a vector space over $\\R$ or $\\C$. Let $A \\subseteq V$ be a non-empty convex set. Then $A$ is a star convex set, and every point in $A$ is a star center."}
{"_id": "6793", "text": "De Morgan's Laws imply Uniquely Complemented Lattice is Boolean Lattice Let $\\left({S, \\wedge, \\vee, \\preceq}\\right)$ be a uniquely complemented lattice. Then the following are equivalent: $(1):\\quad \\forall p, q \\in S: \\neg p \\vee \\neg q = \\neg \\left({p \\wedge q}\\right)$ $(2):\\quad \\forall p, q \\in S: \\neg p \\wedge \\neg q = \\neg \\left({p \\vee q}\\right)$ $(3):\\quad \\forall p, q \\in S: p \\preceq q \\iff \\neg q \\preceq \\neg p$ $(4):\\quad \\left({S, \\wedge, \\vee, \\preceq}\\right)$ is a distributive lattice."}
{"_id": "16034", "text": "Generating Function for mth Terms of Sequence Let $G \\left({z}\\right)$ be the generating function for the sequence $\\left\\langle{a_n}\\right\\rangle$. Let $m \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\omega = e^{2 i \\pi / m} = \\cos \\dfrac {2 \\pi} m + i \\sin \\dfrac {2 \\pi} m$. Then for $r \\in \\Z$ such that $0 \\le r < m$: :$\\displaystyle \\sum_{n \\bmod m \\mathop = r} a_n z^n = \\dfrac 1 m \\sum_{0 \\mathop \\le k \\mathop < m} \\omega^{-k r} G \\left({\\omega^k z}\\right)$"}
{"_id": "9993", "text": "Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x :$\\displaystyle \\int \\frac {\\mathrm d x} {\\sinh^2 a x \\cosh^2 a x} = \\frac {-2 \\coth 2 a x} a + C$"}
{"_id": "17448", "text": "Laplace Transform Exists if Function Piecewise Continuous and of Exponential Order Let $f$ be a real function which is: :piecewise continuous in every closed interval $\\closedint 0 N$ :of exponential order $\\gamma$ for $t > N$ Then the Laplace transform $\\map F s$ of $\\map f t$ exists for all $s > \\gamma$."}
{"_id": "17155", "text": "Zero of Subfield is Zero of Field The zero of $\\struct {K, +, \\times}$ is also $0$."}
{"_id": "18232", "text": "Convergent Sequence in P-adic Numbers has Unique Limit Let $p$ be a prime number. Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic numbers. Let $\\sequence {x_n} $ be a sequence in $\\Q_p$. Then $\\sequence {x_n}$ can have at most one limit."}
{"_id": "19074", "text": "T3 1/2 Space is not necessarily T2 Space Let $T = \\struct {S, \\tau}$ be a be a $T_{3 \\frac 1 2}$ space. Then it is not necessarily the case that $T$ is a $T_2$ (Hausdorff) space."}
{"_id": "2411", "text": "Floor equals Ceiling iff Integer :$\\left \\lfloor {x}\\right \\rfloor = \\begin{cases} \\left \\lceil {x}\\right \\rceil & : x \\in \\Z \\\\ \\left \\lceil {x}\\right \\rceil - 1 & : x \\notin \\Z \\\\ \\end{cases}$ or equivalently: :$\\left \\lceil {x}\\right \\rceil = \\begin{cases} \\left \\lfloor {x}\\right \\rfloor & : x \\in \\Z \\\\ \\left \\lfloor {x}\\right \\rfloor + 1 & : x \\notin \\Z \\\\ \\end{cases}$ where $\\Z$ is the set of integers."}
{"_id": "2520", "text": "Subspace of Real Functions of Differentiability Class Let $\\mathbb J = \\set {x \\in \\R: a < x < b}$ be an open interval of the real number line $\\R$. Let $\\map {\\CC^{\\paren m} } {\\mathbb J}$ be the set of all continuous real functions on $\\mathbb J$ in differentiability class $m$. Then $\\struct {\\map {\\CC^{\\paren m} } {\\mathbb J}, +, \\times}_\\R$ is a subspace of the $\\R$-vector space $\\struct {\\R^{\\mathbb J}, +, \\times}_\\R$."}
{"_id": "140", "text": "Symmetric Difference with Universe :$\\mathbb U * S = \\map \\complement S$ where: :$\\mathbb U$ denotes the universe :$*$ denotes symmetric difference."}
{"_id": "4308", "text": "Sum of Projections/Binary Case Let $H$ be a Hilbert space. Let $P, Q$ be projections. Then $P + Q$ is a projection {{iff}} $\\Rng P \\perp \\Rng Q$."}
{"_id": "15515", "text": "Sum of Reciprocals of Fourth Powers Alternating in Sign {{begin-eqn}} {{eqn | l = \\sum_{n \\mathop = 1}^\\infty \\dfrac {\\paren {-1}^{n + 1} } {n^4}       | r = \\frac 1 {1^4} - \\frac 1 {2^4} + \\frac 1 {3^4} - \\frac 1 {4^4} + \\cdots       | c =  }} {{eqn | r = \\dfrac {7 \\pi^4} {720}       | c =  }} {{end-eqn}}"}
{"_id": "11079", "text": "Sign of Cotangent Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \\cot x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $n \\pi < x < \\paren {n + \\dfrac 1 2} \\pi$ }} {{eqn | l = \\cot x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\\paren {n + \\dfrac 1 2} \\pi < x < \\paren {n + 1} \\pi$ }} {{end-eqn}} where $\\cot$ is the real cotangent function."}
{"_id": "18924", "text": "Complete Graph is Hamiltonian for Order Greater than 2 Let $n \\in \\Z$ be an integer such that $n > 2$. Let $K_n$ denote the complete graph of order $n$. Then $K_n$ is Hamiltonian."}
{"_id": "18666", "text": "Bounded Class is Set Let $B$ be a class. Let it be assumed that $B$ is a subclass of a basic universe $V$. Let $B$ be bounded by a set $x$. Then $B$ is itself a set."}
{"_id": "3476", "text": "Limit Points in Uncountable Fort Space Let $T = \\struct {S, \\tau_p}$ be an uncountable Fort space. Let $U \\subseteq S$ be a countably infinite subset of $S$. Then $p$ is the only limit point of $U$."}
{"_id": "19647", "text": "P-Sequence Space with P-Norm forms Normed Vector Space $P$-Sequence Space with $p$-norm forms normed vector space."}
{"_id": "16408", "text": "Identity of Submonoid is not necessarily Identity of Monoid Let $\\struct {S, \\circ}$ be a monoid whose identity is $e_S$. Let $\\struct {T, \\circ}$ be a submonoid of $\\struct {S, \\circ}$ whose identity is $e_T$. Then it is not necessarily the case that $e_T = e_S$."}
{"_id": "15769", "text": "Modulo Operation/Examples/1.1 mod 1 :$1 \\cdotp 1 \\bmod 1 = 0 \\cdotp 1$"}
{"_id": "2924", "text": "Sequence of Implications of Connectedness Properties Let $P_1$ and $P_2$ be connectedness properties and let: :$P_1 \\implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | Ultraconnected || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | Arc-Connected || | align=\"center\" | $\\implies$ || | align=\"center\" | Path-Connected || |- | align=\"center\" | || | align=\"center\" | || | align=\"center\" | $\\Big\\Downarrow$ || |- | align=\"center\" | Irreducible || | align=\"center\" | $\\implies$ || | align=\"center\" | Connected || |- |}"}
{"_id": "15286", "text": "Hilbert Cube is Compact Let $M = \\struct {I^\\omega, d_2}$ be the Hilbert cube. Then $M$ is a compact space."}
{"_id": "766", "text": "No Group has Two Order 2 Elements A group can not contain exactly two elements of order $2$."}
{"_id": "6495", "text": "Existence of Dedekind Completion Let $\\left({S, \\preceq}\\right)$ be an ordered set. Then there exists a Dedekind completion of $S$. That is, there exists a Dedekind complete ordered set $\\tilde S$ and an order embedding $\\phi: S \\to \\tilde S$ such that: :For all Dedekind complete ordered sets $X$, and for all order embeddings $f: S \\to X$, there exists an order embedding $\\tilde f: \\tilde S \\to X$ such that: ::$\\tilde f \\circ \\phi = f$"}
{"_id": "18131", "text": "Angle of Spherical Triangle from Sides Let $\\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\\cos A = \\cosec b \\cosec c \\paren {\\cos a - \\cos b \\cos c}$"}
{"_id": "9498", "text": "Primitive of x cubed by Root of a squared minus x squared cubed :$\\displaystyle \\int x^3 \\paren {\\sqrt {a^2 - x^2} }^3 \\rd x = \\frac {\\paren {\\sqrt {a^2 - x^2} }^7} 7 - \\frac {a^2 \\paren {\\sqrt {a^2 - x^2} }^5} 5 + C$"}
{"_id": "11110", "text": "Group is Abelian iff Opposite Group is Itself Let $\\left({G, \\circ}\\right)$ be a  group. Let $\\left({G, *}\\right)$ be the opposite group to $({G, \\circ})$. $\\left({G, \\circ}\\right)$ is an Abelian group {{iff}}: :$\\left({G, \\circ}\\right) = \\left({G, *}\\right)$"}
{"_id": "3150", "text": "Sierpiński Space is Ultraconnected Let $T = \\struct {\\set {0, 1}, \\tau_0}$ be a Sierpiński space. Then $T$ is ultraconnected."}
{"_id": "10655", "text": "Open Ball in Real Number Plane under Chebyshev Distance Let $\\R^2$ be the real number plane. Let $d_\\infty: \\R^2 \\times \\R^2 \\to \\R$ be the Chebyshev Distance on $\\R^2$: : $\\displaystyle d_\\infty \\left({x, y}\\right):= \\max \\left\\{ {\\left\\vert{x_1 - y_1}\\right\\vert, \\left\\vert{x_2 - y_2}\\right\\vert}\\right\\}$ where $x = \\left({x_1, x_2}\\right), y = \\left({y_1, y_2}\\right) \\in \\R^2$. For $a \\in \\R^2$, let $B_\\epsilon \\left({a}\\right)$ be the open $\\epsilon$-ball at $a$. Then $B_\\epsilon \\left({a}\\right)$ is the interior of the square centered at $a$ and whose sides are of length $2 \\epsilon$ parallel to the coordinate axes."}
{"_id": "10094", "text": "Steiner-Lehmus Theorem Let $ABC$ be a triangle. Denote the lengths of the angle bisectors through the vertices $A$ and $B$ by $\\omega_\\alpha$ and $\\omega_\\beta$. Let $\\omega_\\alpha = \\omega_\\beta$. Then $ABC$ is an isosceles triangle."}
{"_id": "5832", "text": "Gaussian Integers form Subgroup of Complex Numbers under Addition The set of Gaussian integers $\\Z \\sqbrk i$, under the operation of complex addition, forms a subgroup of the set of additive group of complex numbers $\\struct {\\C, +}$."}
{"_id": "17259", "text": "Polynomials in Integers is not Principal Ideal Domain Let $\\Z \\sqbrk X$ be the ring of polynomials in $X$ over $\\Z$. Then $\\Z \\sqbrk X$ is not a principal ideal domain."}
{"_id": "4282", "text": "Intersection of Orthocomplements is Orthocomplement of Closed Linear Span Let $H$ be a Hilbert space. Let $\\family {M_i}_{i \\mathop \\in I}$ be an $I$-indexed family of closed linear subspaces of $H$. Then: :$\\ds \\bigcap_{i \\mathop \\in I} M_i^\\perp = \\paren {\\vee \\set {M_i : i \\in I} }^\\perp$ where: :$\\perp$ denotes orthocomplementation :$\\vee$ denotes closed linear span."}
{"_id": "14574", "text": "One-Digit Number is Harshad Let $n$ be a $1$-digit positive integer. Then $n$ is a harshad number."}
{"_id": "9081", "text": "Concave Real Function is Continuous Let $f$ be a real function which is concave on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Then $f$ is continuous on $\\left({a \\,.\\,.\\, b}\\right)$."}
{"_id": "8609", "text": "Zero is Limit Point of Integer Reciprocal Space Union with Closed Interval Let $A \\subseteq \\R$ be the set of all points on $\\R$ defined as: :$A := \\set {\\dfrac 1 n : n \\in \\Z_{>0} }$ Let $T = \\struct {A, \\tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Let $B$ be the uncountable set: :$B := A \\cup \\closedint 2 3$ where $\\closedint 2 3$ is a closed interval of $\\R$. $2$ and $3$ are to all intents arbitrary, but convenient. Then $0$ is a limit point of $B$ in $\\R$."}
{"_id": "11495", "text": "Half-Life of Radioactive Substance Let a radioactive element $S$ decay with a rate constant $k$. Then its half-life $T$ is given by: :$T = \\dfrac {\\ln 2} k$ :400px"}
{"_id": "11637", "text": "Second Order ODE/(x^2 + 2 y') y'' + 2 x y' = 0 The second order ODE: :$(1): \\quad \\paren {x^2 + 2 y'} y'' + 2 x y' = 0$ subject to the initial conditions: :$y = 1$ and $y' = 0$ when $x = 0$ has the particular solution: :$y = 1$ or: :$3 y + x^3 = 3$"}
{"_id": "18027", "text": "Left Module Does Not Necessarily Induce Right Module over Ring Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct{G, +_G, \\circ}$ be a left module over $\\struct {R, +_R, \\times_R}$. Let $\\circ' : G \\times R \\to G$ be the binary operation defined by: :$\\forall \\lambda \\in R: \\forall x \\in G: x \\circ' \\lambda = \\lambda \\circ x$ Then $\\struct{G, +_G, \\circ'}$ is not necessarily a right module over $\\struct {R, +_R, \\times_R}$"}
{"_id": "5851", "text": "Pointwise Multiplication on Integer-Valued Functions is Associative Let $f, g, h: S \\to \\Z$ be integer-valued functions. Let $f \\times g: S \\to \\Z$ denote the pointwise product of $f$ and $g$. Then: :$\\paren {f \\times g} \\times h = f \\times \\paren {g \\times h}$"}
{"_id": "13931", "text": "Non-Palindromes in Base 2 by Reverse-and-Add Process Let the number $22$ be expressed in binary: $10110_2$. When the reverse-and-add process is performed on it repeatedly, it never becomes a palindromic number."}
{"_id": "16533", "text": "Elements of Abelian Group whose Order Divides n is Subgroup Let $G$ be an abelian group whose identity element is $e$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer . Let $G_n$ be the subset of $G$ defined as: :$G_n = \\set {x \\in G: \\order x \\divides n}$ where: :$\\order x$ denotes the order of $x$ :$\\divides$ denotes divisibility. Then $G_n$ is a subgroup of $G$."}
{"_id": "14531", "text": "Implicit Function Theorem for Lipschitz Contraction at Point Let $M$ and $N$ be metric spaces. Let $M$ be complete. Let $f : M \\times N \\to M$ be a uniform contraction. Then for all $t\\in N$ there exists a unique $g(t) \\in M$ such that $f(g(t), t) = g(t)$, and if $f$ is Lipschitz continuous at a point $(g(t),t)$, then $g$ is Lipschitz continuous at $t$."}
{"_id": "2038", "text": "Factorial Divisible by Binary Root Let $n \\in \\Z: n \\ge 1$. Let $n$ be expressed in binary notation: :$n = 2^{e_1} + 2^{e_2} + \\cdots + 2^{e_r}$ where $e_1 > e_2 > \\cdots > e_r \\ge 0$. Let $n!$ be the factorial of $n$. Then $n!$ is divisible by $2^{n-r}$, but not by $2^{n-r+1}$."}
{"_id": "12508", "text": "Correctness of Definition of Increasing Mappings Satisfying Inclusion in Lower Closure Let $R = \\left({S, \\preceq}\\right)$ be an ordered set. Let ${\\it Ids}\\left({R}\\right)$ be the set of all ideals in $R$. Let $L = \\left({ {\\it Ids}\\left({R}\\right), \\precsim}\\right)$ be an ordered set where $\\precsim \\mathop = \\subseteq\\restriction_{ {\\it Ids}\\left({R}\\right) \\times {\\it Ids}\\left({R}\\right)}$ Let  :$M = \\left({F, \\preccurlyeq}\\right)$ where :$F = \\left\\{ {f: S \\to {\\it Ids}\\left({R}\\right): f}\\right.$ is increasing mapping $\\left.{\\land \\forall x \\in S: f\\left({x}\\right) \\subseteq x^\\preceq}\\right\\}$ and :$\\preccurlyeq$ is ordering on mappings generated by $\\precsim$ where $x^\\preceq$ denotes the lower closure of $x$. Then :$M$ is an ordered set."}
{"_id": "3242", "text": "Vinogradov's Theorem/Minor Arcs For any $B > 0$: :$\\displaystyle \\int_\\mathcal M \\map F \\alpha^3 \\map e {-\\alpha N} \\rd \\alpha \\ll \\frac {N^2} {\\paren {\\ln N}^{B/2 - 5} }$"}
{"_id": "19712", "text": "Cartesian Plane Rotated with respect to Another Let $\\mathbf r$ be a position vector embedded in a Cartesian plane $\\CC$ with origin $O$. Let $\\CC$ be rotated anticlockwise through an angle $\\varphi$ about the axis of rotation $O$. Let $\\CC'$ denote the Cartesian plane in its new position. Let $\\mathbf r$ be kept fixed during this rotation. Let $\\tuple {x, y}$ denote the components of $\\mathbf r$ with respect to $\\CC$. Let $\\tuple {x', y'}$ denote the components of $\\mathbf r$ with respect to $\\CC'$. Then: {{begin-eqn}} {{eqn | l = x'       | r = x \\cos \\varphi + y \\sin \\varphi }} {{eqn | l = y'       | r = -x \\sin \\varphi + y \\cos \\varphi }} {{end-eqn}}"}
{"_id": "17796", "text": "Primitive of Error Function :$\\displaystyle \\int \\map \\erf x \\rd x = x \\, \\map \\erf x + \\frac 1 {\\sqrt \\pi} e^{-x^2} + C$"}
{"_id": "2818", "text": "Pseudocompact Normal Space is Countably Compact Let $T = \\struct {S, \\tau}$ be a normal space. Then $T$ is pseudocompact {{iff}} $T$ is countably compact."}
{"_id": "10037", "text": "Primitive of x squared over Root of x squared minus a squared/Logarithm Form :$\\displaystyle \\int \\frac {x^2 \\rd x} {\\sqrt {x^2 - a^2} } = \\frac {x \\sqrt {x^2 - a^2} } 2 + \\frac {a^2} 2 \\ln \\size {x + \\sqrt {x^2 - a^2} } + C$"}
{"_id": "9358", "text": "Primitive of Root of a x + b over Power of p x + q :$\\displaystyle \\int \\frac {\\sqrt{a x + b} } {\\left({p x + q}\\right)^n} \\ \\mathrm d x = \\frac {- \\sqrt{a x + b} } {\\left({n - 1}\\right) p \\left({p x + q}\\right)^{n-1} } + \\frac a {2 \\left({n - 1}\\right) p} \\int \\frac {\\mathrm d x} {\\left({p x + q}\\right)^{n-1} \\sqrt{a x + b} }$"}
{"_id": "9703", "text": "Primitive of x over Cosine of a x {{begin-eqn}} {{eqn | l = \\int \\frac {x \\rd x} {\\cos a x}       | r = \\frac 1 {a^2} \\sum_{n \\mathop = 0}^\\infty \\frac {E_n \\paren {a x}^{2 n + 2} } {\\paren {2 n + 2} \\paren {2 n}!} + C       | c =  }} {{eqn | r = \\dfrac 1 {a^2} \\paren {\\frac {\\paren {a x}^2} 2 + \\frac {\\paren {a x}^4} 8 + \\frac {5 \\paren {a x}^6} {144} + \\cdots} + C       | c =  }} {{end-eqn}}"}
{"_id": "15904", "text": "Summation to n of kth Harmonic Number over k+1 :$\\displaystyle \\sum_{k \\mathop = 1}^n \\dfrac {H_k} {k + 1} = \\dfrac { {H_{n + 1} }^2 - H_{n + 1}^{\\left({2}\\right)} } 2$ where: :$H_n$ denotes the $n$th harmonic number :$H_n^{\\left({2}\\right)}$ denotes a general harmonic number."}
{"_id": "17292", "text": "Principle of Finite Induction/Zero-Based Let $S \\subseteq \\N$ be a subset of the natural numbers. Suppose that: :$(1): \\quad 0 \\in S$ :$(2): \\quad \\forall n \\in \\N : n \\in S \\implies n + 1 \\in S$ Then: :$S = \\N$"}
{"_id": "16458", "text": "Symmetry Group of Rectangle is Klein Four-Group The symmetry group of the rectangle is the Klein $4$-group."}
{"_id": "18303", "text": "Root of Equation e^x (x - 1) = e^-x (x + 1) The equation: :$e^x \\paren {x - 1} = e^{-x} \\paren {x + 1}$ has a root: :$x = 1 \\cdotp 19966 \\, 78640 \\, 25773 \\, 4 \\ldots$"}
{"_id": "16508", "text": "Bijection between Power Set of nth Initial Section and Initial Section of nth Power of 2 Let $\\N_n$ be used to denote the first $n$ non-zero natural numbers: :$\\N_n = \\set {1, 2, \\ldots, n}$ Then there exists a bijection between the power set of $\\N_n$ and $\\N_{2^n}$."}
{"_id": "7555", "text": "Fundamental Theorem of Contour Integration Let $D \\subseteq \\C$ be an open set. Let $f: D \\to \\C$ be a continuous function. Suppose that $F: D \\to \\C$ is an antiderivative of $f$. Let $\\gamma: \\closedint a b \\to D$ be a contour in $D$. Then the contour integral: :$\\displaystyle \\int_\\gamma \\map f z \\rd z =  \\map F {\\map \\gamma b} - \\map F {\\map \\gamma a}$"}
{"_id": "2426", "text": "Stabilizer in Group of Transformations Let $X$ be any set with $n$ elements (where $n \\in \\Z_{>0}$). Consider the symmetric group on $n$ letters $S_n$ as a group of transformations on $X$. Let $x \\in X$. Then the stabilizer of $x$ is isomorphic to $S_{n - 1}$."}
{"_id": "17921", "text": "Definite Integral to Infinity of Sine of a x over Hyperbolic Sine of b x :$\\displaystyle \\int_0^\\infty \\frac {\\sin a x} {\\sinh b x} \\rd x = \\frac \\pi {2 b} \\tanh \\frac {a \\pi} {2 b}$"}
{"_id": "17592", "text": "Geometric Mean of two Positive Real Numbers is Between them Let $a, b \\in \\R$ be real numbers such that $0 < a < b$. Let $\\map G {a, b}$ denote the geometric mean of $a$ and $b$. Then: :$a < \\map G {a, b} < b$"}
{"_id": "10840", "text": "Closed Subsets of Symmetry Group of Square Recall the symmetry group of the square:"}
{"_id": "1564", "text": "Primitive Root is Generator of Reduced Residue System Let $a$ be a primitive root of $n$. Then: :$\\left\\{{a, a^2, a^3, \\ldots, a^{\\phi \\left({n}\\right)}}\\right\\}$ where $\\phi \\left({n}\\right)$ is the Euler phi function of $n$, is a reduced residue system of $n$. Thus the first $\\phi \\left({n}\\right)$ powers of $a$ \"generates\" $R$. We say that $a$ is a '''generator''' of $R$."}
{"_id": "18589", "text": "Set is Subset of Power Set of Union Let $x$ be a set of sets. Let $\\displaystyle \\bigcup x$ denote the union of $x$. Let $\\powerset {\\displaystyle \\bigcup x}$ denote the power set of $\\displaystyle \\bigcup x$. Then: :$x \\subseteq \\powerset {\\displaystyle \\bigcup x}$"}
{"_id": "13969", "text": "Absolute Value of Divergent Infinite Product The infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ diverges to $0$ {{iff}}  $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\norm{a_n}$ diverges to $0$."}
{"_id": "7840", "text": "Equivalence of Definitions of Secant of Angle Let $\\theta$ be an angle. {{TFAE|def = Secant of Angle|view = secant}}"}
{"_id": "19495", "text": "Zero of Field is Unique Let $\\struct {F, +, \\times}$ be a field. The zero of $F$ is unique."}
{"_id": "3400", "text": "Second-Countability is not Continuous Invariant Let $T_A = \\struct {A, \\tau_A}$ and $T_B = \\struct {B, \\tau_B}$ be topological spaces. Let $\\phi: T_A \\to T_B$ be a continuous mapping. If $T_A$ is a second-countable space, then it does not necessarily follow that $T_B$ is also second-countable."}
{"_id": "17104", "text": "Absolute Value of Complex Cross Product is Commutative Let $z_1$ and $z_2$ be complex numbers. Let $z_1 \\times z_2$ denote the (complex) cross product of $z_1$ and $z_2$. Then: :$\\size {z_1 \\times z_2} = \\size {z_2 \\times z_1}$ where $\\size {\\, \\cdot \\,}$ denotes the absolute value function."}
{"_id": "4517", "text": "G-Module is Irreducible iff no Non-Trivial Proper Submodules Let $\\left({G, \\circ}\\right)$ be a finite group. Let $\\left({V, \\phi}\\right)$ be a $G$-module. Then $V$ is an irreducible $G$-module {{iff}} $V$ has no non-trivial proper $G$-submodules."}
{"_id": "3264", "text": "Either-Or Topology is Non-Meager Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is non-meager."}
{"_id": "10491", "text": "Relative Complement of Cartesian Product Let $A$ and $B$ be sets. Let $X \\subseteq A$ and $Y \\subseteq B$. Then: :$\\relcomp {A \\mathop \\times B} {X \\times Y} = \\paren {A \\times \\relcomp B Y} \\cup \\paren {\\relcomp A X \\times B}$"}
{"_id": "4457", "text": "Inequality of Hölder Means Let $p, q \\in \\R_{\\ne 0}$ be non-zero real numbers with $p < q$. Let $x_1, x_2, \\ldots, x_n \\ge 0$ be real numbers. If $p < 0$, then we require that $x_1, x_2, \\ldots, x_n > 0$. Then the Hölder mean satisfies the inequality: : $M_p \\left({x_1, x_2, \\ldots, x_n}\\right) \\le M_q \\left({x_1, x_2, \\ldots, x_n}\\right)$ Equality holds if and only if $x_1 = x_2 = \\cdots = x_n$."}
{"_id": "9657", "text": "Primitive of Fourth Power of Sine of a x :$\\displaystyle \\int \\sin^4 a x \\ \\mathrm d x = \\frac {3 x} 8 - \\frac {\\sin 2 a x} {4 a} + \\frac {\\sin 4 a x} {32 a} + C$"}
{"_id": "16810", "text": "Distance between Linear Codewords is Distance Function Let $\\map V {n, p}$ be a master code. Let $d: V \\times V \\to \\Z$ be the mapping defined as: :$\\forall u, v \\in V: \\map d {u, v} =$ the distance between $u$ and $v$ that is, the number of corresponding terms at which $u$ and $v$ are different. Then $d$ defines a distance function in the sense of a metric space."}
{"_id": "5647", "text": "Condition for Cartesian Product Equivalent to Associated Cardinal Number Let $S$ and $T$ be nonempty sets. Let $\\left|{S}\\right|$ denote the cardinal number of $S$. Then: :$S \\times T \\sim \\left|{S \\times T}\\right| \\iff S \\sim \\left|{S}\\right| \\land T \\sim \\left|{T}\\right|$ where $S \\times T$ denotes the cartesian product of $S$ and $T$."}
{"_id": "19151", "text": "Matroid Contains No Loops iff Empty Set is Flat Let $M = \\struct{S, \\mathscr I}$ be a matroid. Then: :$M$ contains no loops {{iff}} the empty set is flat."}
{"_id": "13925", "text": "Integers whose Phi times Tau equal Sigma The positive integers whose Euler $\\phi$ function multiplied by its  $\\tau$ function equals its $\\sigma$ function are: :$1, 3, 14, 42$ {{OEIS|A104905}}"}
{"_id": "2489", "text": "Number of Compositions A $k$-composition of a positive integer $n$ is an ordered $k$-tuple: $c = \\left({c_1, c_2, \\ldots, c_k}\\right)$ such that $c_1 + c_2 + \\cdots + c_k = n$ and $c_i $ are strictly positive integers. The number of $k$-compositions of $n$ is $\\displaystyle \\binom{n-1}{k-1}$ and the total number of compositions of $n$ is $2^{n-1}$ (i.e. for $k = 1, 2, 3, \\ldots, n$)."}
{"_id": "12178", "text": "Area between Radii and Whorls of Archimedean Spiral Let $S$ be the Archimedean spiral defined by the equation: :$r = a \\theta$ Let $\\theta = \\theta_1$ and $\\theta = \\theta_2$ be the two rays from the pole at angles $\\theta_1$ and $\\theta_b$ to the polar axis respectively. Let $R$ be the figure enclosed by: :$\\theta_1$ and $\\theta_2$ :the $n$th turn of $S$ and the $n+1$th turn of $S$ The area $\\mathcal A$ of $R$ is given by: :$\\mathcal A = a^2 \\pi \\left({\\theta_2 - \\theta_1}\\right) \\left({\\theta_2 + \\theta_1 + 2 \\pi \\left({2 n + 1}\\right)}\\right)$ {{Proofread|My algebra may have failed me}}"}
{"_id": "1780", "text": "Cycle does not Contain Subcycles Let $G$ be a cycle graph. Then the only cycle graph that is a subgraph of $G$ is $G$ itself."}
{"_id": "6228", "text": "Central Limit Theorem Let $X_1, X_2, \\ldots$ be a sequence of independent identically distributed random variables with: :Mean $E \\left[{X_i}\\right] = \\mu \\in \\left({-\\infty \\,.\\,.\\, \\infty}\\right)$ :Variance $V \\left({X_i}\\right) = \\sigma^2 > 0$ Let: : $\\displaystyle S_n = \\sum_{i \\mathop = 1}^n X_i$ Then: :$\\displaystyle \\frac {S_n - n \\mu} {\\sqrt {n \\sigma^2} } \\xrightarrow {D} N \\left({0, 1}\\right)$ as $n \\to \\infty$ that is, converges in distribution to a standard normal."}
{"_id": "10121", "text": "Proportion of Power Let $x$ and $y$ be proportional. {{explain|Establish what types of object $x$ and $y$ are. As it stands here, they could be anything.}} Let $n \\in \\Z$.  Then $x^n \\propto y^n$."}
{"_id": "14601", "text": "Lifting The Exponent Lemma for Sums for p=2 Let $x, y \\in \\Z$ be integers with $x + y \\ne 0$. Let $n \\ge 1$ be an odd natural number. Let: :$2 \\divides x + y$ where $\\divides$ denotes divisibility. Then: :$\\map {\\nu_2} {x^n + y^n} = \\map {\\nu_2} {x + y}$ where $\\nu_2$ denotes $2$-adic valuation. </onlyinclude>"}
{"_id": "9824", "text": "Primitive of x by Arccosine of x over a :$\\displaystyle \\int x \\arccos \\frac x a \\ \\mathrm d x = \\left({\\frac {x^2} 2 - \\frac {a^2} 4}\\right) \\arccos \\frac x a - \\frac {x \\sqrt {a^2 - x^2} } 4 + C$"}
{"_id": "18002", "text": "Coproduct on Disjoint Union Let $S_1$ and $S_2$ be sets. Let $S_1 \\sqcup S_2 := \\paren {S_1 \\times \\set 1} \\cup \\paren {S_2 \\times \\set 2}$ be the disjoint union of $S_1$ and $S_2$. Let $i_1: S_1 \\to S_1 \\sqcup S_2$ and $i_2: S_2 \\to S_1 \\sqcup S_2$ be the mappings defined as: :$\\forall s_1 \\in S_1: \\map {i_1} {s_1} = \\tuple {s_1, 1}$ :$\\forall s_2 \\in S_2: \\map {i_2} {s_2} = \\tuple {s_2, 2}$ Then $\\struct {S_1 \\sqcup S_2, i_1, i_2}$ is a coproduct of $S_1$ and $S_2$."}
{"_id": "14176", "text": "Sequence of Palindromic Lucky Numbers The The sequence of lucky numbers which are also palindromic begins: :$1, 3, 7, 9, 33, 99, 111, 141, 151, 171, \\ldots$ {{OEIS|A031161}}"}
{"_id": "14429", "text": "Cotangent of Complex Number/Formulation 3 :$\\cot \\paren {a + b i} = \\dfrac {- \\cot a - \\cot a \\coth^2 b} {\\cot^2 a + \\coth^2 b} + \\dfrac {\\cot^2 a \\coth b - \\coth b} {\\cot^2 a + \\coth^2 b} i$"}
{"_id": "16452", "text": "Order of Element in Group equals its Order in Subgroup Let $G$ be a group. Let $H\\ le G$, where $\\le$ denotes the property of being a subgroup. Let $x \\in H$. Then the order of $x$ in $H$ equals the order of $x$ in $G$."}
{"_id": "8560", "text": "Irrational Number Space is Topological Space Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space formed by the irrational numbers $\\R \\setminus \\Q$ under the usual (Euclidean) topology $\\tau_d$. Then $\\tau_d$ forms a topology."}
{"_id": "4152", "text": "Union of Elements of Power Set Let $S$ be a set. Then: :$\\displaystyle S = \\bigcup_{X \\mathop \\in \\powerset S} X$ where $\\powerset S$ denotes the power set of $S$."}
{"_id": "3843", "text": "Exponential on Real Numbers is Group Isomorphism Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $\\struct {\\R_{> 0}, \\times}$ be the multiplicative group of positive real numbers. Let $\\exp: \\struct {\\R, +} \\to \\struct {\\R_{> 0}, \\times}$ be the mapping: :$x \\mapsto \\map \\exp x$ where $\\exp$ is the exponential function. Then $\\exp$ is a group isomorphism."}
{"_id": "16254", "text": "Real Part of Complex Exponential Function Let $z = x + i y \\in \\C$ be a complex number, where $x, y \\in \\R$. Let $\\exp z$ denote the complex exponential function. Then: :$\\map \\Re {\\exp z} = e^x \\cos y$ where: :$\\Re z$ denotes the real part of a complex number $z$ :$e^x$ denotes the real exponential function of $x$ :$\\cos y$ denotes the real cosine function of $y$."}
{"_id": "19063", "text": "Primitive of Power of x by Cosine of a x/Corollary :$\\displaystyle \\int x^m \\cos a x \\rd x = \\sum_{k \\mathop = 1}^{m + 1} \\paren {m^{\\underline {k - 1} } \\frac {x^{m + 1 - k} } {a^k} \\map {\\sin} {x + \\dfrac {\\pi} 2 \\paren {k - 1} } }$ where $m^{\\underline {k - 1} }$ denotes the $k - 1$th falling factorial of $m$."}
{"_id": "19242", "text": "Trace of Sum of Matrices is Sum of Traces Let $\\mathbf A = \\sqbrk a_n$ and $\\mathbf B = \\sqbrk b_n$ be square matrices of order $n$. let $\\mathbf A + \\mathbf B$ debote the matrix entrywise sum of $\\mathbf A$ and $\\mathbf B$. Then: :$\\map \\tr {\\mathbf A + \\mathbf B} = \\map \\tr {\\mathbf A} + \\map \\tr {\\mathbf B}$ where $\\map \\tr {\\mathbf A}$ denotes the trace of $\\mathbf A$."}
{"_id": "738", "text": "Existence of Unique Subsemigroup Generated by Subset Let $\\struct {S, \\circ}$ be a semigroup. Let $\\O \\subset X \\subseteq S$. Let $\\struct {T, \\circ}$ be the subsemigroup generated by $X$. Then $T = \\gen X$ exists and is unique."}
{"_id": "8735", "text": "Power Reduction Formulas/Hyperbolic Sine Cubed :$\\sinh^3 x = \\dfrac {\\sinh 3 x - 3 \\sinh x} 4$"}
{"_id": "19658", "text": "Probability Measure on Single-Subset Event Space Let $\\EE$ be an experiment whose sample space is $\\Omega$. Let $\\O \\subsetneqq A \\subsetneqq \\Omega$. Let $\\Sigma := \\set {\\O, A, \\Omega \\setminus A, \\Omega}$ be the event space of $\\EE$. Let $\\Pr: \\Sigma \\to \\R$ be a probability measure on $\\struct {\\Omega, \\Sigma}$. Then $\\Pr$ has the form: {{begin-eqn}} {{eqn | n = Pr 1       | l = \\map \\Pr \\O       | r = 0 }} {{eqn | n = Pr 2       | l = \\map \\Pr A       | r = p }} {{eqn | n = Pr 3       | l = \\map \\Pr {\\Omega \\setminus A}       | r = 1 - p }} {{eqn | n = Pr 4       | l = \\map \\Pr \\Omega       | r = 1 }} {{end-eqn}} for some $p \\in \\R$ satisfying $0 \\le p \\le 1$."}
{"_id": "2890", "text": "Countable Product of First-Countable Spaces is First-Countable Let $I$ be an indexing set with countable cardinality. Let $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$ be a family of topological spaces indexed by $I$. Let $\\displaystyle \\struct {S, \\tau} = \\prod_{\\alpha \\mathop \\in I} \\struct {S_\\alpha, \\tau_\\alpha}$ be the product space of $\\family {\\struct {S_\\alpha, \\tau_\\alpha} }_{\\alpha \\mathop \\in I}$. Let each of $\\struct {S_\\alpha, \\tau_\\alpha}$ be first-countable. Then $\\struct {S, \\tau}$ is also first-countable."}
{"_id": "1973", "text": "Lexicographic Order on Products of Well-Ordered Sets Let $S$ be a set which is well-ordered by $\\preceq$. Let $\\preccurlyeq$ be the lexicographic order on the set $T$ of all ordered tuples of $S$. Then: :$(1): \\quad$ For a given $n \\in \\N: n > 0$, $\\preccurlyeq$ is a well-ordering on the set $T_n$ of all ordered $n$-tuples of $S$; :$(2): \\quad \\preccurlyeq$ is '''not''' a well-ordering on the set $T$ of ''all'' ordered tuples of $S$."}
{"_id": "15237", "text": "1 is Limit Point of Sequence in Sierpiński Space Let $T = \\struct {\\set {0, 1}, \\tau_0}$ be a Sierpiński space. The sequence in $T$: :$\\sigma = \\sequence {0, 1, 0, 1, \\ldots}$ has $1$ as a limit point."}
{"_id": "18115", "text": "Intersection of Plane with Sphere is Circle The intersection of a plane with a sphere is a circle."}
{"_id": "504", "text": "Subgroup is Normal iff it contains Product of Inverses A subgroup $H$ of a group $G$ is normal {{iff}}: :$\\forall a, b \\in G: a b \\in H \\implies a^{-1} b^{-1} \\in H$"}
{"_id": "17256", "text": "5th Cyclotomic Ring is not a Unique Factorization Domain Let $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ denote the $5$th cyclotomic ring. Then $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ is not a unique factorization domain. The following elements of $\\struct {\\Z \\sqbrk {i \\sqrt 5}, +, \\times}$ are irreducible: :$2$ :$3$ :$1 + i \\sqrt 5$ :$1 - i \\sqrt 5$"}
{"_id": "17869", "text": "Definite Integral to Infinity of x over Exponential of x minus One :$\\displaystyle \\int_0^\\infty \\frac x {e^x - 1} \\rd x = \\frac {\\pi^2} 6$"}
{"_id": "7275", "text": "Transitive Closure Always Exists (Relation Theory) Let $\\RR$ be a relation on a set $S$. Then the transitive closure $\\RR^+$ of $\\RR$ always exists."}
{"_id": "7329", "text": "Content of Monic Polynomial If $f$ is monic, then $\\cont f = \\dfrac 1 n$ for some integer $n$."}
{"_id": "15252", "text": "Real Number Line is First-Countable Let $\\struct {\\R, \\tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\\struct {\\R, \\tau_d}$ is first-countable."}
{"_id": "15730", "text": "Sum of Geometric Sequence/Examples/Index to Minus 2 Let $x$ be an element of one of the standard number fields: $\\Q, \\R, \\C$ such that $x \\ne 1$. Then the formula for Sum of Geometric Sequence: :$\\displaystyle \\sum_{j \\mathop = 0}^n x^j = \\frac {x^{n + 1} - 1} {x - 1}$ breaks down when $n = -2$: :$\\displaystyle \\sum_{j \\mathop = 0}^{-2} x^j \\ne \\frac {x^{-1} - 1} {x - 1}$"}
{"_id": "15127", "text": "Paracompact Countably Compact Space is Compact Let $T = \\struct {S, \\tau}$ be a countably compact space which is also paracompact. Then $T$ is compact."}
{"_id": "9720", "text": "Primitive of Reciprocal of Square of 1 minus Cosine of a x :$\\displaystyle \\int \\frac {\\mathrm d x} {\\left({1 - \\cos a x}\\right)^2} = \\frac {-1} {2a} \\cot \\frac {a x} 2 - \\frac 1 {6 a} \\cot^3 \\frac {a x} 2 + C$"}
{"_id": "4412", "text": "Product is Left Identity Therefore Left Cancellable Let $\\struct {S, \\circ}$ be a semigroup. Let $e_L \\in S$  be a left identity of $S$. Let $a \\in S$ such that: :$\\exists b \\in S: b \\circ a = e_L$ Then $a$ is left cancellable in $\\struct {S, \\circ}$."}
{"_id": "3129", "text": "Transitivity of Algebraic Extensions Let $E / F / K$ be a tower of field extensions. Let $E$ be algebraic over $F$. Let $F$ be algebraic over $K$. Then $E$ is algebraic over $K$."}
{"_id": "9621", "text": "Primitive of Reciprocal of x by x fourth minus a fourth :$\\displaystyle \\int \\frac {\\mathrm d x} {x \\left({x^4 - a^4}\\right)} = \\frac 1 {4 a^4} {\\ln \\left\\vert{\\frac {x^4 - a^4} {x^4} }\\right\\vert} + C$"}
{"_id": "16812", "text": "Error Detection Capability of Linear Code Let $C$ be a linear code. Let $C$ have a minimum distance $d$. Then $C$ detects $d - 1$ or fewer transmission errors."}
{"_id": "11786", "text": "Linear Second Order ODE/y'' + 3 y' - 10 y = 6 exp 4 x The second order ODE: :$(1): \\quad y'' + 3 y' - 10 y = 6 e^{4 x}$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-5 x} + \\dfrac {e^{4 x} } 3$"}
{"_id": "3383", "text": "Arens-Fort Topology is Topology Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $\\tau$ is a topology on $T$."}
{"_id": "8257", "text": "Upper Bounds for Prime Numbers/Result 3 : $\\forall n \\in \\N_{>1}: p \\left({n}\\right) < 2^n$"}
{"_id": "13254", "text": "Sums of Sequences of Consecutive Squares which are Square The sums of the following sequences of successive squares are themselves square: {{begin-eqn}} {{eqn | l = \\sum_{i \\mathop = 7}^{29} k^2       | r = 7^2 + 8^2 + \\cdots + 29^2       | c =  }} {{eqn | l = \\sum_{i \\mathop = 7}^{39} k^2       | r = 7^2 + 8^2 + \\cdots + 39^2       | c =  }} {{eqn | l = \\sum_{i \\mathop = 7}^{56} k^2       | r = 7^2 + 8^2 + \\cdots + 56^2       | c =  }} {{eqn | l = \\sum_{i \\mathop = 7}^{190} k^2       | r = 7^2 + 8^2 + \\cdots + 190^2       | c =  }} {{end-eqn}}"}
{"_id": "3134", "text": "Cook-Levin Theorem The Boolean Satisfiability Problem is NP-Complete."}
{"_id": "16258", "text": "Imaginary Part of Sine of Complex Number Let $z = x + i y \\in \\C$ be a complex number, where $x, y \\in \\R$. Let $\\sin z$ denote the complex sine function. Then: :$\\Im \\paren {\\sin z} = \\cos x \\sinh y$ where: :$\\Im z$ denotes the imaginary part of a complex number $z$ :$\\sin$ denotes the complex sine function :$\\cos$ denotes the real cosine function :$\\sinh$ denotes the hyperbolic sine function."}
{"_id": "9778", "text": "Primitive of Tangent of a x/Secant Form :$\\ds \\int \\tan a x \\rd x = \\frac {\\ln \\size {\\sec a x} } a + C$"}
{"_id": "5109", "text": "Cartesian Product of Preimage with Image of Relation is Correspondence Let $\\mathcal R \\subseteq S \\times T$ be a relation. Then the restriction of $\\mathcal R$ to $\\Preimg {\\mathcal R} \\times \\Img {\\mathcal R}$ is a correspondence."}
{"_id": "10046", "text": "Primitive of x by Inverse Hyperbolic Cotangent of x over a :$\\displaystyle \\int x \\coth^{-1} \\frac x a \\rd x = \\frac {a x} 2 + \\frac {x^2 - a^2} 2 \\coth^{-1} \\frac x a + C$"}
{"_id": "6880", "text": "Alaoglu's Theorem The closed unit ball of the dual of a normed space is compact with respect to the weak* topology. {{rewrite|Rephrase to match usual presentation}}"}
{"_id": "6464", "text": "Rule of Exportation/Formulation 1/Proof 2 :$\\paren {p \\land q} \\implies r \\dashv \\vdash p \\implies \\paren {q \\implies r}$"}
{"_id": "19360", "text": "Definite Integral from 0 to 2 Pi of Logarithm of a plus b Cosine x :$\\displaystyle \\int_0^{2 \\pi} \\map \\ln {a + b \\cos x} \\rd x = 2 \\pi \\map \\ln {\\frac {a + \\sqrt {a^2 - b^2} } 2}$"}
{"_id": "7062", "text": "Implication is Left Distributive over Disjunction/Formulation 1/Forward Implication :$p \\implies \\paren {q \\lor r} \\vdash \\paren {p \\implies q} \\lor  \\paren {p \\implies r}$"}
{"_id": "13092", "text": "Newton's Formula for Pi $\\pi$ (pi) can be approximated using the formula: :$\\pi = \\dfrac {3 \\sqrt 3} 4 + 24 \\paren {\\dfrac 2 {3 \\times 2^3} - \\dfrac 1 {5 \\times 2^5} - \\dfrac 1 {28 \\times 2^7} - \\dfrac 1 {72 \\times 2^9} - \\dfrac 5 {704 \\times 2^{11} } - \\dfrac 7 {1664 \\times 2^{13} } - \\cdots}$"}
{"_id": "19144", "text": "Distinct Elements are Parallel iff Pair forms Circuit Let $M = \\struct {S, \\mathscr I}$ be a matroid. Let $x, y \\in S : x \\ne y$. Then: :$x$ and $y$ are parallel {{iff}} $\\set {x, y}$ is a circuit"}
{"_id": "7811", "text": "Rational Numbers under Multiplication form Commutative Monoid The set of rational numbers under multiplication $\\struct {\\Q, \\times}$ forms a countably infinite commutative monoid."}
{"_id": "17196", "text": "Field of Quotients of Ring of Polynomial Forms on Reals that yields Complex Numbers Let $\\struct {\\R, +, \\times}$ denote the field of real numbers. Let $X$ be transcendental over $\\R$. Let $\\R \\sqbrk X$ be the ring of polynomials in $X$ over $F$. Consider the field of quotients: :$\\R \\sqbrk X / \\ideal p$ where: :$p = X^2 + 1$ :$\\ideal p$ denotes the ideal generated by $p$. Then $\\R \\sqbrk X / \\ideal p$ is the field of complex numbers."}
{"_id": "13730", "text": "Boundary of Compact Set in Hausdorff Space is Compact Let $X$ be a Hausdorff topological space. Let $K\\subset X$ be a compact subspace of $X$. Then its boundary $\\partial K$ is compact."}
{"_id": "2338", "text": "Area under Arc of Cycloid Let $C$ be a cycloid generated by the equations: :$x = a \\paren {\\theta - \\sin \\theta}$ :$y = a \\paren {1 - \\cos \\theta}$ Then the area under one arc of the cycloid is $3 \\pi a^2$. That is, the area under one arc of the cycloid is three times the area of the generating circle."}
{"_id": "15079", "text": "Compact Subset of Compact Space is not necessarily Closed A compact subset of a compact space not necessarily closed."}
{"_id": "649", "text": "Divisor of Sum of Coprime Integers Let $a, b, c \\in \\Z_{>0}$ such that: :$a \\perp b$ and $c \\divides \\paren {a + b}$. where: :$a \\perp b$ denotes $a$ and $b$ are coprime :$c \\divides \\paren {a + b}$ denotes that $c$ is a divisor of $a + b$. Then $a \\perp c$ and $b \\perp c$. That is, a divisor of the sum of two coprime integers is coprime to both."}
{"_id": "16477", "text": "Subgroup of Subgroup with Prime Index/Corollary Let $\\struct {G, \\circ}$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $K \\subsetneq H$. Let: :$\\index G K = p$ where: :$p$ denotes a prime number :$\\index G K$ denotes the index of $K$ in $G$. Then: :$H = G$"}
{"_id": "15733", "text": "Sum of Sequence of n by 2 to the Power of n :$\\displaystyle \\sum_{j \\mathop = 0}^n j \\, 2^j = n 2^{n + 2} - \\paren {n + 1} 2^{n + 1} + 2$"}
{"_id": "14334", "text": "Difference between Kaprekar Number and Square is Multiple of Repunit Let $n$ be a Kaprekar number of $m$ digits. Then: : $n^2 - n = k R_m$ where: :$R_m$ is the $m$-digit repunit :$k$ is an integer."}
{"_id": "9476", "text": "Primitive of Root of x squared minus a squared cubed over x :$\\displaystyle \\int \\frac {\\paren {\\sqrt {x^2 - a^2} }^3} x \\rd x = \\frac {\\paren {\\sqrt {x^2 - a^2} }^3} 3 - a^2 \\sqrt {x^2 - a^2} + a^3 \\arcsec \\size {\\frac x a} + C$"}
{"_id": "16182", "text": "Combination Theorem for Cauchy Sequences/Quotient Rule Suppose $\\sequence {y_n}$ does not converge to $0$. Then: :$\\exists K \\in \\N: \\forall n > K : y_n \\ne 0$ and the sequences: :$\\sequence { {x_{K + n} } \\paren {y_{K + n} }^{-1} }_{n \\mathop \\in \\N}$ and $\\sequence {\\paren {y_{K + n} }^{-1} {x_{K + n} } }_{n \\mathop \\in \\N}$ are well-defined and Cauchy sequences."}
{"_id": "4374", "text": "Inverse not always Unique for Non-Associative Operation Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $\\circ$ be a non-associative operation. Then for any $x \\in S$, it is possible for $x$ to have more than one inverse element."}
{"_id": "8845", "text": "Biconditional with Contradiction :$p \\iff \\bot \\dashv \\vdash \\neg p$"}
{"_id": "16510", "text": "Symmetry Group of Line Segment is Group The symmetry group of the line segment is a group."}
{"_id": "13942", "text": "Set of Upper Closures of Compact Elements is Basis implies Complete Scott Topological Lattice is Algebraic Let $L = \\left({S, \\preceq, \\tau}\\right)$ be a complete Scott Definition:Topological Lattice. Let $\\mathcal B = \\left\\{ {x^\\succeq: x \\in K\\left({L}\\right)}\\right\\}$ be a basis of $L$ where :$x^\\succeq$ denotes the upper closure of $x$, :$K\\left({L}\\right)$ denotes the compact subset of $L$. Then $L$ is algebraic."}
{"_id": "2148", "text": "Crossbar Theorem Let $\\triangle ABC$ be a triangle. Let $D$ be a point in the interior of $\\triangle ABC$. Then there exists a point $E$ such that $E$ lies on both $AD$ and $BC$."}
{"_id": "19457", "text": "Derivative of Tangent of Function :$\\map {\\dfrac \\d {\\d x} } {\\tan u} = \\sec^2 u \\dfrac {\\d u} {\\d x}$"}
{"_id": "4416", "text": "Non-Zero Rational Numbers Closed under Multiplication The set of non-zero rational numbers is closed under multiplication."}
{"_id": "9603", "text": "Primitive of Reciprocal of x squared by x cubed plus a cubed :$\\displaystyle \\int \\frac {\\d x} {x^2 \\paren {x^3 + a^3} } = \\frac {-1} {a^3 x} - \\frac 1 {6 a^4} \\map \\ln {\\frac {x^2 - a x + a^2} {\\paren {x + a}^2} } - \\frac 1 {a^4 \\sqrt 3} \\arctan \\frac {2 x - a} {a \\sqrt 3}$"}
{"_id": "6819", "text": "False Statement implies Every Statement/Formulation 1/Proof 1 :$\\neg p \\vdash p \\implies q$"}
{"_id": "15452", "text": "Nilpotent Ring Element plus Unity is Unit Let $A$ be a ring with unity. Let $1 \\in A$ be its unity. Let $a \\in A$ be nilpotent. Then $1 + a$ is a unit of $A$."}
{"_id": "16231", "text": "Power Series Expansion for Square of Reciprocal of 1-z {{begin-eqn}} {{eqn | l = \\dfrac 1 {\\paren {1 - z}^2}       | r = \\sum_{n \\mathop = 0}^\\infty \\paren {n + 1} z^n }} {{eqn | r = 1 + 2 z + 3 z^2 + 4 z^3 + \\cdots }} {{end-eqn}}"}
{"_id": "10132", "text": "Condition for Denesting of Square Root Let $a, b \\in \\Q_{\\ge 0}$. Suppose $\\sqrt b \\notin \\Q$. Then: :$\\exists p, q \\in \\Q: \\sqrt {a + \\sqrt b} = \\sqrt p + \\sqrt q$ {{iff}}: :$\\exists n \\in \\Q: a^2 - b = n^2$."}
{"_id": "3100", "text": "Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space Let $T = \\struct {S, \\tau_p}$ be an infinite particular point space. Let $U \\in \\tau_p$ be a countably infinite open set of $T$. Let the elements of $U$ be arranged into a sequence $\\sequence {a_i}$ of distinct terms in $T$. Then while every element $x$ of $U$ such that $x \\ne p$ is a limit point of $U$, there exists no $x \\in U$ such that $x$ is an accumulation point of $\\sequence {a_i}$."}
{"_id": "1815", "text": "Metric Induced by a Pseudometric Let $X$ be a set on which there is a pseudometric $d: X \\times X \\to \\R$. For any $x, y \\in X$, let $x \\sim y$ denote that $d \\left({x, y}\\right) = 0$. Let $\\left[\\!\\left[{x}\\right]\\!\\right]$ denote the equivalence class of $x$ under $\\sim$. Let $X^*$ be the quotient of $X$ by $\\sim$. Then the mapping $d^*: X^* \\times X^* \\to \\R$ defined by: :$d^* \\left({\\left[\\!\\left[{x}\\right]\\!\\right], \\left[\\!\\left[{y}\\right]\\!\\right]}\\right) = d \\left({x, y}\\right)$ is a metric. Hence $\\left({X^*, d^*}\\right)$ is a metric space."}
{"_id": "15953", "text": "Fibonacci Numbers which equal the Square of their Index The only Fibonacci numbers which equal the square of their index are: {{begin-eqn}} {{eqn | l = F_0       | r = 0 }} {{eqn | l = F_1       | r = 1 }} {{eqn | l = F_{12}       | r = 12^2 = 144 }} {{end-eqn}}"}
{"_id": "11654", "text": "Current in Electric Circuit/L, R in Series/Sinusoidal EMF Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let an EMF $E$ be imposed upon $K$ at time $t = 0$ defined by the equation: :$E = E_0 \\sin \\omega t$ The electric current $I$ in $K$ is given by the equation: :$I = \\dfrac {E_0} {\\sqrt {R^2 - L^2 \\omega^2} } \\sin \\left({\\omega t - \\alpha}\\right) + \\left({I_0 - \\dfrac {E_0 L \\omega} {R^2 + L^2 \\omega^2} }\\right) e^{-R t / L}$ where $\\tan \\alpha = \\dfrac {L \\omega} R$."}
{"_id": "19701", "text": "Like Unit Vectors are Equal Let $\\mathbf a$ and $\\mathbf b$ be like vector quantities. Then: :$\\mathbf {\\hat a} = \\mathbf {\\hat b}$ where $\\mathbf {\\hat a}$ and $\\mathbf {\\hat b}$ denote the unit vectors in the direction of $\\mathbf a$ and $\\mathbf b$."}
{"_id": "4506", "text": "Totally Bounded Metric Space is Second-Countable Let $M = \\struct {A, d}$ be a metric space which is totally bounded. Then $M$ is second-countable."}
{"_id": "13222", "text": "Obtuse Triangle Divided into Acute Triangles Let $T$ be an obtuse triangle. Let $T$ be dissected into $n$ acute triangles. Then $n \\ge 7$."}
{"_id": "7036", "text": "Conjunction with Negative Equivalent to Negation of Implication/Formulation 2/Reverse Implication :$\\vdash \\left({\\neg \\left({p \\implies q}\\right)}\\right) \\implies \\left({p \\land \\neg q}\\right)$"}
{"_id": "3943", "text": "Cantor Space is Non-Meager in Itself Let $T = \\struct {\\CC, \\tau_d}$ be the Cantor space. Then $T$ is non-meager in itself."}
{"_id": "15892", "text": "Sum of Harmonic Numbers approaches Harmonic Number of Product of Indices Let $m, n \\in \\Z_{>0}$ be (strictly) positive integers. Let: : $T \\left({m, n}\\right) := H_m + H_n - H_{m n}$ where $H_n$ denotes the $n$th harmonic number. Then as $m$ or $n$ increases, $T \\left({m, n}\\right)$ never increases, and reaches its minimum when $m$ and $n$ approach infinity."}
{"_id": "2351", "text": "Quadrature of Parabola Let $T$ be a parabola. Consider the parabolic segment bounded by an arbitrary chord $AB$. Let $C$ be the point on $T$ where the tangent to $T$ is parallel to $AB$. Let Then the area $S$ of the parabolic segment $ABC$ of $T$ is given by: :$S = \\dfrac 4 3 \\triangle ABC$"}
{"_id": "5442", "text": "Class Equality is Symmetric Let $A$ and $B$ be classes. Let $=$ denote class equality. Then: :$A = B \\implies B = A$"}
{"_id": "16826", "text": "Euler's Equation/Independent of y' Let $y$ be a mapping. Let $J$ a functional be such that :$\\ds J \\sqbrk y = \\int_a^b \\map F {x,y} \\rd x$ Then the corresponding Euler's Equation can be reduced to: :$F_y = 0$  Furthermore, this is an algebraic equation."}
{"_id": "13968", "text": "Absolute Value of Convergent Infinite Product Let the infinite product $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ converge to $a \\in \\mathbb K$. Then $\\displaystyle \\prod_{n \\mathop = 1}^\\infty \\norm {a_n}$  converges to $\\norm{a}$."}
{"_id": "2758", "text": "First-Countability is Hereditary Let $T = \\struct {S, \\tau}$ be a topological space which is first-countable. Let $T_H = \\struct {H, \\tau_H}$, where $\\O \\subset H \\subseteq S$, be a subspace of $T$. Then $T_H$ is first-countable."}
{"_id": "5945", "text": "Boundedness of Metric Space by Open Ball Let $M = \\struct {X, d}$ be a metric space. Let $M' = \\struct {Y, d_Y}$ be a subspace of $M$. Then $M'$ is bounded in $M$ {{iff}}: :$\\exists x \\in M, \\epsilon \\in \\R_{>0}: Y \\subseteq \\map {B_\\epsilon} x$ where $\\map {B_\\epsilon} x$ is the open $\\epsilon$-ball of $x$. Simply put: a subspace is bounded {{iff}} it can be fitted inside an open ball."}
{"_id": "3360", "text": "Uncountable Fort Space is not First-Countable Let $T = \\struct {S, \\tau_p}$ be a Fort space on an uncountable set $S$. Then $T$ is not a first-countable space."}
{"_id": "18794", "text": "Definite Integral of Fourier Series at Ends of Interval Let $f: \\R \\to \\R$ be a real function defined in the open interval $\\openint {-\\pi} \\pi$. Let $f$ fulfil the Dirichlet conditions in $\\openint {-\\pi} \\pi$. Let $a_0, a_1, \\dotsc; b_1, \\dotsc$ be the Fourier coefficients of $f$ in $\\openint {-\\pi} \\pi$. Consider the real function: :$\\map F x = \\displaystyle \\int_{-\\pi}^x \\map f t \\rd t - \\dfrac {a_0} 2 x$ Then: :$\\map F \\pi = \\map F {-\\pi} = \\dfrac {a_0 \\pi} 2$"}
{"_id": "14088", "text": "Pluperfect Digital Invariant has less than 61 Digits Let $n \\in \\Z_{>0}$ be a pluperfect digital invariant. Then $n$ has less than $61$ digits."}
{"_id": "15392", "text": "Rouché's Theorem Let $f$ and $g$ be complex-valued functions which are holomorphic in the interior of some simply connected region $D$. Let $\\left\\vert{g \\left({z}\\right)}\\right\\vert < \\left\\vert{f \\left({z}\\right)}\\right\\vert$ on the boundary of $D$. Then $f$ and $f + g$ have the same number of zeroes in the interior of $D$ counted up to multiplicity."}
{"_id": "10731", "text": "Superset of Neighborhood in Topological Space is Neighborhood Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $x \\in S$. Let $N$ be a neighborhood of $x$ in $T$. Let $N \\subseteq N' \\subseteq S$. Then $N'$ is a neighborhood of $x$ in $T$."}
{"_id": "6887", "text": "Squeeze Theorem for Filter Bases Let $\\struct {S, \\le, \\tau}$ be a linearly ordered space. Let $F_1$, $F_2$, and $F_3$ be filter bases in $S$. Let: :$\\forall T \\in F_1: \\exists M \\in F_2: \\forall x \\in M: \\exists y \\in T: y \\le x$ That is: :for each $T \\in F_1$, $F_2$ has an element $M$ such that all elements of $M$ succeed some element of $T$. Similarly, let: :$\\forall U \\in F_3: \\exists N \\in F_2: \\forall x \\in N: \\exists y \\in U: x \\le y$ That is: :for each $U \\in F_3$, $F_2$ has an element $N$ such that all elements of $N$ precede some element of $U$. Let $F_1$ and $F_3$ each converge to a point $p \\in S$. Then $F_2$ converges to $p$."}
{"_id": "10832", "text": "Rational Numbers under Multiplication do not form Group The algebraic structure $\\struct {\\Q, \\times}$ consisting of the set of rational numbers $\\Q$ under multiplication $\\times$ is not a group."}
{"_id": "14848", "text": "Consecutive Primes of form 4n+1 The sequence of $16$ consecutive prime numbers beginning from $207 \\, 622 \\, 273$ are all of the form $4 n + 1$."}
{"_id": "8615", "text": "Local Connectedness is not Preserved under Continuous Mapping Let $\\struct {A, \\tau_1}$ and $\\struct {B, \\tau_2}$ be topological spaces. Let $f: A \\to B$ be a continuous mapping. Let $\\struct {\\Img f, \\tau_3}$ be the image of $f$ with the subspace topology of $B$. Let $\\struct {A, \\tau_1}$ be locally connected. Then it is not necessarily the case that $\\struct {\\Img f, \\tau_3}$ is also locally connected."}
{"_id": "16669", "text": "Condition for Element of Quotient Group of Additive Group of Reals by Integers to be of Finite Order Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\R / \\Z$ denote the quotient group of $\\struct {\\R, +}$ by $\\struct {\\Z, +}$. Let $x + \\Z$ denote the coset of $\\Z$ by $x \\in \\R$. Then $x + \\Z$ is of finite order {{iff}} $x$ is rational."}
{"_id": "7550", "text": "Probability Generating Function of One :$\\Pi_X \\left({1}\\right) = 1$"}
{"_id": "9701", "text": "Primitive of Reciprocal of Cosine of a x/Logarithm of Secant plus Tangent Form :$\\ds \\int \\frac {\\d x} {\\cos a x} = \\frac 1 a \\ln \\size {\\sec a x + \\tan a z}$"}
{"_id": "19656", "text": "Discrete Uniform Distribution gives rise to Probability Measure Let $\\EE$ be an experiment. Let the probability space $\\struct {\\Omega, \\Sigma, \\Pr}$ be defined as: :$\\Omega = \\set {\\omega_1, \\omega_2, \\ldots, \\omega_n}$ :$\\Sigma = \\powerset \\Omega$ :$\\forall A \\in \\Sigma: \\map \\Pr A = \\dfrac 1 n \\card A$ where: :$\\powerset \\Omega$ denotes the power set of $\\Omega$ :$\\card A$ denotes the cardinality of $A$. Then $\\Pr$ is a probability measure on $\\struct {\\Omega, \\Sigma}$."}
{"_id": "3643", "text": "Norm of Unit of Normed Division Algebra Let $\\struct {A_F, \\oplus}$ be a normed division algebra. Let the unit of $\\struct {A_F, \\oplus}$ be $1_A$. Then: :$\\norm {1_A} = 1$ where $\\norm {1_A}$ denotes the norm of $1_A$."}
{"_id": "15264", "text": "Alexandroff Extension of Rational Number Space is Sequentially Compact Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Let $p$ be a new element not in $\\Q$. Let $\\Q^* := \\Q \\cup \\set p$. Let $T^* = \\struct {\\Q^*, \\tau^*}$ be the Alexandroff extension on $\\struct {\\Q, \\tau_d}$. Then $T^*$ is a sequentially compact space."}
{"_id": "14804", "text": "Summation over Finite Set is Well-Defined Let $\\mathbb A$ be one of the standard number systems $\\N, \\Z, \\Q, \\R, \\C$. Let $S$ be a finite set. Let $f: S \\to \\mathbb A$ be a mapping. Let $n$ be the cardinality of $S$. let $\\N_{<n}$ be an initial segment of the natural numbers. Let $g, h: \\N_{<n} \\to S$ be bijections. Then we have an equality of indexed summations of the compositions $f \\circ g$ and $f \\circ h$: :$\\displaystyle\\sum_{i \\mathop = 0}^{n - 1} \\map f {\\map g i} = \\sum_{i \\mathop = 0}^{n - 1} \\map f {\\map h i}$ That is, the definition of summation over a finite set does not depend on the choice of the bijection $g: S \\to \\N_{< n}$."}
{"_id": "292", "text": "Complement of Symmetric Relation Let $\\mathcal R \\subseteq S \\times S$ be a relation. Then $\\mathcal R$ is symmetric {{iff}} its complement $\\relcomp {S \\times S} {\\mathcal R} \\subseteq S \\times S$ is also symmetric."}
{"_id": "4594", "text": "Diffuse Measure of Countable Set Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Suppose that for all $x \\in X$, the singleton $\\left\\{{x}\\right\\}$ is in $\\Sigma$. Suppose further that $\\mu$ is a diffuse measure. Let $E \\in \\Sigma$ be a countable measurable set. Then $\\mu \\left({E}\\right) = 0$."}
{"_id": "12782", "text": "Supremum of Ideals is Upper Adjoint implies Lattice is Continuous Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a bounded below up-complete lattice. Let $\\mathit{Ids}\\left({L}\\right)$ be the set of all ideals in $L$. Let $P = \\left({\\mathit{Ids}\\left({L}\\right), \\precsim}\\right)$ be an ordered set where $\\mathord \\precsim = \\subseteq\\restriction_{\\mathit{Ids}\\left({L}\\right)\\times \\mathit{Ids}\\left({L}\\right)}$ Let $f: \\mathit{Ids}\\left({L}\\right) \\to S$ be a mapping such that :$\\forall I \\in \\mathit{Ids}\\left({L}\\right): f\\left({I}\\right) = \\sup I$ Let $f$ be an upper adjoint of a Galois connection. Then $L$ is continuous."}
{"_id": "5579", "text": "Orbit of Subgroup under Coset Action is Coset Space Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $\\powerset G$ be the power set of $G$. Let $H \\in \\powerset G$ be a subgroup of $G$. Let $*$ be the group action on $H$ defined as: :$\\forall g \\in G: g * H = g \\circ H$ where $g \\circ H$ is the (left) coset of $g$ by $H$. Then the orbit of $H$ in $\\powerset G$ is the (left) coset space of $H$: :$\\Orb H = G / H^l$"}
{"_id": "9872", "text": "Primitive of x by Exponential of a x by Cosine of b x :$\\displaystyle \\int x e^{a x} \\cos b x \\ \\mathrm d x = \\frac {x e^{a x} \\left({a \\cos b x + b \\sin bx}\\right)} {a^2 + b^2} - \\frac {e^{a x} \\left({\\left({a^2 - b^2}\\right) \\cos b x - 2 a b \\sin bx}\\right)} {\\left({a^2 + b^2}\\right)^2} + C$"}
{"_id": "1783", "text": "Three-Way Exclusive Or and Equivalence Let $p \\iff q$ be the biconditional operator, and $p \\oplus q$ be the exclusive or operator. Then: : $p \\iff q \\iff r \\dashv \\vdash p \\oplus q \\oplus r$"}
{"_id": "15164", "text": "Connected Space is not necessarily Locally Connected Let $T = \\struct {S, \\tau}$ be a topological space which is connected. Then it is not necessarily the case that $T$ is also a locally connected space."}
{"_id": "13948", "text": "Prime between n and 9 n divided by 8 Let $n \\in \\Z$ be an integer such that $n \\ge 48$. Then there exists a prime number $p$ such that $n < p < \\dfrac {9 n} 8$."}
{"_id": "19681", "text": "Scalar Multiplication by Zero gives Zero Vector Let $\\mathbf a$ be a vector quantity. Let $0 \\mathbf a$ denote the scalar product of $\\mathbf a$ with $0$. Then: :$0 \\mathbf a = \\bszero$ where $\\bszero$ denotes the zero vector."}
{"_id": "8347", "text": "Affine Group of One Dimension is Group Let $\\map {\\operatorname {Af}_1} \\R$ be the $1$-dimensional affine group on $\\R$. Then $\\map {\\operatorname {Af}_1} \\R$ is a group."}
{"_id": "423", "text": "Opposite Group is Group Let $\\struct {G, \\circ}$ be a group. Let $\\struct {G, *}$ be the opposite group to $G$. Then $\\struct {G, *}$ is a group."}
{"_id": "9390", "text": "Chapman-Kolmogorov Equation Let $X$ be a discrete state-space Markov chain with $n$-step transition probability matrix: :$\\mathbf P^{\\paren n} = \\sqbrk {\\map {p^{\\paren n} } {j, k} }_{j, k \\mathop \\in S}$ where: :$\\map {p^{\\paren n} } {j, k} = \\condprob {X_{m + n} = k} {X_m = j} = {p_{j k} }^{\\paren n}$ is the $n$-step transition probability. Then: :$\\mathbf P^{\\paren {n + m} } = \\mathbf P^{\\paren n} \\mathbf P^{\\paren m}$ or equivalently: :$\\displaystyle {p_{i j} }^{\\paren {n + m} } = \\sum_{k \\mathop \\in S} {p_{i k} }^{\\paren n} {p_{k j} }^{\\paren m}$"}
{"_id": "3312", "text": "Higher-Aleph Complement Topology is Topology Let $T = \\left({S, \\tau}\\right)$ be an $\\aleph_n$ complement space. Then $\\tau$ is a topology on $T$."}
{"_id": "8789", "text": "Inverse Secant of Imaginary Number :$\\sec^{-1} x = \\pm i \\sech^{-1} x$"}
{"_id": "7823", "text": "Equivalence of Definitions of Tangent of Angle Let $\\theta$ be an angle. {{TFAE|def = Tangent of Angle|view = Tangent of $\\theta$}}"}
{"_id": "5720", "text": "Set of All Mappings of Cartesian Product Let $R$, $S$, and $T$ be sets. Then: :$R^{S \\times T} \\sim \\left({ R^S }\\right)^T$"}
{"_id": "3257", "text": "Separation Properties in Open Extension of Particular Point Topology Let $T = \\struct {S, \\tau_p}$ be a particular point space such that $S$ is not a singleton or a doubleton. Let $T^*_{\\bar q} = \\struct {S^*_q, \\tau^*_{\\bar q} }$ be an open extension space of $T$. Then: :$T^*_{\\bar q}$ is a $T_0$ (Kolmogorov) space. :$T^*_{\\bar q}$ is a $T_4$ (space. :$T^*_{\\bar q}$ is not a $T_1$ (Fréchet) space. :$T^*_{\\bar q}$ is not a $T_5$ (space."}
{"_id": "3665", "text": "Unordered Pairs Exist Let $A$ and $B$ be classes, which may be either sets or proper classes. Then: :$\\forall A, B: \\left\\{{A, B}\\right\\} \\in U$ where $U$ is the universal class."}
{"_id": "16434", "text": "Subgroup Generated by Infinite Order Element is Infinite Let $G$ be a group. Let $a \\in G$ be of infinite order. Let $\\gen a$ be the subgroup generated  by $a$. Then $\\gen a$ is of infinite order."}
{"_id": "11147", "text": "Mellin Transform of Heaviside Step Function/Corollary :$\\mathcal M \\left\\{ {u\\left({c - t}\\right)}\\right\\} \\left({s}\\right) = \\dfrac {c^s} s$ for $c > 0, \\Re \\left({s}\\right) > 0$"}
{"_id": "15924", "text": "Rectangles with Equal Bases and Equal Altitudes are Congruent Let $ABCD$ and $EFGH$ be rectangles. Then $ABCD$ and $EFGH$ are congruent if: : the base of $ABCD$ equals the base of $EFGH$ : the altitude of $ABCD$ equals the altitude of $EFGH$."}
{"_id": "3108", "text": "Non-Trivial Particular Point Topology is not T3 Let $T = \\struct {S, \\tau_p}$ be a particular point space such that $S$ is not a singleton. Then $T$ is not a $T_3$ space."}
{"_id": "9531", "text": "Commutativity of Parameters of Beta Function :$\\map \\Beta {x, y} = \\map \\Beta {y, x}$"}
{"_id": "13883", "text": "Triangular Numbers in Geometric Sequence The numbers: :$1, 6, 36$ are the smallest triangular numbers in geometric sequence."}
{"_id": "741", "text": "Existence of Unique Subgroup Generated by Subset Let $\\struct {G, \\circ}$ be a group. Let $\\O \\subset S \\subseteq G$. Let $\\struct {H, \\circ}$ be the subgroup generated by $S$. Then $H = \\gen S$ exists and is unique. Also, $\\struct {H, \\circ}$ is the intersection of all of the subgroups of $G$ which contain the set $S$: :$\\displaystyle \\gen S = \\bigcap_i {H_i}: S \\subseteq H_i \\le G$"}
{"_id": "4050", "text": "Convergent Sequence in Set of Integers Let $\\left \\langle {x_n}\\right \\rangle_{n \\in \\N}$ be a sequence in the set $\\Z$ of integers considered as a subspace of the real number line $\\R$ under the Euclidean metric. Then $\\left \\langle {x_n}\\right \\rangle_{n \\in \\N}$ converges in $\\R$ to a limit {{iff}}: :$\\exists k \\in \\N: \\forall m \\in \\N: m > k: x_m = x_k$ That is, {{iff}} the sequence reaches some value of $\\Z$ and \"stays there\"."}
{"_id": "2588", "text": "Additive Group of Rationals is Subgroup of Complex Let $\\struct {\\Q, +}$ be the additive group of rational numbers. Let $\\struct {\\C, +}$ be the additive group of complex numbers. Then $\\struct {\\Q, +}$ is a normal subgroup of $\\struct {\\C, +}$."}
{"_id": "15623", "text": "Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors Let $\\mathbf a$ and $\\mathbf b$ be vectors in a vector space of $3$ dimensions: Let $\\mathbf a \\times \\mathbf b$ denote the vector cross product of $\\mathbf a$ with $\\mathbf b$. Then $\\left\\lvert{\\mathbf a \\times \\mathbf b}\\right\\rvert$ equals the area of the parallelogram two of whose sides are $\\mathbf a$ and $\\mathbf b$."}
{"_id": "18934", "text": "Edgeless Graphs of Order n are Isomorphic Let $n \\in \\Z_{>0}$ be a positive integer. Let $G_1 = \\struct {\\map V {G_1}, \\map E {G_1} }$ and $G_2 = \\struct {\\map V {G_2}, \\map E {G_2} }$ be edgeless graphs of order $n$. Then $G_1$ and $G_2$ are isomorphic."}
{"_id": "13601", "text": "Fourth Power is Sum of 2 Triangular Numbers Let $n \\in \\Z$ be an integer. Then: :$\\exists a, b \\in \\Z_{\\ge 0}: n^4 = T_a + T_b$ where $T_a$ and $T_b$ are triangular numbers. That is, the $4$th power of an integer equals the sum of two triangular numbers."}
{"_id": "16096", "text": "Sum of Squares of Complex Moduli of Sum and Differences of Complex Numbers Let $\\alpha, \\beta \\in \\C$ be complex numbers. Then: :$\\left\\lvert{\\alpha + \\beta}\\right\\rvert^2 + \\left\\lvert{\\alpha - \\beta}\\right\\rvert^2 = 2 \\left\\lvert{\\alpha}\\right\\rvert^2 + 2 \\left\\lvert{\\beta}\\right\\rvert^2$"}
{"_id": "1777", "text": "Commutativity of Incidence Matrix with its Transpose for Symmetric Design Let $A$ be the incidence matrix of a symmetric design. Then: : $A A^\\intercal = A^\\intercal A$ where $A^\\intercal$ is the transpose of $A$."}
{"_id": "9179", "text": "Primitive of Product of Cosecant and Cotangent :$\\ds \\int \\csc x \\cot x \\rd x = -\\csc x + C$ where $C$ is an arbitrary constant."}
{"_id": "7300", "text": "Vector Space with Standard Affine Structure is Affine Space Let $E$ be a vector space. Let $\\left({\\mathcal E, E, +, -}\\right)$ be the standard affine structure on $E$. Then with this structure, $\\mathcal E$ is an affine space."}
{"_id": "14284", "text": "219 Fedorov Groups There are $219$ Fedorov groups, if chiral copies are considered the same."}
{"_id": "17149", "text": "Equivalence of Definitions of Division Ring {{TFAE|def = Division Ring}} A '''division ring''' is a ring with unity $\\struct {R, +, \\circ}$ with the following properties:"}
{"_id": "13732", "text": "Group is Hausdorff iff has Closed Discrete Subgroup A topological group is Hausdorff {{iff}} it has a closed discrete subgroup."}
{"_id": "4030", "text": "Image of Set Difference under Mapping/Corollary 1 Let $f: S \\to T$ be a mapping. Let $S_1 \\subseteq S_2 \\subseteq S$. Then: :$\\relcomp {f \\sqbrk {S_2} } {f \\sqbrk {S_1} } \\subseteq f \\sqbrk {\\relcomp {S_2} {S_1} }$ where $\\complement$ (in this context) denotes relative complement."}
{"_id": "4583", "text": "Measure Invariant on Generator is Invariant Let $\\left({X, \\Sigma, \\mu}\\right)$ be a $\\sigma$-finite measure space. Let $\\theta: X \\to X$ be an $\\Sigma / \\Sigma$-measurable mapping. Suppose that $\\Sigma$ is generated by $\\mathcal G \\subseteq \\mathcal P \\left({X}\\right)$. Also, let $\\mathcal G$ satisfy the following: :$(1):\\quad \\forall G, H \\in \\mathcal G: G \\cap H \\in \\mathcal G$ :$(2):\\quad$ There exists an exhausting sequence $\\left({G_n}\\right)_{n \\in \\N} \\uparrow X$ in $\\mathcal G$ such that: :::$\\quad \\forall n \\in \\N: \\mu \\left({G_n}\\right) < +\\infty$ Suppose furthermore that, for all $G \\in \\mathcal G$, $\\mu$ satisfies: :$(3):\\quad \\mu \\left({\\theta^{-1} \\left({G}\\right) }\\right) = \\mu \\left({G}\\right)$ Then $\\mu$ is a $\\theta$-invariant measure."}
{"_id": "16803", "text": "Generator of Vector Space is Basis iff Cardinality equals Dimension :$G$ is a basis for $E$ {{iff}} $\\card G = n$."}
{"_id": "15274", "text": "Hilbert Sequence Space is not Sigma-Compact Let $A$ be the set of all real sequences $\\sequence {x_i}$ such that the series $\\displaystyle \\sum_{i \\mathop \\ge 0} x_i^2$ is convergent. Let $\\ell^2 = \\struct {A, d_2}$ be the Hilbert sequence space on $\\R$. Then $\\ell^2$ is not $\\sigma$-compact."}
{"_id": "9608", "text": "Primitive of Reciprocal of x squared by x cubed plus a cubed squared :$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\left({x^3 + a^3}\\right)^2} = \\frac {-1} {a^6 x} - \\frac {x^2} {3 a^6 \\left({x^3 + a^3}\\right)} - \\frac 4 {3 a^6} \\int \\frac {x \\ \\mathrm d x} {x^3 + a^3}$"}
{"_id": "15331", "text": "Integral Operator is Linear/Corollary 2 :$\\forall \\alpha \\in \\R: \\map T {\\alpha f} = \\alpha \\map T f$"}
{"_id": "14351", "text": "Completely Irreducible implies Meet Irreducible Let $L = \\left({S, \\wedge, \\preceq}\\right)$ be a meet semilattice. Let $p \\in S$. Then if $p$ is completely irreducible, then $p$ is meet irreducible."}
{"_id": "10013", "text": "Primitive of x by Hyperbolic Secant of a x :$\\displaystyle \\int x \\sech a x \\rd x = \\frac 1 {a^2} \\paren {\\frac {\\paren {a x}^2} 2 - \\frac {\\paren {a x}^4} 8 + \\frac {5 \\paren {a x}^6} {144} - \\cdots + \\frac {E_{2 n} \\paren {a x}^{2 n + 2} } {\\paren {2 n + 2} \\paren {2 n}!} + \\cdots} + C$"}
{"_id": "18911", "text": "Simple Graph whose Vertices Incident to All Edges Let $G = \\struct {V, E}$ be a simple graph whose vertices are incident to all its edges. Then $G$ is either: :the star graph $S_2$, which is also the complete graph $K_2$ :an edgeless graph of any order."}
{"_id": "16005", "text": "Representation of Integers in Golden Mean Number System The positive integers $n$ are represented in the golden mean number system in their simplest form $S_n$ as follows: :{| border=\"1\" |- ! align=\"right\" style = \"padding: 2px 10px\" | $n$  ! align=\"left\" style = \"padding: 2px 10px\" | $S_n$  |- | align=\"right\" style = \"padding: 2px 10px\" | $1$ | align=\"left\" style = \"padding: 2px 10px\" | $1$ |- | align=\"right\" style = \"padding: 2px 10px\" | $2$ | align=\"left\" style = \"padding: 2px 10px\" | $10 \\cdotp 01$ |- | align=\"right\" style = \"padding: 2px 10px\" | $3$ | align=\"left\" style = \"padding: 2px 10px\" | $100 \\cdotp 01$ |- | align=\"right\" style = \"padding: 2px 10px\" | $4$ | align=\"left\" style = \"padding: 2px 10px\" | $101 \\cdotp 01$ |- | align=\"right\" style = \"padding: 2px 10px\" | $5$ | align=\"left\" style = \"padding: 2px 10px\" | $1000 \\cdotp 1001$ |- | align=\"right\" style = \"padding: 2px 10px\" | $6$ | align=\"left\" style = \"padding: 2px 10px\" | $1010 \\cdotp 0001$ |- | align=\"right\" style = \"padding: 2px 10px\" | $7$ | align=\"left\" style = \"padding: 2px 10px\" | $10000 \\cdotp 0001$ |- | align=\"right\" style = \"padding: 2px 10px\" | $8$ | align=\"left\" style = \"padding: 2px 10px\" | $10001 \\cdotp 0001$ |- | align=\"right\" style = \"padding: 2px 10px\" | $9$ | align=\"left\" style = \"padding: 2px 10px\" | $10010 \\cdotp 0101$ |- | align=\"right\" style = \"padding: 2px 10px\" | $10$ | align=\"left\" style = \"padding: 2px 10px\" | $10100 \\cdotp 0101$ |- | align=\"right\" style = \"padding: 2px 10px\" | $11$ | align=\"left\" style = \"padding: 2px 10px\" | $10101 \\cdotp 0101$ |- | align=\"right\" style = \"padding: 2px 10px\" | $12$ | align=\"left\" style = \"padding: 2px 10px\" | $100000 \\cdotp 101001$ |- | align=\"right\" style = \"padding: 2px 10px\" | $13$ | align=\"left\" style = \"padding: 2px 10px\" | $100010 \\cdotp 001001$ |- | align=\"right\" style = \"padding: 2px 10px\" | $14$ | align=\"left\" style = \"padding: 2px 10px\" | $100100 \\cdotp 001001$ |- | align=\"right\" style = \"padding: 2px 10px\" | $15$ | align=\"left\" style = \"padding: 2px 10px\" | $100101 \\cdotp 001001$ |- | align=\"right\" style = \"padding: 2px 10px\" | $16$ | align=\"left\" style = \"padding: 2px 10px\" | $101000 \\cdotp 100001$ |}"}
{"_id": "9787", "text": "Primitive of x by Square of Tangent of a x :$\\displaystyle \\int x \\tan^2 a x \\ \\mathrm d x = \\frac {x \\tan a x} a + \\frac 1 {a^2} \\ln \\left\\vert{\\cos a x}\\right\\vert - \\frac {x^2} 2 + C$"}
{"_id": "7557", "text": "Probability Generating Function of Shifted Random Variable Let $X$ be a discrete random variable whose probability generating function is $\\Pi_X \\left({s}\\right)$. Let $k \\in \\Z_{\\ge 0}$ be a positive integer. Let $Y$ be a discrete random variable such that $Y = X + m$. Then :$\\Pi_Y \\left({s}\\right) = s^m \\Pi_X \\left({s}\\right)$. where $\\Pi_Y \\left({s}\\right)$ is the probability generating function of $Y$."}
{"_id": "7046", "text": "Rule of Material Implication/Formulation 2/Reverse Implication : $\\vdash \\left({\\neg p \\lor q}\\right) \\implies \\left({p \\implies q}\\right)$"}
{"_id": "19064", "text": "Homeomorphism may Exist between Non-Comparable Topologies Let $S$ be a set. Let $T_1 = \\struct {S, \\tau_1}$ and $T_2 = \\struct {S, \\tau_2}$ be topological spaces defined on the underlying set $S$. Let $\\tau_1$ and $\\tau_2$ be non-comparable. Then it may possibly be the case that $T_1$ and $T_2$ are homeomorphic."}
{"_id": "1461", "text": "Equivalence of Definitions of Connected Topological Space {{TFAE|def = Connected Topological Space}} Let $T = \\struct {S, \\tau}$ be a topological space."}
{"_id": "6390", "text": "Double Negation Elimination implies Law of Excluded Middle Let the Law of Double Negation Elimination be supposed to hold: :$\\neg \\neg p \\vdash p$ Then the Law of Excluded Middle likewise holds: :$\\vdash p \\lor \\neg p$"}
{"_id": "5920", "text": "Multiindices under Addition form Commutative Monoid Let $Z$ be the set of multiindices. Let $+$ denote the addition of multiindices. Then $\\left({Z, +}\\right)$ is a commutative monoid."}
{"_id": "7668", "text": "Four Color Theorem for Finite Maps implies Four Color Theorem for Infinite Maps Suppose that any finite planar graph can be assigned a proper vertex $k$-coloring such that $k \\le 4$. Then the same is true of any infinite planar graph."}
{"_id": "19720", "text": "Partial Derivatives of tan^2 (x^2 - y^2) Let: :$\\map f {x, y} = \\map {\\tan^2} {x^2 - y^2}$ Then: {{begin-eqn}} {{eqn | l = \\map {f_1} {x, y}       | r = 4 x \\map \\tan {x^2 - y^2} \\map {\\sec^2} {x^2 - y^2} }} {{eqn | l = \\map {f_2} {1, 2}       | r = 8 \\tan 3 \\sec^2 3 }} {{end-eqn}}"}
{"_id": "8599", "text": "Rational Number Space is not Scattered Let $\\struct {\\Q, \\tau_d}$ be the rational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\Q, \\tau_d}$ is not scattered."}
{"_id": "15492", "text": "Definite Integral from 0 to Pi of Cosine of m x by Cosine of n x Let $m, n \\in \\Z$ be integers. Then: :$\\displaystyle \\int_0^\\pi \\cos m x \\cos n x \\rd x = \\begin{cases} 0 & : m \\ne n \\\\ \\dfrac \\pi 2 & : m = n \\end{cases}$ That is: :$\\displaystyle \\int_0^\\pi \\cos m x \\cos n x \\rd x = \\dfrac \\pi 2 \\delta_{m n}$ where $\\delta_{m n}$ is the Kronecker delta."}
{"_id": "15387", "text": "Ring Epimorphism Preserves Unity Let $A$ be a ring with unity $1$. Let $B$ be a ring. Let $f: A \\to B$ be a ring epimorphism. Then $\\map f 1$ is a unity of $B$."}
{"_id": "12480", "text": "Floor Function/Examples/Floor of One Half :$\\floor {\\dfrac 1 2} = 0$"}
{"_id": "9380", "text": "Primitive of x over Power of x squared plus a squared :$\\displaystyle \\int \\frac {x \\ \\mathrm d x} {\\left({x^2 + a^2}\\right)^n} = \\frac {-1} {2 \\left({n - 1}\\right) \\left({x^2 + a^2}\\right)^{n - 1} }$"}
{"_id": "16056", "text": "Generating Function for Sequence of Sum over k to n of Reciprocal of k by n-k Let $\\sequence {a_n}$ be the sequence whose terms are defined as: :$\\forall n \\in \\Z_{\\ge 0}: a_n = \\displaystyle \\sum_{k \\mathop = 1}^{n - 1} \\dfrac 1 {k \\paren {n - k} }$ Then $\\sequence {a_n}$ has the generating function $\\map G z$ such that: :$\\map G z = \\paren {\\ln \\dfrac 1 {1 - z} }^2$ and whose terms are: :$a_n = \\dfrac {2 H_{n - 1} } n$"}
{"_id": "2746", "text": "Exterior of Finite Union equals Intersection of Exteriors Let $T$ be a topological space. Let $n \\in \\N$. Let $\\forall i \\in \\closedint 1 n: H_i \\subseteq T$. Then: :$\\displaystyle \\paren {\\bigcup_{i \\mathop = 1}^n H_i}^e = \\bigcap_{i \\mathop = 1}^n H_i^e$ where $H_i^e$ denotes the exterior of $H_i$."}
{"_id": "15991", "text": "Fibonacci Number of Even Index by Golden Mean Modulo 1 Let $n \\in \\Z$ be an integer. Then: :$F_{2 n} \\phi \\bmod 1 = 1 - \\phi^{-2 n}$ :$F_n$ denotes the $n$th Fibonacci number :$\\phi$ is the golden mean: $\\phi = \\dfrac {1 + \\sqrt 5} 2$"}
{"_id": "1854", "text": "Overflow Theorem Let $F$ be a set of first-order formulas which has finite models of arbitrarily large size. Then $F$ has an infinite model."}
{"_id": "6124", "text": "Equivalence of Definitions of Homeomorphic Topological Spaces Let $T_\\alpha = \\left({S_\\alpha, \\tau_\\alpha}\\right)$ and $T_\\beta = \\left({S_\\beta, \\tau_\\beta}\\right)$ be topological spaces. Let $f: T_\\alpha \\to T_\\beta$ be a bijection. {{TFAE|def = Homeomorphism (Topological Spaces)|view = homeomorphism}}"}
{"_id": "12494", "text": "Exponential Sequence is Eventually Strictly Positive Let $\\sequence {E_n}$ be the sequence of real functions $E_n: \\R \\to \\R$ defined as: :$\\map {E_n} x = \\paren {1 + \\dfrac x n}^n$ Then, for each $x \\in \\R$ and for sufficiently large $n \\in \\N$, $\\map {E_n} x$ is positive. That is: :$\\forall x \\in \\R: \\forall n \\in \\N: n \\ge \\ceiling {\\size x} \\implies \\map {E_n} x > 0$ where $\\ceiling x$ denotes the ceiling of $x$."}
{"_id": "17786", "text": "Integral Representation of Dirichlet Beta Function in terms of Gamma Function :$\\displaystyle \\map \\beta s = \\frac 1 {\\map \\Gamma s} \\int_0^\\infty \\frac {x^{s - 1} e^{-x} } {1 + e^{-2 x} } \\rd x$"}
{"_id": "12386", "text": "Way Below Compact is Topological Compact Let $T = \\left({S, \\tau}\\right)$ be a topological space. Let $L = \\left({\\tau, \\preceq}\\right)$ be an ordered set where $\\preceq \\mathop = \\subseteq\\restriction_{\\tau \\times \\tau}$ Let $x \\in \\tau$. Then :$x$ is compact in $L$ (it means: $x \\ll x$) {{iff}} :$T_x$ is compact (topologically) where $T_x = \\left({x, \\tau_x}\\right)$ denotes the topological subspace of $x$."}
{"_id": "16916", "text": "Cardinality of Mapping Let $S$ be a finite set whose cardinality is $n$: :$\\card S = n$ Let $T$ be a non-empty set Let $f: S \\to T$ be a mapping. Then: :$\\card f = n$"}
{"_id": "11492", "text": "Orthogonal Trajectories/Cardioids Consider the one-parameter family of curves of cardioids given in polar form as: :$(1): \\quad r = c \\paren {1 + \\cos \\theta}$ Its family of orthogonal trajectories is given by the equation: :$r = c \\paren {1 - \\cos \\theta}$ :600px"}
{"_id": "3518", "text": "Circumscribing Regular Pentagon about Circle About a given circle, it is possible to circumscribe a regular pentagon. {{:Euclid:Proposition/IV/12}}"}
{"_id": "18125", "text": "Distance Between Points of Same Latitude along Parallel of Latitude Let $J$ and $K$ be points on Earth's surface that have the same latitude. Let $JK$ be the length of the arc joining $JK$ measured along the parallel of latitude on which they both lie. Let $R$ denote the center of the parallel of latitude holding $J$ and $K$. Let $\\operatorname {Long}_J$ and $\\operatorname {Long}_K$ denote the longitude of $J$ and $K$ respectively, measured in degrees. Let $\\operatorname {Lat}_J$ denote the latitude of $J$ (and $K$). Then:  :$JK \\approx 60 \\times \\size {\\operatorname {Long}_J - \\operatorname {Long}_K} \\cos \\operatorname {Lat}_J$ nautical miles"}
{"_id": "8594", "text": "Irrational Number Space is not Weakly Sigma-Locally Compact Let $\\struct {\\R \\setminus \\Q, \\tau_d}$ be the irrational number space under the Euclidean topology $\\tau_d$. Then $\\struct {\\R \\setminus \\Q, \\tau_d}$ is not weakly $\\sigma$-locally compact."}
{"_id": "2160", "text": "PGF of Sum of Random Number of Discrete Random Variables Let $\\left({\\Omega, \\Sigma, \\Pr}\\right)$ be a probability space. Let: :$N, X_1, X_2, \\ldots$ be independent discrete random variables such that the $X$'s have the same probability distribution. Let: * $\\Pi_N \\left({s}\\right)$ be the PGF of $N$ * $\\Pi_X \\left({s}\\right)$ be the PGF of each of the $X$'s. Let: :$Z = X_1 + X_2 + \\ldots + X_N$ Then: :$\\Pi_Z \\left({s}\\right) = \\Pi_N \\left({\\Pi_X \\left({s}\\right)}\\right)$"}
{"_id": "13253", "text": "Successive Solutions of Phi of n equals Phi of n + 2 $7$ and $8$ are two successive integers which are solutions to the equation: :$\\map \\phi n = \\map \\phi {n + 2}$"}
{"_id": "15112", "text": "Existence of Weakly Locally Compact Space which is not Strongly Locally Compact There exists at least one example of a weakly locally compact topological space which is not also a strongly locally compact space."}
{"_id": "19244", "text": "Cosine of Integer Multiple of Argument/Formulation 1 {{begin-eqn}} {{eqn | l = \\cos n \\theta       | r = \\dfrac 1 2 \\paren {\\paren {2 \\cos \\theta }^n - \\dfrac n 1 \\paren {2 \\cos \\theta }^{n - 2} + \\dfrac n 2 \\dbinom {n - 3} 1 \\paren {2 \\cos \\theta }^{n - 4} - \\dfrac n 3 \\dbinom {n - 4} 2 \\paren {2 \\cos \\theta }^{n - 6} + \\cdots }       | c =  }} {{eqn | r = \\dfrac 1 2 \\paren {\\paren {2 \\cos \\theta }^n + \\sum_{k \\mathop \\ge 1} \\paren {-1 }^k \\dfrac n k \\dbinom {n - \\paren {k + 1 } } {k - 1} \\paren {2 \\cos \\theta }^{n - 2 k } }       | c =  }} {{end-eqn}}"}
{"_id": "16947", "text": "Greatest Common Divisor divides Lowest Common Multiple Let $a, b \\in \\Z$ such that $a b \\ne 0$. Then: :$\\gcd \\set {a, b} \\divides \\lcm \\set {a, b}$ where: :$\\lcm$ denotes lowest common multiple :$\\gcd$ denotes greatest common divisor. :$\\divides$ denotes divisibility."}
{"_id": "5538", "text": "Ordinal Multiplication via Cantor Normal Form/Infinite Exponent Let $x$ and $y$ be ordinals. Let $x > 1$. Let $y \\ge \\omega$ where $\\omega$ denotes the minimal infinite successor set. Let $\\left\\langle{a_i}\\right\\rangle$ be a sequence of ordinals that is strictly decreasing on $1 \\le i \\le n$. Let $\\left\\langle{b_i}\\right\\rangle$ be a sequence of ordinals such that $0 < b_i < x$ for all $1 \\le i \\le n$. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^n \\left({x^{a_i} b_i}\\right) \\times x^y = x^{a_1 \\mathop + y}$"}
{"_id": "2554", "text": "Sum of Ring Products is Subring of Commutative Ring Let $\\left({R, +, \\circ}\\right)$ be a commutative ring. Let $\\left({S, +, \\circ}\\right)$ and $\\left({T, +, \\circ}\\right)$ be subrings of $\\left({R, +, \\circ}\\right)$. Let $S T$ be defined as: :$\\displaystyle S T = \\left\\{{\\sum_{i \\mathop = 1}^n s_i \\circ t_i: s_1 \\in S, t_i \\in T, i \\in \\left[{1 \\,.\\,.\\ n}\\right]}\\right\\}$ Then $S T$ is a subring of $\\left({R, +, \\circ}\\right)$."}
{"_id": "14727", "text": "Smallest Fermat Pseudoprime to Bases 2, 3, 5 and 7 The smallest Fermat pseudoprime to bases $2$, $3$, $5$ and $7$ is $29 \\, 341$."}
{"_id": "802", "text": "Subset has 2 Conjugates then Normal Subgroup Let $G$ be a group. Let $S$ be a subset of $G$. Let $S$ have exactly two conjugates in $G$. Then $G$ has a proper non-trivial normal subgroup."}
{"_id": "9036", "text": "Equivalence of Definitions of Complex Inverse Hyperbolic Cotangent {{TFAE|def = Complex Inverse Hyperbolic Cotangent}} Let $S$ be the subset of the complex plane: :$S = \\C \\setminus \\left\\{{-1 + 0 i, 1 + 0 i}\\right\\}$"}
{"_id": "18626", "text": "Power Set of Transitive Set is Transitive Let $S$ be a transitive set. Then its power set $\\powerset S$ is also a transitive set."}
{"_id": "17558", "text": "Infimum is not necessarily Smallest Element Let $\\struct {S, \\preceq}$ be an ordered set. Let $T$ admit a infimum in $S$. Then the infimum of $T$ in $S$ is not necessarily the smallest element of $T$."}
{"_id": "16736", "text": "Cauchy Sequence is Bounded/Normed Division Ring/Proof 1 Let $\\struct {R, \\norm {\\,\\cdot\\,} }$ be a normed division ring. Every Cauchy sequence in $R$ is bounded."}
{"_id": "11622", "text": "Bernoulli's Equation/x y^2 y' + y^3 = x cosine x The first order ODE: :$(1): \\quad x y^2 y' + y^3 = x \\cos x$ has the general solution: :$y^3 = 3 \\sin x + \\dfrac {9 \\cos x} x - \\dfrac {18 \\sin x} {x^2} - \\dfrac {18 \\cos x} {x^3} + \\dfrac C {x^3}$"}
{"_id": "16956", "text": "Minimum Area of Triangle whose Vertices are Lattice Points Let $T$ be a triangle embedded in a cartesian plane. Let the vertices of $T$ be lattice points which are not all on the same straight line. Then the area of $T$ is such that: :$\\map \\Area T \\ge \\dfrac 1 2$"}
{"_id": "12888", "text": "Set of Meet Irreducible Elements Excluded Top is Order Generating Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a continuous complete lattice. Let $X = \\mathit{IRR}\\left({L}\\right) \\setminus \\left\\{ {\\top}\\right\\}$ where $\\mathit{IRR}\\left({L}\\right)$ denotes the set of all meet irreducible element of $S$, :$\\top$ denotes the top of $L$. Then $X$ is order generating."}
{"_id": "18835", "text": "Non-Empty Set of Natural Numbers with no Greatest Element is Denumerable Let $A$ be a class of natural numbers. Let $A$ have no greatest element. Then $A$ is a denumerable class."}
{"_id": "721", "text": "Floor defines Equivalence Relation Let $x \\in \\R$ be a real number. Let $\\floor x$ denote the floor function of $x$. Let $\\RR$ be the relation defined on $\\R$ such that: :$\\forall x, y, \\in \\R: \\tuple {x, y} \\in \\RR \\iff \\floor x = \\floor y$ Then $\\RR$ is an equivalence, and $\\forall n \\in \\Z$, the $\\RR$-class of $n$ is the half-open interval $\\hointr n {n + 1}$."}
{"_id": "4584", "text": "Sigma-Algebras with Independent Generators are Independent Let $\\struct {\\Omega, \\EE, \\Pr}$ be a probability space. Let $\\Sigma, \\Sigma'$ be sub-$\\sigma$-algebras of $\\EE$. Suppose that $\\GG, \\HH$ are $\\cap$-stable generators for $\\Sigma, \\Sigma'$, respectively. Suppose that, for all $G \\in \\GG, H \\in \\HH$: :$(1): \\quad \\map \\Pr {G \\cap H} = \\map \\Pr G \\map \\Pr H$ Then $\\Sigma$ and $\\Sigma'$ are $\\Pr$-independent."}
{"_id": "5918", "text": "Ordering on Multiindices is Partial Order Let $Z$ be the set of multiindices indexed by a set $J$. The ordering on $Z$ is a partial ordering."}
{"_id": "5755", "text": "Group Epimorphism Induces Bijection between Subgroups/Corollary Let $H \\le G$ denote that $H$ is a subgroup of $G$. Then: :$\\forall H \\le G, K \\subseteq H: \\phi \\sqbrk H \\cong H / K$ where $H / K$ denotes the quotient group of $H$ by $K$."}
{"_id": "10240", "text": "Power of Two is Even-Times Even Only Let $a > 2$ be a power of $2$. Then $a$ is even-times even only. {{:Euclid:Proposition/IX/32}}"}
{"_id": "13977", "text": "Product of Absolutely Convergent Products is Absolutely Convergent Let $\\struct {\\mathbb K, \\norm{\\,\\cdot\\,}}$ be a valued field. Let $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_n$ converge absolutely. Let $\\displaystyle \\prod_{n \\mathop = 1}^\\infty b_n$ converge absolutely. Then $\\displaystyle \\prod_{n \\mathop = 1}^\\infty a_nb_n$ converges absolutely."}
{"_id": "15998", "text": "Representations for 1 in Golden Mean Number System Then there are infinitely many ways to express the number $1$ in the golden mean number system."}
{"_id": "19689", "text": "Order 2 Square Matrices with Zero Diagonals do not form Ring Let $S$ be the set of square matrices of order $2$ whose diagonal elements are zero. Then the algebraic structure $\\struct {S, +, \\times}$ is not a ring. Note that $\\times$ denotes conventional matrix multiplication."}
{"_id": "5637", "text": "Well-Founded Induction Let $\\struct {A, \\prec}$ be a foundational relation. Let $\\prec^{-1} \\sqbrk x$ denote the preimage of $x$ for each $x \\in A$. Let $B$ be a class such that $B \\subseteq A$. Suppose that: :$(1): \\quad \\forall x \\in A: \\paren {\\prec^{-1} \\sqbrk x \\subseteq B \\implies x \\in B}$ Then: :$A = B$ That is, if a property passes from the preimage of $x$ to $x$, then this property is true for all $x \\in A$."}
{"_id": "4549", "text": "Relation Induced by Quotient Set is Equivalence Let $S$ be a set. Let $\\mathcal R$ be an equivalence relation on $S$. Let $S / \\mathcal R$ be the quotient set of $S$ determined by $\\mathcal R$. Let $\\mathcal R'$ be the relation induced by $S / \\mathcal R$ on $S$. Then $\\mathcal R' = \\mathcal R$."}
{"_id": "9302", "text": "Primitive of x squared over Root of a x + b :$\\displaystyle \\int \\frac {x^2 \\ \\mathrm d x} {\\sqrt{a x + b} } = \\frac {2 \\left({3 a^2 x^2 - 4 a b x + 8 b^2}\\right) \\sqrt{a x + b} } {15 a^3}$"}
{"_id": "11823", "text": "Linear Second Order ODE/y'' - 2 y' = 12 x - 10 The second order ODE: :$(1): \\quad y'' - 2 y' = 12 x - 10$ has the general solution: :$y = C_1 + C_2 e^{2 x} + 2 x - 3 x^2$"}
{"_id": "14189", "text": "2-Digit Permutable Primes The $2$-digit permutable primes are: :$11, 13, 17, 37, 79$ and their anagrams, and no other."}
{"_id": "15511", "text": "Sum of Sequence of Binomial Coefficients by Sum of Powers of Integers Let $n, k \\in \\Z_{\\ge 0}$ be positive integers. Let $S_k = \\displaystyle \\sum_{i \\mathop = 1}^n i^k$. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^k \\binom {k + 1} i S_i = \\paren {n + 1}^{k + 1} - \\paren {n + 1}$"}
{"_id": "2492", "text": "Construction of Inverse Completion/Cartesian Product with Cancellable Elements Let $\\left({S \\times C, \\oplus}\\right)$ be the external direct product of $\\left({S, \\circ}\\right)$ and $\\left({C, \\circ {\\restriction_C}}\\right)$, where $\\oplus$ is the operation on $S \\times C$ induced by $\\circ$ on $S$ and $\\circ {\\restriction_C}$ on $C$. That is: :$\\forall \\left({x, y}\\right), \\left({u, v}\\right) \\in S \\times C: \\left({x, y}\\right) \\oplus \\left({u, v}\\right) = \\left({x \\circ u, y \\mathop{\\circ {\\restriction_C}} v}\\right)$ Then $\\left({S \\times C, \\oplus}\\right)$ is a commutative semigroup."}
{"_id": "13554", "text": "Simple Variable End Point Problem/Endpoints on Curves Let $y$, $F$, $\\phi$ and $\\psi$ be smooth real functions. Let $J = J \\sqbrk y$ be a functional of the form: :$\\ds J \\sqbrk y = \\int_{x_0}^{x_1} \\map F {x, y, y'} \\rd x$ Let $P_0$, $P_1$ be the endpoints of the curve $y$. Suppose $P_0$, $P_1$ lie on curves $y = \\map {\\phi} x$, $y = \\map {\\psi} x$. Then the extremum of $J \\sqbrk y$ is a curve which satisfies the following system of Euler and transversality equations: {{begin-eqn}} {{eqn | l = F_y - \\dfrac {\\d} {\\d x} F_{y'}       | r = 0 }} {{eqn | l = \\bigvalueat {\\paren {F + \\paren {\\phi' - y'} F_{y'} } } {x \\mathop = x_0}       | r = 0 }} {{eqn | l = \\bigvalueat {\\paren {F + \\paren {\\psi' - y'} F_{y'} } } {x \\mathop = x_1} = 0       | r = 0 }} {{end-eqn}} {{explain|meaning of square brackets}}"}
{"_id": "18206", "text": "Sample Matrix Independence Test Let $V$ be a vector space of real or complex-valued functions on a set $J$. Let $f_1, \\ldots, f_n$ be functions in $V$. Let '''samples''' $x_1, \\ldots, x_n$ from $J$ be given. Define '''sample matrix'''  :$\\displaystyle S = \\begin{bmatrix} f_1(x_1)   &  \\cdots      & f_n(x_1) \\\\ \\vdots     & \\ddots       & \\vdots \\\\ f_1(x_n)   & \\cdots       & f_n(x_n) \\\\ \\end{bmatrix}$ Let $S$ be invertible. Then $f_1, \\ldots, f_n$ are linearly independent in $V$."}
{"_id": "16393", "text": "Difference of Images under Mapping not necessarily equal to Image of Difference Let $f: S \\to T$ be a mapping. The image of the set difference of two subsets of $S$ is not necessarily equal to the set difference of the images. That is: Let $S_1$ and $S_2$ be subsets of $S$. Then it is not always the case that: :$f \\sqbrk {S_1} \\setminus f \\sqbrk {S_2} = f \\sqbrk {S_1 \\setminus S_2}$ where $\\setminus$ denotes set difference."}
{"_id": "18352", "text": "Equation of Cissoid of Diocles/Cartesian Form The cissoid of Diocles can be defined by the Cartesian equation: :$x \\paren {x^2 + y^2} = 2 a y^2$"}
{"_id": "5544", "text": "Ordinal Exponentiation via Cantor Normal Form/Corollary Let $x$ and $y$ be ordinals. Let $x$ be a limit ordinal and let $y > 0$. Let $\\langle a_i \\rangle$ be a sequence of ordinals that is strictly decreasing on $1 \\le i \\le n$. Let $\\langle b_i \\rangle$ be a sequence of natural numbers. Then: :$\\displaystyle \\left({ \\sum_{i \\mathop = 1}^n x^{a_i} \\times b_i }\\right)^{x^y} = x^{a_1 \\mathop \\times x^y}$ {{explain|what \"Cantor normal form\" is in the exposition of the above.}}"}
{"_id": "3894", "text": "Arens-Fort Space is Non-Meager Let $T = \\struct {S, \\tau}$ be the Arens-Fort space. Then $T$ is a non-meager space."}
{"_id": "15859", "text": "Chu-Vandermonde Identity for Gaussian Binomial Coefficients {{begin-eqn}} {{eqn | l = \\binom {r + s} n_q       | r = \\sum_k \\binom r k_q \\binom s {n - k}_q q^{\\left({r - k}\\right) \\left({n - k}\\right)}       | c =  }} {{eqn | r = \\sum_k \\binom r k_q \\binom s {n - k}_q q^{\\left({s - n + k}\\right) k}       | c =  }} {{end-eqn}}"}
{"_id": "4988", "text": "Conditions for Homogeneity/Straight Line The line $L = \\alpha_1 x_1 + \\alpha_2 x_2 = \\beta$ is homogeneous {{iff}} $\\beta = 0$."}
{"_id": "14661", "text": "Sum of Two Rational 4th Powers but not Two Integer 4th Powers $5906$ is the smallest integer which can be expressed as the sum of two rational $4$th powers, but not two integer $4$th powers."}
{"_id": "3274", "text": "Uncountable Finite Complement Topology is not Perfectly T4 Let $T = \\struct {S, \\tau}$ be a finite complement topology on an uncountable set $S$. Then $T$ is not a perfectly $T_4$ space."}
{"_id": "15952", "text": "Fibonacci Numbers which equal their Index The only Fibonacci numbers which equal their index are: {{begin-eqn}} {{eqn | l = F_0       | r = 0 }} {{eqn | l = F_1       | r = 1 }} {{eqn | l = F_5       | r = 5 }} {{end-eqn}}"}
{"_id": "13721", "text": "Numbers whose Cyclic Permutations of 3-Digit Multiples are Multiples Let $n$ be a two-digit positive integer with the following property: :Let $m$ be a $3$-digit multiple of $n$. :Then any cyclic permutation of the digits of $m$ is also a multiple of $n$. Then $n$ is either $27$ or $37$."}
{"_id": "15426", "text": "Existence of Homomorphism between Localizations of Ring at Elements Let $A$ be a commutative ring with unity. Let $f, g \\in A$. {{TFAE}} :$(1): \\quad$ There exists an $A$-algebra homomorphism $h : A_f \\to A_g$ between localizations, the '''induced homomorphism'''. :$(2): \\quad f$ divides some power of $g$. :$(3): \\quad$ There is an inclusion of vanishing sets: $\\map V f \\subseteq \\map V g$. That is, every prime ideal containing $f$ also contains $g$. :$(4): \\quad$ There is an inclusion of principal open subsets: $\\map D f \\supseteq \\map D g$"}
{"_id": "18242", "text": "Arcsecant Logarithmic Formulation Let $x$ be a real number.  Let $x \\in \\hointl {-\\infty} {-1} \\cup \\hointr 1 {\\infty}$. Then:  :$\\displaystyle \\arcsec x = -i \\map \\Ln {i \\sqrt {1 - \\frac 1 {x^2} } + \\frac 1 x}$ where: :$\\arcsec$ is the arcsecant function :$\\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\\hointl {-\\pi} \\pi$."}
{"_id": "18301", "text": "P-adic Expansion is a Cauchy Sequence in P-adic Norm/Represents a P-adic Number Let $\\struct {\\Q_p, \\norm {\\,\\cdot\\,}_p}$ be the $p$-adic Numbers as the Quotient of Cauchy Sequences. Then the sequence of partial sums of the series: :$\\displaystyle \\sum_{n \\mathop = m}^\\infty d_n p^n$ represents an element of $\\struct {\\Q_p,\\norm {\\,\\cdot\\,}_p}$."}
{"_id": "11869", "text": "Position of Cart attached to Wall by Spring under Damping/Overdamped/x = x0 at t = 0 Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = \\dfrac {x_0} {m_1 - m_2} \\left({m_1 e^{m_2 t} - m_2 e^{m_1 t} }\\right)$"}
{"_id": "9093", "text": "Inverse of Strictly Decreasing Strictly Convex Real Function is Strictly Convex Let $f$ be a real function which is strictly convex on the open interval $I$. Let $J = f \\sqbrk I$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is strictly convex on $J$."}
{"_id": "1838", "text": "Measure Space from Outer Measure Suppose $\\mu^*$ is an outer measure on a set $X$. Let $\\mathfrak M(\\mu^*)$ be the collection of $\\mu^*$-measurable sets. Let $\\mu$ be the restriction of $\\mu^*$ to $\\mathfrak M(\\mu^*)$. Then $(X, \\mathfrak M(\\mu^*), \\mu)$ is a measure space."}
{"_id": "12769", "text": "Summation over k of Ceiling of mk+x over n Let $m, n \\in \\Z$ such that $n > 0$. Let $x \\in \\R$. Then: :$\\displaystyle \\sum_{0 \\mathop \\le k \\mathop < n} \\left \\lceil{\\dfrac {m k + x} n}\\right \\rceil = \\dfrac {\\left({m + 1}\\right) \\left({n - 1}\\right)} 2 - \\dfrac {d - 1} 2 + d \\left \\lceil{\\dfrac x d}\\right \\rceil$ where: :$\\left \\lceil{x}\\right \\rceil$ denotes the ceiling of $x$ :$d$ is the greatest common divisor of $m$ and $n$."}
{"_id": "4749", "text": "Pointwise Minimum of Measurable Functions is Measurable Let $\\left({X, \\Sigma}\\right)$ be a measurable space. Let $f, g: X \\to \\overline{\\R}$ be $\\Sigma$-measurable functions. Then the pointwise minimum $\\min \\left({f, g}\\right): X \\to \\overline{\\R}$ is also $\\Sigma$-measurable."}
{"_id": "10201", "text": "Numbers between which exists one Mean Proportional are Similar Plane Let $a, b \\in \\Z$ such that the geometric mean is an integer. Then $a$ and $b$ are similar plane numbers. {{:Euclid:Proposition/VIII/20}}"}
{"_id": "3877", "text": "Closure in Infinite Particular Point Space is not Compact Let $T = \\struct {S, \\tau_p}$ be an infinite particular point space. Let $A \\in \\tau_p$ be open in $T$. Let $A^-$ be the closure of $A$. Then $A^-$ is not compact."}
{"_id": "14216", "text": "Reciprocal of 8 The reciprocal of $8$ can be expressed as the summation of the powers of $2$ multiplied by the reciprocal  powers of $10$: :$\\dfrac 1 8 = \\displaystyle \\sum_{k \\mathop \\ge 0} \\dfrac {2^k} {10^{k + 1} }$ <pre> .1   2    4     8     16      32       64       128        256         512         1024          .... ------------- .12499999744 ------------- </pre>"}
{"_id": "19209", "text": "Singleton is Independent implies Rank is One/Corollary :$\\set x$ is an independent subset {{iff}} $\\map \\rho {\\set x} = 1$"}
{"_id": "3957", "text": "Derivative of Hyperbolic Secant Function :$\\map {\\dfrac \\d {\\d x} } {\\sech u} = -\\sech u \\tanh u \\dfrac {\\d u} {\\d x}$"}
{"_id": "4423", "text": "Min Semigroup on Toset is Semilattice Let $\\struct {S, \\preceq}$ be a totally ordered set.  Then the min semigroup $\\struct {S, \\min}$ is a semilattice."}
{"_id": "12646", "text": "Sum over k to n of Stirling Number of the Second Kind of k with m by m+1^n-k Let $m, n \\in \\Z_{\\ge 0}$. :$\\displaystyle \\sum_{k \\mathop \\le n} {k \\brace m} \\paren {m + 1}^{n - k} = {{n + 1} \\brace {m + 1}}$ where $\\displaystyle {k \\brace m}$ etc. denotes a Stirling number of the second kind."}
{"_id": "4865", "text": "Congruence Relation on Ring induces Ring Let $\\struct {R, +, \\circ}$ be a ring. Let $\\EE$ be a congruence relation on $R$ for both $+$ and $\\circ$. Let $R / \\EE$ be the quotient set of $R$ by $\\EE$. Let $+_\\EE$ and $\\circ_\\EE$ be the operations induced on $R / \\EE$ by $+$ and $\\circ$ respectively. Then $\\struct {R / \\EE, +_\\EE, \\circ_\\EE}$ is a ring."}
{"_id": "15578", "text": "Power Series Expansion for Exponential of Sine of x :$e^{\\sin x} = 1 + x + \\dfrac {x^2} 2 - \\dfrac {x^4} 8 - \\dfrac {x^5} {15} + \\cdots$ for all $x \\in \\R$."}
{"_id": "13227", "text": "Proper Filter is Included in Ultrafilter in Boolean Lattice Let $L = \\left({S, \\vee, \\wedge, \\preceq}\\right)$ be a Boolean lattice. Let $F$ be a proper filter on $L$. Then there exists ultrafilter $G$ on $L$: $F \\subseteq G$"}
{"_id": "13839", "text": "Integer as Sum of 5 Non-Zero Squares Let $n \\in \\Z$ be an integer such that $n > 33$. Then $n$ can be expressed as the sum of $5$ non-zero squares."}
{"_id": "4230", "text": "Equidistance is Independent of Betweenness Let $\\mathcal G$ be a formal systematic treatment of geometry containing only: : The language and axioms of first-order logic, and the disciplines preceding it : The undefined terms of Tarski's Geometry (excluding equidistance)  : Some or all of Tarski's Axioms of Geometry. In $\\mathcal G$, equidistance $\\equiv$ is necessarily an undefined term with respect to betweenness $\\mathsf B$."}
{"_id": "16020", "text": "Existence of Interval of Convergence of Power Series/Corollary 2 Let $\\displaystyle S \\left({x}\\right) = \\sum_{n \\mathop = 0}^\\infty a_n x^n$ be a power series. Let $S \\left({x}\\right)$ be convergent at $x = x_0$. Then $S \\left({x}\\right)$ is convergent for all $x$ such that $\\left\\lvert{x}\\right\\rvert < \\left\\lvert{x_0}\\right\\rvert$."}
{"_id": "16104", "text": "Equation of Line in Complex Plane/Formulation 2 Let $\\C$ be the complex plane. Let $L$ be the infinite straight line in $\\C$ which is the locus of the equation: :$l x + m y = 1$ Then $L$ may be written as: :$\\map \\Re {a z} = 1$ where $a$ is the point in $\\C$ defined as: :$a = l - i m$"}
{"_id": "5561", "text": "Epimorphism into Projective Object Splits Let $\\mathbf C$ be a metacategory. Let $P \\in \\mathbf C_0$ be a projective object of $\\mathbf C$. Let $e: E \\twoheadrightarrow P$ be an epimorphism. Then $e$ is a split epimorphism, i.e. it admits a retraction $f: P \\to E$."}
{"_id": "19464", "text": "Derivative of Exponential of Function :$\\map {\\dfrac \\d {\\d x} } {e^u} = e^u \\dfrac {\\d u} {\\d x}$"}
{"_id": "10526", "text": "Taxicab Metric on Real Vector Space is Metric The taxicab metric on the real vector space $\\R^n$ is a metric."}
{"_id": "5415", "text": "Commutative B-Algebra is Entropic Structure Let $\\left({G, *}\\right)$ be a commutative $B$-algebra. Then $\\left({G, *}\\right)$ is an entropic structure."}
{"_id": "14449", "text": "Substitution in Big-O Estimate/General Result Let $X$ and $Y$ be topological spaces. Let $V$ be a normed vector space over $\\R$ or $\\C$ with norm $\\norm {\\,\\cdot\\,}$. Let $x_0 \\in X$ and $y_0 \\in Y$. Let $f: X \\to Y$ be a function with $\\map f {x_0} = y_0$ that is continuous at $x_0$. Let $g, h: Y \\to V$ be functions. Suppose $\\map g y = \\map O {\\map h y}$ as $y \\to y_0$, where $O$ denotes big-O notation. Then $\\map {\\paren {g \\circ f} } x = \\map O {\\map {\\paren {h \\circ f} } x}$ as $x \\to x_0$."}
{"_id": "5852", "text": "Pointwise Multiplication on Complex-Valued Functions is Associative Let $f, g, h: S \\to \\C$ be complex-valued functions. Let $f \\times g: S \\to \\C$ denote the pointwise product of $f$ and $g$. Then: :$\\paren {f \\times g} \\times h = f \\times \\paren {g \\times h}$"}
{"_id": "11548", "text": "First Order ODE/(x^2 - 2 y^2) dx + x y dy = 0 is a homogeneous differential equation with solution: :$y^2 = x^2 + C x^4$"}
{"_id": "11299", "text": "Relative Prime Modulo Tensor is Zero Let $p \\in \\Z_{>0}$ and $q \\in \\Z_{>0}$ be positive coprime integers. Let $\\Z / p \\Z$ and $\\Z / q \\Z$ be $\\Z$-modules. {{explain|It is not a good idea to use the same notation for both a ring and a module. Either $\\Z / p \\Z$ is a ring or it is a module. Please consider taking the advice in the explain template at the bottom of this page.}} Then: :$\\Z / p \\Z \\otimes_\\Z \\Z / q\\Z = 0$ where $\\otimes_\\Z$ denotes tensor product over integers."}
{"_id": "1578", "text": "Congruence of Sum with Constant Let $a, b, z \\in \\R$. Let $a$ be congruent to $b$ modulo $z$: : $a \\equiv b \\pmod z$ Then: :$\\forall c \\in \\R: a + c \\equiv b + c \\pmod z$"}
{"_id": "5699", "text": "Infinite Ramsey's Theorem implies Finite Ramsey's Theorem :$\\forall l, n, r \\in \\N: \\exists m \\in \\N: m \\to \\left({l}\\right)_r^n$ where $\\alpha \\to \\left({\\beta}\\right)^n_r$ means that: :for any assignment of $r$-colors to the $n$-subsets of $\\alpha$ ::there is a particular color $\\gamma$ and a subset $X$ of $\\alpha$ of size $\\beta$ such that all $n$-subsets of $X$ are $\\gamma$."}
{"_id": "5979", "text": "Equivalent Subobjects have Isomorphic Domains Let $\\mathbf C$ be a metacategory. Let $C$ be an object of $\\mathbf C$. Let $\\mathbf{Sub}_{\\mathbf C} \\left({C}\\right)$ be the category of subobjects of $C$. Let $m, m'$ be equivalent subobjects of $C$. Then there exists an isomorphism $f: m \\to m'$."}
{"_id": "15092", "text": "Existence of Regular Space which is not Tychonoff There exists at least one example of a topological space which is a regular space, but is not also a Tychonoff space."}
{"_id": "8470", "text": "Element in Image of Preimage under Mapping Let $f: S \\to T$ be a mapping. Then: :$\\forall y \\in T: \\in f \\sqbrk {f^{-1} \\sqbrk y} = \\set y$"}
{"_id": "19362", "text": "Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Secant of p x :$\\displaystyle \\int_0^\\infty \\frac {e^{-a x} - e^{-b x} } {x \\sec p x} \\rd x = \\frac 1 2 \\map \\ln {\\frac {b^2 + p^2} {a^2 + p^2} }$"}
{"_id": "9806", "text": "Primitive of Cube of Cosecant of a x :$\\ds \\int \\csc^3 a x \\rd x = \\frac {-\\csc a x \\cot a x} {2 a} + \\frac 1 {2 a} \\ln \\size {\\tan \\dfrac {a x} 2} + C$"}
{"_id": "18896", "text": "Complement of Complete Graph is Edgeless Graph Let $K_n$ denote the complete graph of order $n$. Then the complement of $K_n$ is the $n$-edgeless graph $N_n$."}
{"_id": "12970", "text": "Anning's Theorem In any base greater than $1$, the fraction: :$\\dfrac {101 \\, 010 \\, 101} {110 \\, 010 \\, 011}$ has the property that if the two $1$'s in the center of the numerator and the denominator are replaced by the same odd number of $1$'s, the value of the fraction remains the same. For example: :$\\dfrac {101 \\, 010 \\, 101} {110 \\, 010 \\, 011} = \\dfrac {1 \\, 010 \\, 111 \\, 110 \\, 101} {1 \\, 100 \\, 111 \\, 110 \\, 011} = \\dfrac {9091} {9901}$ (in base $10$)."}
{"_id": "14837", "text": "Summation of Zero/Set Let $S$ be a set. Let $0: S \\to \\mathbb A$ be the zero mapping. Then the summation with finite support of $0$ over $S$ equals zero: :$\\displaystyle \\sum_{s \\mathop \\in S} \\map 0 s = 0$"}
{"_id": "3061", "text": "Subset of Indiscrete Space is Everywhere Dense Let $T = \\struct {S, \\set {\\O, S} }$ be an indiscrete topological space. Let $H \\subseteq S$ such that $H \\ne \\O$. Then $H$ is everywhere dense."}
{"_id": "8787", "text": "Inverse Cotangent of Imaginary Number :$\\cot^{-1} \\paren {i x} = - i \\coth^{-1} x$"}
{"_id": "9326", "text": "Primitive of Power of Root of a x + b :$\\displaystyle \\int \\left({\\sqrt{a x + b} }\\right)^m \\ \\mathrm d x = \\frac {2 \\left({\\sqrt{a x + b} }\\right)^{m + 2} } {a \\left({m + 2}\\right)} + C$"}
{"_id": "2527", "text": "Transpose of Linear Transformation is a Linear Transformation Let $R$ be a commutative ring. Let $G$ and $H$ be $R$-modules. Let $G^*$ and $H^*$ be the algebraic duals of $G$ and $H$ respectively. Let $\\map {\\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$. Let $u \\in \\map {\\LL_R} {G, H}$. Let $u^t: H^* \\to G^*$ be the transpose of $u$. Then $u^t: H^* \\to G^*$ is itself a linear transformation."}
{"_id": "17540", "text": "Convergence of P-Series/Divergence if p between 0 and 1 Let $0 < \\map \\Re p \\le 1$. Then the $p$-series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty n^{-p}$ diverges."}
{"_id": "15917", "text": "Distance between Two Parallel Straight Lines is Everywhere the Same Let $AB$ and $CD$ be parallel straight lines. Let perpendiculars $EF$ and $GH$ be drawn from $AB$ to $CD$, where $E, G$ are on $AB$ and $F, H$ are on $CD$. Then $EF = GH$. That is, the distance between  $AB$ and $CD$ is the same everywhere along their length."}
{"_id": "11230", "text": "Union Distributes over Intersection/Family of Sets/Corollary Let $I$ and $J$ be indexing sets. Let $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$ and $\\family {B_\\beta}_{\\beta \\mathop \\in J}$ be indexed families of subsets of a set $S$. Then: :$\\displaystyle \\bigcap_{\\tuple{\\alpha, \\beta} \\mathop \\in I \\times J} \\paren {A_\\alpha \\cup B_\\beta} = \\paren {\\bigcap_{\\alpha \\mathop \\in I} A_\\alpha} \\cup \\paren {\\bigcap_{\\beta \\mathop \\in J} B_\\beta}$ where $\\displaystyle \\bigcap_{\\alpha \\mathop \\in I} A_\\alpha$ denotes the intersection of $\\family {A_\\alpha}_{\\alpha \\mathop \\in I}$."}
{"_id": "4780", "text": "Order Topology on Natural Numbers is Discrete Topology Let $\\le$ be the standard ordering on the natural numbers $\\N$. Then the order topology $\\tau$ on $\\N$ is the discrete topology."}
{"_id": "19148", "text": "Set with Two Parallel Elements is Dependent Let $M = \\struct{S, \\mathscr I}$ be a matroid. Let $A \\subseteq S$. Let $x, y \\in S$. Let $x, y$ be parallel elements. If $x, y \\in A$ then $A$ is dependent."}
{"_id": "5262", "text": "Klein Four-Group as Order 2 Matrices Let $G$ be the set of order $2$ square matrices: :$G = \\set {I, A, B, C}$ where: :$I = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}, \\quad A = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}, \\quad B = \\begin{bmatrix} -1 & 0 \\\\ 0 & 1 \\end{bmatrix}, \\quad C = \\begin{bmatrix} -1 & 0 \\\\ 0 & -1 \\end{bmatrix}$ Then the algebraic structure $\\struct {G, \\times}$, where $\\times$ denotes (conventional) matrix multiplication, forms the Klein four-group."}
{"_id": "6737", "text": "Conjunction Equivalent to Negation of Implication of Negative/Formulation 1/Proof :$p \\land q \\dashv \\vdash \\neg \\left({p \\implies \\neg q}\\right)$"}
{"_id": "2359", "text": "Leibniz's Formula for Pi :$\\dfrac \\pi 4 = 1 - \\dfrac 1 3 + \\dfrac 1 5 - \\dfrac 1 7 + \\dfrac 1 9 - \\cdots \\approx 0 \\cdotp 78539 \\, 81633 \\, 9744 \\ldots$"}
{"_id": "5960", "text": "Indiscrete Space is Hereditarily Compact Let $\\struct {S, \\tau}$ be an indiscrete topological space. Then $\\struct {S, \\tau}$ is hereditarily compact."}
{"_id": "9692", "text": "Primitive of Power of x by Sine of a x :$\\displaystyle \\int x^m \\sin a x \\rd x = \\frac {-x^m \\cos a x} a + \\frac {m x^{m - 1} \\sin a x} {a^2} - \\frac {m \\paren {m - 1} } {a^2} \\int x^{m - 2} \\sin a x \\rd x$"}
{"_id": "18682", "text": "Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element Every bounded subset of $N$ has a greatest element."}
{"_id": "14735", "text": "Pentagonal and Hexagonal Numbers The sequence of positive integers which are simultaneously pentagonal and hexagonal begins: :$1, 40 \\, 755, 1 \\, 533 \\, 776 \\, 805, 57 \\, 722 \\, 156 \\, 241 \\, 751, \\ldots$ {{OEIS|A046180}}"}
{"_id": "566", "text": "Principle of Counting Let $T$ be a set such that $T \\sim \\N_n$. Then: : $\\forall m \\in \\N: n \\ne m \\implies T \\nsim \\N_m$"}
{"_id": "7610", "text": "Hilbert's Basis Theorem/Corollary Let $A$ be a Noetherian ring. Let $n \\ge 1$ be an integer. Let $A \\sqbrk {x_1, \\ldots, x_n}$ be the ring of polynomial forms over $A$ in the indeterminates $x_1, \\ldots, x_n$. Then $A \\sqbrk {x_1, \\ldots, x_n}$ is also a Noetherian ring."}
{"_id": "6129", "text": "Existence and Uniqueness of Direct Limit of Sequence of Groups Let $\\sequence {G_n}_{n \\mathop \\in \\N}$ be a sequence of groups. Let $\\sequence {g_n}_{n \\mathop \\in \\N}: g_n: G_n \\to G_{n + 1}$ be a sequence of group homomorphisms. Then their direct limit $G_\\infty$ exists and is unique up to unique isomorphism."}
{"_id": "17473", "text": "Integral to Infinity of Dirac Delta Function Let $\\map \\delta x$ denote the Dirac delta function. Then: :$\\displaystyle \\int_0^{+ \\infty} \\map \\delta x \\rd x = 1$"}
{"_id": "1236", "text": "Condition for Continuity on Interval Let $f$ be a real function defined on an interval $\\mathbb I$. Then $f$ is continuous on $\\mathbb I$ {{iff}}: :$\\forall x \\in \\mathbb I: \\forall \\epsilon > 0: \\exists \\delta > 0: y \\in \\mathbb I \\land \\size {x - y} < \\delta \\implies \\size {\\map f x - \\map f y} < \\epsilon$"}
{"_id": "13952", "text": "Compact Convergence Implies Local Uniform Convergence if Weakly Locally Compact Let $T = \\left({S, \\tau}\\right)$ be a weakly locally compact topological space. Let $M = \\left({A, d}\\right)$ be a metric space. Let $\\left\\langle{f_n}\\right\\rangle$ be a sequence of mappings $f_n : X\\to M$. Let $f_n$ converge compactly to $f:X\\to M$. Then $f_n$ converges locally uniformly to $f$."}
{"_id": "17508", "text": "Gamma Function of Minus 5 over 2 :$\\map \\Gamma {-\\dfrac 5 2} = -\\dfrac {8 \\sqrt \\pi} {15}$"}
{"_id": "13188", "text": "Rational Power is of Exponential Order Epsilon Let $r = \\dfrac p q$ be a rational number, with $p, q \\in \\Z: q \\ne 0, r > 0$. Then:  :$t \\mapsto t^r$ is of exponential order $\\epsilon$ for any  $\\epsilon > 0$ arbitrarily small in magnitude."}
{"_id": "3323", "text": "Countable Complement Space is Connected Let $T = \\struct {S, \\tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is a connected space."}
{"_id": "13038", "text": "Power of 2 is Almost Perfect Let $n \\in \\Z_{>0}$ be a power of $2$: :$n = 2^k$ for some $k \\in \\Z_{>0}$. Then $n$ is almost perfect."}
{"_id": "7347", "text": "Cardinality of Finite Set is Well-Defined Let $S$ be a finite set. Then there is a unique natural number $n$ such that $S \\sim \\N_n$, where: :$\\sim$ represents set equivalence and: :$\\N_n = \\set {0, 1, \\dotsc, n - 1}$ is the initial segment of $\\N$ determined by $n$."}
{"_id": "11499", "text": "Temperature of Body under Newton's Law of Cooling Let $B$ be a body in an environment whose ambient temperature is $H_a$. Let $H$ be the temperature of $B$ at time $t$. Let $H_0$ be the temperature of $B$ at time $t = 0$. Then: :$H = H_a - \\paren {H_0 - H_a} e^{-k t}$ where $k$ is some positive constant."}
{"_id": "12478", "text": "Floor Function/Examples/Floor of Root 2 :$\\floor {\\sqrt 2} = 1$"}
{"_id": "8368", "text": "Group Action on Subgroup by Left Regular Representation Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $*: H \\times G \\to G$ be the operation defined as: :$\\forall \\left({h, g}\\right) \\in H \\times G: h * g = \\lambda_h \\left({g}\\right)$ where $\\lambda_h \\left({g}\\right)$ is the left regular representation of $g$ by $h$. Then $*$ is a group action."}
{"_id": "7494", "text": "Composition of Affine Transformations is Affine Transformation Let $\\EE$, $\\FF$ and $\\GG$ be affine spaces with difference spaces $E$, $F$ and $G$ respectively. Let $\\LL: \\EE \\to \\FF$ and $\\MM: \\FF \\to \\GG$ be affine transformations. Let $L$ and $M$ be the tangent maps of $\\LL$ and $\\MM$ respectively. Then the composition $\\MM \\circ \\LL: \\EE \\to \\FF$ is an affine transformation with tangent map $M \\circ L$."}
{"_id": "9673", "text": "Derivative of Periodic Function Let $f: \\R \\to \\R$ be a real function. Let $f$ be differentiable on all of $\\R$. Then if $f$ is periodic with period $L$, then its derivative is also periodic with period $L$."}
{"_id": "17474", "text": "Integral to Infinity of Dirac Delta Function by Continuous Function Let $\\map \\delta x$ denote the Dirac delta function. Let $g$ be a continuous real function. Then: :$\\displaystyle \\int_0^{+ \\infty} \\map \\delta x \\, \\map g x \\rd x = \\map g 0$"}
{"_id": "2220", "text": "Diagonal Relation is Many-to-One The diagonal relation is many-to-one. That is: :$\\forall x \\in \\Dom {\\Delta_S}: \\tuple {x, y_1} \\in \\Delta_S \\land \\tuple {x, y_2} \\in \\Delta_S \\implies y_1 = y_2$ where $\\Delta_S$ is the diagonal relation on a set $S$."}
{"_id": "2326", "text": "Zero of Power Set with Union Let $S$ be a set and let $\\powerset S$ be its power set. Consider the algebraic structure $\\struct {\\powerset S, \\cup}$, where $\\cup$ denotes set union. Then $S$ serves as the zero element for $\\struct {\\powerset S, \\cup}$."}
{"_id": "11340", "text": "Characterization of Derivative by Local Basis Let $T = \\struct {S, \\tau}$ be a topological space. Let $A$ be a subset of $S$. Let $x$ be a point of $T$. Let $\\BB \\subseteq \\tau$ be a local basis at $x$. Then :$x \\in A'$ {{iff}}: :for every $U \\in \\BB$, there exists a point $y$ of $T$ such that $y \\in A \\cap U$ and $x \\ne y$ where: :$A'$ denotes the derivative of $A$."}
{"_id": "2547", "text": "Triangular Matrices forms Subring of Square Matrices Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\map {\\mathcal M_R} n$ be the order $n$ square matrix space over a ring $R$. Let $\\struct {\\map {\\mathcal M_R} n, +, \\times}$ denote the ring of square matrices of order $n$ over $R$. Let $\\map {U_R} n$ be the set of upper triangular matrices of order $n$ over $R$. Then $\\map {U_R} n$ forms a subring of $\\struct {\\map {\\mathcal M_R} n, +, \\times}$. Similarly, let $\\map {L_R} n$ be the set of lower triangular matrices of order $n$ over $R$. Then $\\map {L_R} n$ forms a subring of $\\struct {\\map {\\mathcal M_R} n, +, \\times}$."}
{"_id": "9163", "text": "Laplace Transform of Identity Mapping Let $\\laptrans f$ denote the Laplace transform of a function $f$. Let $\\map {I_\\R} t$ denote the identity mapping on $\\R$ for $t > 0$. Then: :$\\laptrans {\\map {I_\\R} t} = \\dfrac 1 {s^2}$ for $\\map \\Re s > 0$."}
{"_id": "11154", "text": "Linear Combination of Mellin Transforms Let $\\mathcal M$ be the Mellin transform. Let $f \\left({t}\\right)$, $g \\left({t}\\right)$ be functions such that $\\mathcal M \\left\\{ {f \\left({t}\\right)}\\right\\} \\left({s}\\right)$ and $\\mathcal M \\left\\{ {f \\left({t}\\right)}\\right\\} \\left({s}\\right)$ exist. Let $\\lambda \\in \\C$ be a constant. Then: :$\\mathcal M \\left\\{ {\\lambda f \\left({t}\\right) + g \\left({t}\\right)}\\right\\} \\left({s}\\right) = \\lambda \\mathcal M \\left\\{ {f \\left({t}\\right)}\\right\\}\\left({s}\\right) + \\mathcal M \\left\\{ {g \\left({t}\\right)}\\right\\} \\left({s}\\right)$ everywhere all the above expressions are defined."}
{"_id": "12013", "text": "Ordering on Mappings Implies Galois Connection Let $\\left({S, \\preceq}\\right)$, $\\left({T, \\precsim}\\right)$ be ordered sets. Let $g: S \\to T$ and $d: T \\to S$ be mappings such that: :$g$ and $d$ are increasing mappings  and :$d \\circ g \\preceq I_S$ and $I_T \\precsim g \\circ d$ Then :$\\left({g, d}\\right)$ is Galois connection. where :$\\preceq, \\precsim$ denote the orderings on mappings :$I_S$ denotes the identity mapping of $S$ :$\\circ$ denotes the composition of mappings."}
{"_id": "17737", "text": "Product of Divisors is Divisor of Product Let $a, b, c, d \\in \\Z$ be integers such that $a, c \\ne 0$. Let $a \\divides b$ and $c \\divides d$, where $\\divides$ denotes divisibility. Then: :$a c \\divides b d$"}
{"_id": "10586", "text": "Metric Space Completeness is not Preserved by Homeomorphism Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $\\phi: M_1 \\to M_2$ be a homeomorphism. If $M_1$ is complete then it is not necessarily the case that so is $M_2$."}
{"_id": "19285", "text": "Effect of Elementary Column Operations on Determinant Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ denote the determinant of $\\mathbf A$. Take the elementary column operations: {{begin-axiom}} {{axiom | n = \\text {ECO} 1         | t = For some $\\lambda$, multiply column $i$ by $\\lambda$         | m = \\kappa_i \\to \\lambda \\kappa_i }} {{axiom | n = \\text {ECO} 2         | t = For some $\\lambda$, add $\\lambda$ times column $j$ to column $i$         | m = \\kappa_i \\to \\kappa_i + \\lambda \\kappa_j }} {{axiom | n = \\text {ECO} 3         | t = Exchange columns $i$ and $j$         | m = \\kappa_i \\leftrightarrow \\kappa_j }} {{end-axiom}} Applying $\\text {ECO} 1$ has the effect of multiplying $\\map \\det {\\mathbf A}$ by $\\lambda$. Applying $\\text {ECO} 2$ has no effect on $\\map \\det {\\mathbf A}$. Applying $\\text {ECO} 3$ has the effect of multiplying $\\map \\det {\\mathbf A}$ by $-1$."}
{"_id": "7165", "text": "Biconditional as Disjunction of Conjunctions/Formulation 2/Proof 2 : $\\vdash \\left({p \\iff q}\\right) \\iff \\left({\\left({p \\land q}\\right) \\lor \\left({\\neg p \\land \\neg q}\\right)}\\right)$"}
{"_id": "16950", "text": "Join of Sets of Integer Multiples is Set of Integer Multiples of GCD Let $m, n \\in \\Z$. Let $m \\Z$ denote the set of integer multiples of $m$ Let $r \\in \\Z$ such that: :$m \\Z \\subseteq r \\Z$ and: :$n \\Z \\subseteq r \\Z$ Then: :$\\gcd \\set {m, n} \\Z \\subseteq r \\Z$ where $\\gcd$ denotes greatest common divisor."}
{"_id": "16827", "text": "Euler's Equation/Integrated wrt Length Element Let $y$ be a real mapping belonging to $C^2$ differentiability class. Assume that: :$\\ds J \\sqbrk y = \\int_a^b \\map f {x, y, y'} \\rd s$ where :$\\rd s = \\sqrt {1 + y'^2} \\rd x$ Then Euler's Equation can be reduced to: :$f_y - f_x y' - f_{y'} y' y'' - f \\dfrac {y''} {\\paren {1 + y'^2}^{\\frac 3 2} } = 0$"}
{"_id": "5294", "text": "Well-Ordered Induction Let $\\left({A,\\prec}\\right)$ be a strict well-ordering. For all $x \\in A$, let the $\\prec$-initial segment of $x$ be a small class. Let $B$ be a class such that $B \\subseteq A$. Let: :$(1): \\quad \\forall x \\in A: \\left({ \\left({ A \\cap \\prec^{-1} \\left({ x }\\right) }\\right) \\subseteq B \\implies x \\in B }\\right)$ Then: :$A = B$ That is, if a property passes from the initial segment of $x$ to $x$, then this property is true for all $x \\in A$."}
{"_id": "18365", "text": "Regiomontanus' Angle Maximization Problem Let $AB$ be a line segment. Let $AB$ be produced to $P$. Let $PQ$ be constructed perpendicular to $AB$. Then the angle $AQB$ is greatest when $PQ$ is tangent to a circle passing through $A$, $B$ and $Q$."}
{"_id": "4892", "text": "Zero Vector is Unique Let $\\struct {\\mathbf V, +, \\circ}_{\\mathbb F}$ be a vector space over $\\mathbb F$, as defined by the vector space axioms. Then the zero vector in $\\mathbf V$ is unique: :$\\exists! \\mathbf 0 \\in \\mathbf V: \\forall \\mathbf x \\in \\mathbf V: \\mathbf x + \\mathbf 0 = \\mathbf x$"}
{"_id": "3263", "text": "Either-Or Topology is not Separable Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is not a separable space."}
{"_id": "2984", "text": "Range of Characters Let $G$ be a finite abelian group of order $m$. Let $\\chi: G \\to \\C^\\times$ be a character on $G$. Then for any $g \\in G$, $\\map \\chi g$ is an $m$th root of unity. If $e$ is the identity of $G$ then $\\map \\chi g = 1$."}
{"_id": "5328", "text": "Lexicographic Order Initial Segments Let $\\operatorname{Le}$ denote the lexicographic order for the set $\\left({\\operatorname{On} \\times \\operatorname{On} }\\right)$. Let the ordinal number $1$ denote the successor of $\\varnothing$. Then the initial segment of $ \\left({1, \\varnothing}\\right)$ with respect to the lexicographic order $\\operatorname{Le}$ is a proper class. This initial segment shall be denoted $\\left({\\operatorname{On} \\times \\operatorname{On} }\\right)_{\\left({1, \\varnothing}\\right)}$. {{explain|The connection between these two statements is not clear.}}"}
{"_id": "15226", "text": "Limit Points of Sequence in Indiscrete Space on Uncountable Set Let $S$ be an uncountable set. Let $T = \\struct {S, \\set {\\O, S} }$ be the indiscrete topological space on $S$. Let $\\sequence {s_n}$ be a sequence in $T$. Then every sequence in $T$ has an uncountable number of limit points."}
{"_id": "123", "text": "Equivalence of Definitions of Symmetric Difference Let $S$ and $T$ be sets. {{TFAE|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}"}
{"_id": "10480", "text": "Open Real Interval is Subset of Closed Real Interval Let $a, b \\in \\R$ be real numbers. Then: :$\\openint a b \\subseteq \\closedint a b$ where: :$\\openint a b$ is the open interval between $a$ and $b$ :$\\closedint a b$ is the closed interval between $a$ and $b$."}
{"_id": "4739", "text": "Relation Isomorphism Preserves Equivalence Relations Let $\\struct {S, \\RR_1}$ and $\\struct {T, \\RR_2}$ be relational structures. Let $\\struct {S, \\RR_1}$ and $\\struct {T, \\RR_2}$ be (relationally) isomorphic. Then $\\RR_1$ is an equivalence relation {{iff}} $\\RR_2$ is also an equivalence relation."}
{"_id": "12570", "text": "Sum over k of -2 Choose k :$\\displaystyle \\sum_{k \\mathop = 0}^n \\binom {-2} k = \\left({-1}\\right)^n \\left\\lceil {\\dfrac {n + 1} 2}\\right\\rceil$ where: :$\\dbinom {-2} k$ is a binomial coefficient :$\\left\\lceil {x}\\right\\rceil$ denotes the ceiling of $x$."}
{"_id": "7814", "text": "Composite of Epimorphisms is Epimorphism Let: :$\\struct {S_1, \\circ_1, \\circ_2, \\ldots, \\circ_n}$ :$\\struct {S_2, *_1, *_2, \\ldots, *_n}$ :$\\struct {S_3, \\oplus_1, \\oplus_2, \\ldots, \\oplus_n}$ be algebraic structures. Let: :$\\phi: \\struct {S_1, \\circ_1, \\circ_2, \\ldots, \\circ_n} \\to \\struct {S_2, *_1, *_2, \\ldots, *_n}$ :$\\psi: \\struct {S_2, *_1, *_2, \\ldots, *_n} \\to \\struct {S_3, \\oplus_1, \\oplus_2, \\ldots, \\oplus_n}$ be epimorphisms. Then the composite of $\\phi$ and $\\psi$ is also an epimorphism."}
{"_id": "4279", "text": "Axiom of Choice Implies Zorn's Lemma Acceptance of the Axiom of Choice implies the truth of Zorn's Lemma."}
{"_id": "7397", "text": "Norm of Eisenstein Integer Let $\\alpha$ be an Eisenstein integer. That is, $\\alpha = a + b \\omega$ for some $a, b \\in \\Z$, where $\\omega = e^{2\\pi i /3}$. Then: :$\\cmod \\alpha^2 = a^2 - a b + b^2$ where $\\cmod {\\, \\cdot \\,}$ denotes the modulus of a complex number."}
{"_id": "8893", "text": "Conditions on Rational Solution to Polynomial Equation Let $P$ be the polynomial equation: :$a_n z^n + a_{n - 1} z^{n - 1} + \\cdots + a_1 z + a_0 = 0$ where $a_0, \\ldots, a_n$ are integers. Let $\\dfrac p q$ be a root of $P$ expressed in canonical form. Then $p$ is a divisor of $a_0$ and $q$ is a divisor of $a_n$."}
{"_id": "15681", "text": "Negated Upper Index of Binomial Coefficient/Complex Numbers For all $z, w \\in \\C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers: :$\\dbinom z w = \\dfrac {\\sin \\left({\\pi \\left({w - z - 1}\\right)}\\right)} {\\sin \\left({\\pi z}\\right)} \\dbinom {w - z - 1} w$ where $\\dbinom z w$ is a binomial coefficient."}
{"_id": "18097", "text": "Hamiltonian of Standard Lagrangian is Total Energy Let $P$ be a physical system of classical particles. Let $L$ be a standard Lagrangian associated with $P$. Then the Hamiltonian of $P$ is the total energy of $P$."}
{"_id": "12115", "text": "Normal to Cycloid passes through Bottom of Generating Circle Let $C$ be a cycloid generated by the equations: : $x = a \\left({\\theta - \\sin \\theta}\\right)$ : $y = a \\left({1 - \\cos \\theta}\\right)$ Then the normal to $C$ at a point $P$ on $C$ passes through the bottom of the generating circle of $C$."}
{"_id": "15725", "text": "Argument of x to the n Equals n Times The Argument Let $z$ be a complex number. Then: :$\\forall n \\in \\N_{>0}: \\map \\arg {z^n} = n \\map \\arg z$"}
{"_id": "18220", "text": "Permutation is Product of Transpositions Let $S_n$ denote the symmetric group on $n$ letters. Every element of $S_n$ can be expressed as a product of transpositions."}
{"_id": "5031", "text": "Idempotent Elements for Integer Multiplication There are exactly two integers which are idempotent with respect to multiplication: :$0 \\times 0 = 0$ :$1 \\times 1 = 1$"}
{"_id": "3724", "text": "Zero Vector is Linearly Dependent Let $G$ be a group whose identity is $e$. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {G, +_G, \\circ}_R$ be a unitary $R$-module. Then the singleton set $\\set e$ consisting of the zero vector is linearly dependent."}
{"_id": "6782", "text": "Power Set with Union and Intersection forms Boolean Algebra Let $S$ be a set, and let $\\powerset S$ be its power set. Denote with $\\cup$, $\\cap$ and $\\complement$ the operations of union, intersection and complement on $\\powerset S$, respectively. Then $\\struct {\\powerset S, \\cup, \\cap, \\complement}$ is a Boolean algebra."}
{"_id": "17997", "text": "Composite of Surjection on Injection is not necessarily Either Let $f$ be an injection. Let $g$ be a surjection. Let $g \\circ f$ denote the composition of $g$ with $f$. Then it is not necessarily the case that $g \\circ f$ is either a surjection or an injection."}
{"_id": "540", "text": "Diagonal Relation is Universally Congruent The diagonal relation $\\Delta_S$ on a set $S$ is universally congruent with every operation on $S$."}
{"_id": "6179", "text": "Discrete Space is Totally Disconnected Let $T = \\struct {S, \\tau}$ be a topological space where $\\tau$ is the discrete topology on $S$. Then $T$ is totally disconnected."}
{"_id": "19219", "text": "Sine of Integer Multiple of Argument/Formulation 2 {{begin-eqn}} {{eqn | l = \\sin n \\theta       | r = \\cos^n \\theta \\paren {\\paren {\\tan \\theta} - \\dbinom n 3 \\paren {\\tan \\theta}^3 + \\dbinom n 5 \\paren {\\tan \\theta}^5 - \\cdots}       | c =  }} {{eqn | r = \\cos^n \\theta \\sum_{k \\mathop \\ge 0} \\paren {-1}^k \\dbinom n {2 k + 1} \\paren {\\tan^{2 k + 1} \\theta}       | c =  }} {{end-eqn}}"}
{"_id": "15612", "text": "Taylor Series of Logarithm of Gamma Function Let $\\gamma$ denote the Euler-Mascheroni constant. Let $\\map \\zeta s$ denote the Riemann zeta function. Let $\\map \\Gamma z$ denote the gamma function. Let $\\Log$ denote the natural logarithm. Then $\\map \\Log {\\map \\Gamma z}$ has the power series expansion: {{begin-eqn}} {{eqn | l = \\map \\Log {\\map \\Gamma z}       | r = -\\map \\gamma {z - 1} + \\sum_{k \\mathop = 2}^\\infty \\frac {\\paren {-1}^k \\map \\zeta k} k \\paren {z - 1}^k  }} {{end-eqn}} which is valid for all $z \\in \\C$ such that $\\cmod {z - 1} < 1$."}
{"_id": "11355", "text": "Space is First-Countable iff Character not greater than Aleph 0 Let $T$ be a topological space. $T$ is first-countable {{iff}}: :$\\chi \\left({T}\\right) \\leq \\aleph_0$ where $\\chi \\left({T}\\right)$ denotes the character of $T$."}
{"_id": "9615", "text": "Primitive of Reciprocal of x squared by x fourth plus a fourth :$\\displaystyle \\int \\frac {\\mathrm d x} {x^2 \\left({x^4 + a^4}\\right)} = \\frac {-1} {a^4 x} - \\frac {-1} {4 a^5 \\sqrt 2} \\ln \\left({\\frac {x^2 - a x \\sqrt 2 + a^2} {x^2 + a x \\sqrt 2 + a^2} }\\right) + \\frac 1 {2 a^5 \\sqrt 2} \\left({\\arctan \\left({1 - \\frac {x \\sqrt 2} a}\\right) - \\arctan \\left({1 + \\frac {x \\sqrt 2} a}\\right)}\\right)$"}
{"_id": "2144", "text": "Condition for Independence of Discrete Random Variables Let $\\struct {\\Omega, \\Sigma, \\Pr}$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\\struct {\\Omega, \\Sigma, \\Pr}$. Then $X$ and $Y$ are independent {{iff}} there exist functions $f, g: \\R \\to \\R$ such that the joint mass function of $X$ and $Y$ satisfies: :$\\forall x, y \\in \\R: \\map {p_{X, Y} } {x, y} = \\map f x \\map g y$"}
{"_id": "8275", "text": "Integers under Multiplication do not form Group The set of integers under multiplication $\\struct {\\Z, \\times}$ does not form a group."}
{"_id": "1384", "text": "Finite Union of Bounded Subsets Let $M = \\left({A, d}\\right)$ be a metric space. Then the union of any finite number of bounded subsets of $M$ is itself bounded."}
{"_id": "11549", "text": "First Order ODE/x^2 y' - 3 x y - 2 y^2 = 0 is a homogeneous differential equation with solution: :$y = C x^2 \\paren {x + y}$"}
{"_id": "2753", "text": "Intersection of Exteriors contains Exterior of Union Let $T$ be a topological space. Let $\\mathbb H$ be a set of subsets of $T$. That is, let $\\mathbb H \\subseteq \\powerset T$ where $\\powerset T$ is the power set of $T$. Then: :$\\displaystyle \\paren {\\bigcup_{H \\mathop \\in \\mathbb H} H}^e \\subseteq \\bigcap_{H \\mathop \\in \\mathbb H} H^e $ where $H^e$ denotes the exterior of $H$."}
{"_id": "11642", "text": "Cycloid has Tautochrone Property Consider a wire bent into the shape of an arc of a cycloid $C$ and inverted so that its cusps are uppermost and on the same horizontal line. Let a bead $B$ be released from some point on the wire. The time taken for $B$ to reach the lowest point of $C$ is: :$T = \\pi \\sqrt {\\dfrac a g}$ independently of the point at which $B$ is released from. That is, a cycloid is a tautochrone."}
{"_id": "13158", "text": "Ratio of 2016 to Aliquot Sum $2016$ has the property that its ratio to its aliquot sum is $4 : 9$."}
{"_id": "18641", "text": "Natural Number is Union of its Successor Let $n \\in \\N$ be a natural number as defined by the von Neumann construction. Then: :$\\map \\bigcup {n^+} = n$"}
{"_id": "5729", "text": "Smallest Positive Integer Combination is Greatest Common Divisor Let $a, b \\in \\Z_{>0}$ be (strictly) positive integers. Let $d \\in \\Z_{>0}$ be the smallest positive integer such that: : $d = a s + b t$ where $s, t \\in \\Z$. Then: :$(1): \\quad d \\divides a \\land d \\divides b$ :$(2): \\quad c \\divides a \\land c \\divides b \\implies c \\divides d$ where $\\divides$ denotes divisibility. That is, $d$ is the greatest common divisor of $a$ and $b$."}
{"_id": "16768", "text": "Infinite Number of Even Fermat Pseudoprimes Despite their relative rarity, there exist an infinite number of even Fermat pseudoprimes."}
{"_id": "15992", "text": "Fibonacci Number of Odd Index by Golden Mean Modulo 1 Let $n \\in \\Z$ be an integer. Then: :$F_{2 n + 1} \\phi \\bmod 1 = \\phi^{-2 n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number :$\\phi$ is the golden mean: $\\phi = \\dfrac {1 + \\sqrt 5} 2$"}
{"_id": "4759", "text": "Integral of Positive Simple Function Well-Defined Let $\\left({X, \\Sigma, \\mu}\\right)$ be a measure space. Let $f: X \\to \\R, f \\in \\mathcal{E}^+$ be a positive simple function. Then the $\\mu$-integral of $f$, $I_\\mu \\left({f}\\right)$, is well-defined. That is, for any two standard representations for $f$, say: :$\\displaystyle f = \\sum_{i \\mathop = 0}^n a_i \\chi_{E_i} = \\sum_{j \\mathop = 0}^m b_j \\chi_{F_j}$ it holds that: :$\\displaystyle \\sum_{i \\mathop = 0}^n a_i \\mu \\left({E_i}\\right) = \\sum_{j \\mathop = 0}^m b_j \\mu \\left({F_j}\\right)$"}
{"_id": "17317", "text": "Subrings of Integers are Sets of Integer Multiples/Examples/Even Integers Let $2 \\Z$ be the set of even integers. Then $\\struct {2 \\Z, +, \\times}$ is a subring of $\\struct {\\Z, +, \\times}$."}
{"_id": "11771", "text": "Linear Second Order ODE/y'' + 8 y = 0 The second order ODE: :$(1): \\quad y'' + 8 y = 0$ has the general solution: :$y = C_1 \\cos 2 \\sqrt 2 x + C_2 \\sin 2 \\sqrt 2 x$"}
{"_id": "9598", "text": "Primitive of Reciprocal of x by Half Integer Power of a x squared plus b x plus c Let $a \\in \\R_{\\ne 0}$. Then: :$\\displaystyle \\int \\frac {\\d x} {x \\paren {a x^2 + b x + c}^{n + \\frac 1 2} } = \\frac 1 {\\paren {2 n - 1} c \\paren {a x^2 + b x + c}^{n - \\frac 1 2} } + \\frac 1 c \\int \\frac {\\d x} {x \\paren {a x^2 + b x + c}^{n - \\frac 1 2} } - \\frac b {2 c} \\int \\frac {\\d x} {\\paren {a x^2 + b x + c}^{n + \\frac 1 2} }$"}
{"_id": "16178", "text": "Three Times Sum of Cubes of Three Indeterminates Plus 6 Times their Product :$3 \\paren {a^3 + b^3 + c^3 + 6 a b c} = \\paren {a + b + c}^3 + \\paren {a + b \\omega + c \\omega^2}^3 + \\paren {a + b \\omega^2 + c \\omega}^3$ where: : $\\omega = -\\dfrac 1 2 + \\dfrac {\\sqrt 3} 2$"}
{"_id": "5549", "text": "Morphism in Preorder Category is Epic Let $\\mathbf P$ be a preorder category. Let $f \\in \\mathbf P_1$ be a morphism. Then $f$ is epic."}
{"_id": "19586", "text": "Topologies on Set with More than One Element may not be Homeomorphic Let $S$ be a set which contains at least $2$ elements. Let $\\tau_1$ and $\\tau_2$ be topologies on $S$. Then it is not necessarily the case that $\\struct {S, \\tau_1}$ and $\\struct {S, \\tau_2}$ are homeomorphic."}
{"_id": "3267", "text": "Either-Or Topology is Scattered Let $T = \\struct {S, \\tau}$ be the either-or space. Then $T$ is a scattered space."}
{"_id": "3902", "text": "Modified Fort Space is not Zero Dimensional Let $T = \\struct {S, \\tau_{a, b} }$ be a modified Fort space. Then $T$ is not zero dimensional."}
{"_id": "10698", "text": "Equivalence of Definitions of Topologically Equivalent Metrics {{TFAE|def = Topologically Equivalent Metrics}} Let $M_1 = \\struct {A, d_1}$ and $M_2 = \\struct {A, d_2}$ be metric spaces on the same underlying set $A$."}