id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-21800 | Order of Hadamard Matrix | Let $\mathbf A$ be a Hadamard matrix.
Then the order of $\mathbf A$ must either be $2$ or an integer multiple of $4$. | Let $\mathbf A$ be of order $n$.
{{ProofWanted}} | Let $\mathbf A$ be a [[Definition:Hadamard Matrix|Hadamard matrix]].
Then the [[Definition:Order of Square Matrix|order]] of $\mathbf A$ must either be $2$ or an [[Definition:Integer Multiple|integer multiple]] of $4$. | Let $\mathbf A$ be of [[Definition:Order of Square Matrix|order]] $n$.
{{ProofWanted}} | Order of Hadamard Matrix | https://proofwiki.org/wiki/Order_of_Hadamard_Matrix | https://proofwiki.org/wiki/Order_of_Hadamard_Matrix | [
"Hadamard Matrices"
] | [
"Definition:Hadamard Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Integral Multiple/Real Numbers"
] | [
"Definition:Matrix/Square Matrix/Order"
] |
proofwiki-21801 | Riemann-Stieltjes Integral with Step Integrator | Let $a < c < b$ be real numbers.
Let $f$ be a real function that is bounded on $\closedint a b$.
Let $\alpha$ be a real function on $\closedint a b$ such that:
:$\forall x \in \hointr a c: \map \alpha x = \map \alpha a$
:$\forall x \in \hointl c b: \map \alpha x = \map \alpha b$
Suppose that:
:Either $f$ is left-contin... | Let $\epsilon > 0$ be arbitrary.
The construction of $P_\epsilon$ will depend on which of $f$ and $\alpha$ are continuous on which sides at $c$. | Let $a < c < b$ be [[Definition:Real Number|real numbers]].
Let $f$ be a [[Definition:Real Function|real function]] that is [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$.
Let $\alpha$ be a [[Definition:Real Function|real function]] on $\closedint a b$ such that:
:$\forall x \in \hointr a c: ... | Let $\epsilon > 0$ be arbitrary.
The construction of $P_\epsilon$ will depend on which of $f$ and $\alpha$ are [[Definition:One-Sided Continuity|continuous on which sides]] at $c$. | Riemann-Stieltjes Integral with Step Integrator | https://proofwiki.org/wiki/Riemann-Stieltjes_Integral_with_Step_Integrator | https://proofwiki.org/wiki/Riemann-Stieltjes_Integral_with_Step_Integrator | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Definition:Continuous Real Function/Left-Continuous",
"Definition:Continuous Real Function/Right-Continuous",
"Definition:Riemann-Stieltjes Integral"
] | [
"Definition:Continuous Real Function/One Side"
] |
proofwiki-21802 | Hankel Matrix is Symmetric | Let $\mathbf H$ be a Hankel matrix.
Then $\mathbf H$ is a symmetric matrix. | Recall the definition of symmetric matrix:
$\mathbf A$ is '''symmetric''' {{iff}}:
:$\mathbf A = \mathbf A^\intercal$
where $\mathbf A^\intercal$ is the transpose of $\mathbf A$.
By definition of transpose of $\mathbf A$:
:$\mathbf A^\intercal_{i j}: = \mathbf A_{j i}$
Recall the definition of Hankel matrix:
{{:Definit... | Let $\mathbf H$ be a [[Definition:Hankel Matrix|Hankel matrix]].
Then $\mathbf H$ is a [[Definition:Symmetric Matrix|symmetric matrix]]. | Recall the definition of [[Definition:Symmetric Matrix|symmetric matrix]]:
$\mathbf A$ is '''[[Definition:Symmetric Matrix|symmetric]]''' {{iff}}:
:$\mathbf A = \mathbf A^\intercal$
where $\mathbf A^\intercal$ is the [[Definition:Transpose of Matrix|transpose of $\mathbf A$]].
By definition of [[Definition:Transpose ... | Hankel Matrix is Symmetric | https://proofwiki.org/wiki/Hankel_Matrix_is_Symmetric | https://proofwiki.org/wiki/Hankel_Matrix_is_Symmetric | [
"Hankel Matrices",
"Symmetric Matrices"
] | [
"Definition:Hankel Matrix",
"Definition:Symmetric Matrix"
] | [
"Definition:Symmetric Matrix",
"Definition:Symmetric Matrix",
"Definition:Transpose of Matrix",
"Definition:Transpose of Matrix",
"Definition:Hankel Matrix",
"Definition:Matrix/Diagonal/Antidiagonal"
] |
proofwiki-21803 | Differential Equation governing Simple Harmonic Motion | Let $B$ be a body undergoing simple harmonic motion in a straight line.
Then its motion can be described using the equation:
:$\dfrac {\d^2 x} {\d x^2} = -\omega^2 x$ | Consider the equation governing simple harmonic motion:
:$(1): \quad x = A \map \cos {\omega t + \phi}$
Differentiating $2$ times {{WRT|Differentiation}} $t$:
{{begin-eqn}}
{{eqn | n = 2
| l = \dfrac {\d x} {\d t}
| r = -\omega A \map \sin {\omega t + \phi}
| c = Derivative of Cosine Function
}}
{{eqn... | Let $B$ be a [[Definition:Body|body]] undergoing [[Definition:Simple Harmonic Motion|simple harmonic motion]] in a [[Definition:Straight Line|straight line]].
Then its [[Definition:Motion|motion]] can be described using the [[Definition:Differential Equation|equation]]:
:$\dfrac {\d^2 x} {\d x^2} = -\omega^2 x$ | Consider the [[Definition:Equation|equation]] governing [[Definition:Simple Harmonic Motion|simple harmonic motion]]:
:$(1): \quad x = A \map \cos {\omega t + \phi}$
[[Definition:Differentiation|Differentiating]] $2$ times {{WRT|Differentiation}} $t$:
{{begin-eqn}}
{{eqn | n = 2
| l = \dfrac {\d x} {\d t}
... | Differential Equation governing Simple Harmonic Motion | https://proofwiki.org/wiki/Differential_Equation_governing_Simple_Harmonic_Motion | https://proofwiki.org/wiki/Differential_Equation_governing_Simple_Harmonic_Motion | [
"Simple Harmonic Motion",
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Body",
"Definition:Simple Harmonic Motion",
"Definition:Line/Straight Line",
"Definition:Motion",
"Definition:Differential Equation"
] | [
"Definition:Equation",
"Definition:Simple Harmonic Motion",
"Definition:Differentiation",
"Derivative of Cosine Function",
"Derivative of Sine Function",
"Definition:Differential Equation/Ordinary",
"Definition:Differential Equation/Order"
] |
proofwiki-21804 | Meet-Irreducible Open Set Induces Completely Prime Filter | Let $\struct{S, \tau}$ be a topological space.
Let:
:$W \in \tau : W \ne S : W$ is meet-irreducible
Let:
:$\FF = \set{U \in \tau : U \nsubseteq W}$.
Then:
:$\FF$ is a completely prime filter in the complete lattice $\struct{\tau, \subseteq}$ | === $\FF$ satisfies {{Ordered-set-filter-axiom|1}} ===
We have {{hypothesis}}:
:$W \ne S$
From {{Open-set-axiom|3}}:
:$S \in \tau$
By definition of $\FF$:
:$S \in \FF$
By definition of empty set:
:$\FF \ne \O$
It follows that $\FF$ satisfies {{Ordered-set-filter-axiom|1}}.
{{qed|lemma}} | Let $\struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let:
:$W \in \tau : W \ne S : W$ is [[Definition:Meet-Irreducible Open Set|meet-irreducible]]
Let:
:$\FF = \set{U \in \tau : U \nsubseteq W}$.
Then:
:$\FF$ is a [[Definition:Completely Prime Filter|completely prime filter]] in the [[D... | === $\FF$ satisfies {{Ordered-set-filter-axiom|1}} ===
We have {{hypothesis}}:
:$W \ne S$
From {{Open-set-axiom|3}}:
:$S \in \tau$
By definition of $\FF$:
:$S \in \FF$
By definition of [[Definition:Empty Set|empty set]]:
:$\FF \ne \O$
It follows that $\FF$ satisfies {{Ordered-set-filter-axiom|1}}.
{{qed|lemma}} | Meet-Irreducible Open Set Induces Completely Prime Filter | https://proofwiki.org/wiki/Meet-Irreducible_Open_Set_Induces_Completely_Prime_Filter | https://proofwiki.org/wiki/Meet-Irreducible_Open_Set_Induces_Completely_Prime_Filter | [
"Meet-Irreducible Open Sets",
"Completely Prime Filters"
] | [
"Definition:Topological Space",
"Definition:Meet-Irreducible Open Set",
"Definition:Completely Prime Filter",
"Definition:Complete Lattice"
] | [
"Definition:Empty Set"
] |
proofwiki-21805 | Completely Prime Filter Induces Meet-Irreducible Open Set | Let $\struct{S, \tau}$ be a topological space.
Let $\FF$ be a completely prime filter in the complete lattice $\struct{\tau, \subseteq}$.
Let $W = \bigcup \set{U \in \tau : U \notin \FF}$.
Then:
:$W$ is a meet-irreducible open set | By definition of completely prime filter:
:$\FF$ is a proper subset of $\tau$
Hence:
:$\set{U \in \tau : U \notin \FF} \ne \O$
By {{Open-set-axiom|1}}:
:$W \in \tau$
From Filter Contains Greatest Element:
:$S \in \FF$
By definition of completely prime filter:
:$W \notin \FF$
and so:
:$W \ne S$
Let $U_1, U_2 \in \tau$:
... | Let $\struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\FF$ be a [[Definition:Completely Prime Filter|completely prime filter]] in the [[Definition:Complete Lattice|complete lattice]] $\struct{\tau, \subseteq}$.
Let $W = \bigcup \set{U \in \tau : U \notin \FF}$.
Then:
:$W$ is a [[Def... | By definition of [[Definition:Completely Prime Filter|completely prime filter]]:
:$\FF$ is a [[Definition:Proper Subset|proper subset]] of $\tau$
Hence:
:$\set{U \in \tau : U \notin \FF} \ne \O$
By {{Open-set-axiom|1}}:
:$W \in \tau$
From [[Filter Contains Greatest Element]]:
:$S \in \FF$
By definition of [[Defi... | Completely Prime Filter Induces Meet-Irreducible Open Set | https://proofwiki.org/wiki/Completely_Prime_Filter_Induces_Meet-Irreducible_Open_Set | https://proofwiki.org/wiki/Completely_Prime_Filter_Induces_Meet-Irreducible_Open_Set | [
"Meet-Irreducible Open Sets",
"Completely Prime Filters"
] | [
"Definition:Topological Space",
"Definition:Completely Prime Filter",
"Definition:Complete Lattice",
"Definition:Meet-Irreducible Open Set"
] | [
"Definition:Completely Prime Filter",
"Definition:Proper Subset",
"Filter Contains Greatest Element",
"Definition:Completely Prime Filter",
"Definition:Contrapositive Statement",
"Definition:Contrapositive Statement",
"Characterization of Meet-Irreducible Open Set",
"Definition:Meet-Irreducible Open S... |
proofwiki-21806 | Fontené Theorems/Third | Let $\triangle ABC$ be a triangle.
Let $P$ be an arbitrary point in the plane of $\triangle ABC$.
Let the isogonal conjugate of $P$ {{WRT}} to $\triangle ABC$ be denoted $P^{-1}$.
Let $O$ be the circumcenter of $\triangle ABC$.
Then the pedal circle of $P$ is tangent to the Feuerbach circle of $\triangle ABC$ {{iff}} $... | {{explain|Too sketchy to be very useful}}
By the Second Fontené Theorem we can prove that the second intersection $Q'$ of the circle $O'$ and the circle $E$ is the anti-Steiner point of $OP^{-1}$.
This means $Q' = Q$ {{iff}} $O P = O P^{-1}$
That is:
:$O$, $P$ and $P^{-1}$ are collinear.
{{qed}} | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $P$ be an arbitrary [[Definition:Point|point]] in the [[Definition:Plane|plane]] of $\triangle ABC$.
Let the [[Definition:Isogonal Conjugate|isogonal conjugate]] of $P$ {{WRT}} to $\triangle ABC$ be denoted $P^{-1}$.
Let $O$ be the [[Definitio... | {{explain|Too sketchy to be very useful}}
By the [[Second Fontené Theorem]] we can prove that the second [[Definition:Intersection (Geometry)|intersection]] $Q'$ of the [[Definition:Tangent Circles|circle]] $O'$ and the [[Definition:Circle|circle]] $E$ is the [[Definition:Anti-Steiner Point|anti-Steiner point]] of $OP... | Fontené Theorems/Third | https://proofwiki.org/wiki/Fontené_Theorems/Third | https://proofwiki.org/wiki/Fontené_Theorems/Third | [
"Fontené Theorems",
"Feuerbach Circles",
"Pedal Circles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Point",
"Definition:Plane Surface",
"Definition:Isogonal Conjugate",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Pedal Circle",
"Definition:Tangent Circles",
"Definition:Feuerbach Circle",
"Definition:Collinear/Points"
] | [
"Fontené Theorems/Second",
"Definition:Intersection (Geometry)",
"Definition:Tangent Circles",
"Definition:Circle",
"Definition:Anti-Steiner Point",
"Definition:Collinear/Points"
] |
proofwiki-21807 | System of Open Neighborhoods are Equal Iff Singleton Closures are Equal | Let $T = \struct{S, \tau}$ be a topological space.
Let $x, y \in S$ such that $x \ne y$.
Let $\map \UU x$ and $\map \UU y$ denote the system of open neighborhoods of $x$ and $y$ respectively.
Let $\set x^-$ and $\set y^-$ denote the topological closures of $\set x$ and $\set y$ respectively.
Then:
:$\map \UU x = \map \... | We have
{{begin-eqn}}
{{eqn | l = \map \UU x = \map \UU y
| o = \leadstoandfrom
| r = \tau \setminus \map \UU x = \tau \setminus \map \UU y
| c = Equal Relative Complements iff Equal Subsets
}}
{{eqn | o = \leadsto
| r = \bigcup \paren{\tau \setminus \map \UU x} = \bigcup \paren{\tau \setminus \... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x, y \in S$ such that $x \ne y$.
Let $\map \UU x$ and $\map \UU y$ denote the [[Definition:System of Open Neighborhoods|system of open neighborhoods]] of $x$ and $y$ respectively.
Let $\set x^-$ and $\set y^-$ denote the [[Defin... | We have
{{begin-eqn}}
{{eqn | l = \map \UU x = \map \UU y
| o = \leadstoandfrom
| r = \tau \setminus \map \UU x = \tau \setminus \map \UU y
| c = [[Equal Relative Complements iff Equal Subsets]]
}}
{{eqn | o = \leadsto
| r = \bigcup \paren{\tau \setminus \map \UU x} = \bigcup \paren{\tau \setmin... | System of Open Neighborhoods are Equal Iff Singleton Closures are Equal | https://proofwiki.org/wiki/System_of_Open_Neighborhoods_are_Equal_Iff_Singleton_Closures_are_Equal | https://proofwiki.org/wiki/System_of_Open_Neighborhoods_are_Equal_Iff_Singleton_Closures_are_Equal | [
"Systems of Open Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:System of Open Neighborhoods",
"Definition:Closure (Topology)"
] | [
"Equal Relative Complements iff Equal Subsets",
"Union of Open Sets Not in System of Open Neighborhoods is Complement of Singleton Closure",
"Equal Relative Complements iff Equal Subsets",
"Open Set Not in System of Open Neighborhoods Iff Subset of Complement of Singleton Closure",
"Open Set Not in System o... |
proofwiki-21808 | Fontené Theorems/Second | Let $\triangle ABC$ be a triangle.
:480px
Let $P$ be a point moving on a fixed straight line through the circumcenter $O$ of $\triangle ABC$.
Then the pedal circle of $P$ with respect to passes through a fixed point $F$ on the Feuerbach circle of $\triangle ABC$. | {{explain|Most of the below completely meaningless because absolutely nothing is explained.}}
By the First Fontené Theorem, the point of contact $Q$ of the circle $E$ and the circle $O'$ is the reflection of a point $F$ which lies on $O P$ with respect to the line $B_1 C_1$.
It is easy to show that $O$ is the orthocent... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
:[[File:Fontene2.gif|480px]]
Let $P$ be a [[Definition:Point|point]] moving on a fixed [[Definition:Straight Line|straight line]] through the [[Definition:Circumcenter of Triangle|circumcenter]] $O$ of $\triangle ABC$.
Then the [[Definition:Pedal ... | {{explain|Most of the below completely meaningless because absolutely nothing is explained.}}
By the [[First Fontené Theorem]], the point of contact $Q$ of the [[Definition:Circle|circle]] $E$ and the [[Definition:Circle|circle]] $O'$ is the reflection of a point $F$ which lies on $O P$ with respect to the line $B_1 C... | Fontené Theorems/Second | https://proofwiki.org/wiki/Fontené_Theorems/Second | https://proofwiki.org/wiki/Fontené_Theorems/Second | [
"Fontené Theorems",
"Pedal Circles",
"Feuerbach Circles"
] | [
"Definition:Triangle (Geometry)",
"File:Fontene2.gif",
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Pedal Circle",
"Definition:Point",
"Definition:Feuerbach Circle"
] | [
"Fontené Theorems/First",
"Definition:Circle",
"Definition:Circle",
"Definition:Orthocenter",
"Definition:Anti-Steiner Point"
] |
proofwiki-21809 | Open Set Not in System of Open Neighborhoods Iff Subset of Complement of Singleton Closure | Let $T = \struct{S, \tau}$ be a topological space.
Let $x \in S$.
Let $\map \UU x$ denote the system of open neighborhoods of $x$.
Let $\set x^-$ denote the topological closure of $\set x$.
Then for all $U \in \tau$:
:$U \notin \map \UU x$ {{iff}} $U \subseteq S \setminus \set x^-$ | First note that by definition of closed set:
:$S \setminus U$ is a closed set
We have:
{{begin-eqn}}
{{eqn | l = U \notin \map \UU x
| o = \leadstoandfrom
| r = x \notin U
| c = {{Defof|System of Open Neighborhoods}}
}}
{{eqn | o = \leadstoandfrom
| r = \set x \subseteq S \setminus U
| c =... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $\map \UU x$ denote the [[Definition:System of Open Neighborhoods|system of open neighborhoods]] of $x$.
Let $\set x^-$ denote the [[Definition:Closure (Topology)|topological closure]] of $\set x$.
Then for all ... | First note that by definition of [[Definition:Closed Set (Topology)|closed set]]:
:$S \setminus U$ is a [[Definition:Closed Set (Topology)|closed set]]
We have:
{{begin-eqn}}
{{eqn | l = U \notin \map \UU x
| o = \leadstoandfrom
| r = x \notin U
| c = {{Defof|System of Open Neighborhoods}}
}}
{{eqn ... | Open Set Not in System of Open Neighborhoods Iff Subset of Complement of Singleton Closure | https://proofwiki.org/wiki/Open_Set_Not_in_System_of_Open_Neighborhoods_Iff_Subset_of_Complement_of_Singleton_Closure | https://proofwiki.org/wiki/Open_Set_Not_in_System_of_Open_Neighborhoods_Iff_Subset_of_Complement_of_Singleton_Closure | [
"Systems of Open Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:System of Open Neighborhoods",
"Definition:Closure (Topology)"
] | [
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Closure of Subset of Closed Set of Topological Space is Subset",
"Relative Complement inverts Subsets of Relative Complement",
"Category:Systems of Open Neighborhoods"
] |
proofwiki-21810 | Complement of Closed Set is Open Set | Let $T = \struct{S, \tau}$ be a topological space.
Let $F \subseteq S$ be a closed set in $T$.
Then:
:$S \setminus F \in \tau$ | By definition of closed set:
:$\exists U \in \tau : F = S \setminus U$
From Relative Complement inverts Subsets of Relative Complement:
:$U = S \setminus F$
The result follows.
{{qed}}
Category:Closed Sets
Category:Open Sets
7rzvy2si8mmcbrrrx4adcclprcor1c0 | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $F \subseteq S$ be a [[Definition:Closed Set (Topology)|closed set]] in $T$.
Then:
:$S \setminus F \in \tau$ | By definition of [[Definition:Closed Set (Topology)|closed set]]:
:$\exists U \in \tau : F = S \setminus U$
From [[Relative Complement inverts Subsets of Relative Complement]]:
:$U = S \setminus F$
The result follows.
{{qed}}
[[Category:Closed Sets]]
[[Category:Open Sets]]
7rzvy2si8mmcbrrrx4adcclprcor1c0 | Complement of Closed Set is Open Set | https://proofwiki.org/wiki/Complement_of_Closed_Set_is_Open_Set | https://proofwiki.org/wiki/Complement_of_Closed_Set_is_Open_Set | [
"Closed Sets",
"Open Sets"
] | [
"Definition:Topological Space",
"Definition:Closed Set/Topology"
] | [
"Definition:Closed Set/Topology",
"Relative Complement inverts Subsets of Relative Complement",
"Category:Closed Sets",
"Category:Open Sets"
] |
proofwiki-21811 | Union of Open Sets Not in System of Open Neighborhoods is Complement of Singleton Closure | Let $T = \struct{S, \tau}$ be a topological space.
Let $x \in S$.
Let $\map \UU x$ denote the system of open neighborhoods of $x$.
Let $\set x^-$ denote the topological closure of $\set x$.
Then:
:$\bigcup \set{U \in \tau : U \notin \map \UU x} = S \setminus \set x^-$ | From Open Set Not in System of Open Neighborhoods Iff Subset of Complement of Singleton Closure:
:$\forall U \in \tau : \paren{U \notin \map \UU x \iff U \subseteq S \setminus \set x^-}$
From Union of Subsets is Subset:
:$\bigcup \set{U \in \tau : U \notin \map \UU x} \subseteq S \setminus \set x^-$
From Topological Cl... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $\map \UU x$ denote the [[Definition:System of Open Neighborhoods|system of open neighborhoods]] of $x$.
Let $\set x^-$ denote the [[Definition:Closure (Topology)|topological closure]] of $\set x$.
Then:
:$\bigc... | From [[Open Set Not in System of Open Neighborhoods Iff Subset of Complement of Singleton Closure]]:
:$\forall U \in \tau : \paren{U \notin \map \UU x \iff U \subseteq S \setminus \set x^-}$
From [[Union of Subsets is Subset]]:
:$\bigcup \set{U \in \tau : U \notin \map \UU x} \subseteq S \setminus \set x^-$
From [[... | Union of Open Sets Not in System of Open Neighborhoods is Complement of Singleton Closure | https://proofwiki.org/wiki/Union_of_Open_Sets_Not_in_System_of_Open_Neighborhoods_is_Complement_of_Singleton_Closure | https://proofwiki.org/wiki/Union_of_Open_Sets_Not_in_System_of_Open_Neighborhoods_is_Complement_of_Singleton_Closure | [
"Systems of Open Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:System of Open Neighborhoods",
"Definition:Closure (Topology)"
] | [
"Open Set Not in System of Open Neighborhoods Iff Subset of Complement of Singleton Closure",
"Union of Subsets is Subset",
"Topological Closure is Closed",
"Definition:Closed Set/Topology",
"Complement of Closed Set is Open Set",
"Set is Subset of Union",
"Definition:Set Equality",
"Category:Systems ... |
proofwiki-21812 | Helly's Theorem | Let $A_1, A_2, \ldots, A_r \in \R^n$ be convex sets in real Euclidean $n$-space such that $r > n$.
Let $A_1, A_2, \ldots, A_r$ have the property that every collection of $n + 1$ of $A_1, A_2, \ldots, A_r$ have a point in common.
Then all of $A_1, A_2, \ldots, A_r$ have a point in common. | {{ProofWanted}}
{{Namedfor|Eduard Helly|cat = Helly}} | Let $A_1, A_2, \ldots, A_r \in \R^n$ be [[Definition:Convex Set (Vector Space)|convex sets]] in [[Definition:Real Euclidean Space|real Euclidean $n$-space]] such that $r > n$.
Let $A_1, A_2, \ldots, A_r$ have the property that every [[Definition:Collection|collection]] of $n + 1$ of $A_1, A_2, \ldots, A_r$ have a [[De... | {{ProofWanted}}
{{Namedfor|Eduard Helly|cat = Helly}} | Helly's Theorem | https://proofwiki.org/wiki/Helly's_Theorem | https://proofwiki.org/wiki/Helly's_Theorem | [
"Helly's Theorem",
"Convex Sets (Vector Spaces)",
"Linear Algebra"
] | [
"Definition:Convex Set (Vector Space)",
"Definition:Euclidean Space/Real",
"Definition:Collection",
"Definition:Point",
"Definition:Point"
] | [] |
proofwiki-21813 | Reflexive Riesz Lemma | Let $X$ be a reflexive normed vector space.
Let $Y$ be a proper closed linear subspace of $X$.
Then $\exists x_\alpha \in X$ such that:
:$\norm {x_\alpha} = 1$
with:
:$\ds \map d {x_\alpha, Y} = 1$
where $d$ denotes distance to a set. | By Existence of Distance Functional, there exists a bounded linear functional $f$ on $X$ such that:
:$\map f y = 0$ for each $y \in Y$
with norm $1$:
:$\norm f = 1$
By the easy direction (sufficiency) of James's Theorem, $\exists x_\alpha \in X$ such that:
:$\norm {x_\alpha} = 1$
and $f$ attains its norm at $x_\alpha$:... | Let $X$ be a [[Definition:Reflexive Space|reflexive]] [[Definition:Normed Vector Space|normed vector space]].
Let $Y$ be a [[Definition:Proper Subset|proper]] [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Then $\exists x_\alpha \in X$ such that:
:$\norm {x_\alpha} = 1$
with:
:$\ds \map d {x... | By [[Existence of Distance Functional]], there exists a [[Definition:Bounded Linear Functional|bounded linear functional]] $f$ on $X$ such that:
:$\map f y = 0$ for each $y \in Y$
with [[Definition:Operator Norm|norm]] $1$:
:$\norm f = 1$
By the easy direction (sufficiency) of [[James's Theorem]], $\exists x_\alpha \i... | Reflexive Riesz Lemma | https://proofwiki.org/wiki/Reflexive_Riesz_Lemma | https://proofwiki.org/wiki/Reflexive_Riesz_Lemma | [
"Riesz's Lemma",
"Functional Analysis"
] | [
"Definition:Reflexive Space",
"Definition:Normed Vector Space",
"Definition:Proper Subset",
"Definition:Closed Linear Subspace",
"Definition:Distance/Sets/Metric Spaces"
] | [
"Existence of Distance Functional",
"Definition:Bounded Linear Functional",
"Definition:Operator Norm",
"James's Theorem",
"Definition:Operator Norm",
"Definition:Infimum",
"Definition:Distance/Sets/Metric Spaces"
] |
proofwiki-21814 | Integral Representation of Bernoulli Number | Bernoulli numbers can be expressed in integral form as follows:
:$\ds \size {B_{2 n} } = 4 n \int_0^\infty \frac {t^{2 n - 1} } {e^{2 \pi t} - 1} \rd t$
where:
:$B_n$ are the Bernoulli numbers
:$n$ is a positive integer. | {{begin-eqn}}
{{eqn | l = \map \zeta s \map \Gamma s
| r = \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \rd t
| c = Integral Representation of Riemann Zeta Function in terms of Gamma Function
}}
{{eqn | ll= \leadsto
| l = \map \zeta {2 n} \map \Gamma {2 n}
| r = \int_0^\infty \frac {t^{2 n - 1} } ... | [[Definition:Bernoulli Numbers|Bernoulli numbers]] can be expressed in integral form as follows:
:$\ds \size {B_{2 n} } = 4 n \int_0^\infty \frac {t^{2 n - 1} } {e^{2 \pi t} - 1} \rd t$
where:
:$B_n$ are the [[Definition:Bernoulli Numbers|Bernoulli numbers]]
:$n$ is a [[Definition:Positive Integer|positive integer]]. | {{begin-eqn}}
{{eqn | l = \map \zeta s \map \Gamma s
| r = \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \rd t
| c = [[Integral Representation of Riemann Zeta Function in terms of Gamma Function]]
}}
{{eqn | ll= \leadsto
| l = \map \zeta {2 n} \map \Gamma {2 n}
| r = \int_0^\infty \frac {t^{2 n - 1... | Integral Representation of Bernoulli Number | https://proofwiki.org/wiki/Integral_Representation_of_Bernoulli_Number | https://proofwiki.org/wiki/Integral_Representation_of_Bernoulli_Number | [
"Bernoulli Numbers",
"Definite Integrals involving Exponential Function",
"Gamma Function",
"Riemann Zeta Function",
"Analytic Number Theory"
] | [
"Definition:Bernoulli Numbers",
"Definition:Bernoulli Numbers",
"Definition:Positive/Integer"
] | [
"Integral Representation of Riemann Zeta Function in terms of Gamma Function",
"Riemann Zeta Function at Even Integers",
"Gamma Function Extends Factorial",
"Exponent Combination Laws/Power of Product"
] |
proofwiki-21815 | Equivalence of Definitions of Generic Point of Topological Space | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $x \in S$ be an element of $S$.
{{TFAE|def=Generic Point of Topological Space}} | Let $\set x^-$ denote the closure of $\set x$ in $T$.
Let $\map \UU x$ denote the system of open neighborhoods of $x$.
We have:
{{begin-eqn}}
{{eqn | o =
| r = \set x^- = S
}}
{{eqn | ll = \leadstoandfrom
| o =
| r = S \setminus \set x^- = \O
| c = Set Difference with Superset is Empty Set
}}
... | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$ be an [[Definition:Element|element]] of $S$.
{{TFAE|def=Generic Point of Topological Space}} | Let $\set x^-$ denote the [[Definition:Closure (Topology)|closure]] of $\set x$ in $T$.
Let $\map \UU x$ denote the [[Definition:System of Open Neighborhoods|system of open neighborhoods]] of $x$.
We have:
{{begin-eqn}}
{{eqn | o =
| r = \set x^- = S
}}
{{eqn | ll = \leadstoandfrom
| o =
| r = S ... | Equivalence of Definitions of Generic Point of Topological Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generic_Point_of_Topological_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generic_Point_of_Topological_Space | [
"Generic Points"
] | [
"Definition:Topological Space",
"Definition:Element"
] | [
"Definition:Closure (Topology)",
"Definition:System of Open Neighborhoods",
"Set Difference with Superset is Empty Set",
"Union of Open Sets Not in System of Open Neighborhoods is Complement of Singleton Closure",
"Set is Subset of Union",
"Subset of Empty Set iff Empty",
"Category:Generic Points"
] |
proofwiki-21816 | Filter Containing Complements is Not Proper | Let $L = \struct {S, \lor, \land, \preceq}$ be a bounded lattice.
Let $F \subseteq S$ be a filter on $L$.
Suppose there exist $a, b \in F$ such that:
:$b$ is a complement of $a$.
Then:
:$F = S$ | By filter axiom $\paren 2$:
:$\exists c \in F: c \preceq a \land c \preceq b$
By definition of complement:
:$b \land a = \bot$
Thus, by definition of meet:
:$c \preceq \bot$
Therefore:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = x
| o = \succeq
| r = \bot
| c = {{Defof|Bottom of Lattice}}
}}... | Let $L = \struct {S, \lor, \land, \preceq}$ be a [[Definition:Bounded Lattice|bounded lattice]].
Let $F \subseteq S$ be a [[Definition:Filter|filter]] on $L$.
Suppose there exist $a, b \in F$ such that:
:$b$ is a [[Definition:Complement (Lattice Theory)|complement]] of $a$.
Then:
:$F = S$ | By [[Axiom:Filter Axioms|filter axiom $\paren 2$]]:
:$\exists c \in F: c \preceq a \land c \preceq b$
By definition of [[Definition:Complement (Lattice Theory)|complement]]:
:$b \land a = \bot$
Thus, by definition of [[Definition:Meet (Order Theory)|meet]]:
:$c \preceq \bot$
Therefore:
{{begin-eqn}}
{{eqn | q = \for... | Filter Containing Complements is Not Proper | https://proofwiki.org/wiki/Filter_Containing_Complements_is_Not_Proper | https://proofwiki.org/wiki/Filter_Containing_Complements_is_Not_Proper | [
"Bounded Lattices",
"Filter Theory"
] | [
"Definition:Bounded Lattice",
"Definition:Filter",
"Definition:Complement (Lattice Theory)"
] | [
"Axiom:Filter Axioms",
"Definition:Complement (Lattice Theory)",
"Definition:Meet (Order Theory)",
"Axiom:Filter Axioms",
"Definition:Set Equality",
"Category:Bounded Lattices",
"Category:Filter Theory"
] |
proofwiki-21817 | Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace | Let $T = \struct{S, \tau}$ be a topological space.
Let $U \in \tau$.
Let $F = S \setminus U$.
Then:
:$U$ is a meet-irreducible open set {{iff}} $F$ is a closed irreducible subspace | We prove the contrapositive statement:
:$F$ is not a closed irreducible subspace {{iff}} $U$ is not a meet-irreducible open set | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U \in \tau$.
Let $F = S \setminus U$.
Then:
:$U$ is a [[Definition:Meet-Irreducible Open Set|meet-irreducible open set]] {{iff}} $F$ is a [[Definition:Closed Set (Topology)|closed]] [[Definition:Irreducible Space|irreducible]]... | We prove the [[Definition:Contrapositive Statement|contrapositive statement]]:
:$F$ is not a [[Definition:Closed Set (Topology)|closed]] [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspace]] {{iff}} $U$ is not a [[Definition:Meet-Irreducible Open Set|meet-irreducible open set]] | Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace | https://proofwiki.org/wiki/Meet-Irreducible_Open_Set_iff_Complement_is_Closed_Irreducible_Subspace | https://proofwiki.org/wiki/Meet-Irreducible_Open_Set_iff_Complement_is_Closed_Irreducible_Subspace | [
"Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace",
"Meet-Irreducible Open Sets",
"Irreducible Spaces"
] | [
"Definition:Topological Space",
"Definition:Meet-Irreducible Open Set",
"Definition:Closed Set/Topology",
"Definition:Irreducible Space",
"Definition:Topological Subspace"
] | [
"Definition:Contrapositive Statement",
"Definition:Closed Set/Topology",
"Definition:Irreducible Space",
"Definition:Topological Subspace",
"Definition:Meet-Irreducible Open Set",
"Definition:Closed Set/Topology",
"Definition:Irreducible Space",
"Definition:Topological Subspace",
"Definition:Meet-Ir... |
proofwiki-21818 | Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace/Necessary Condition | Let $T = \struct{S, \tau}$ be a topological space.
Let $U \in \tau$.
Let $F = S \setminus U$.
Let $F$ be a closed irreducible subspace.
Then:
:$U$ is a meet-irreducible open set | By definition of closed set:
:$F$ is a closed set
By definition of closed irreducible subspace there exists proper closed subsets $F_1, F_2$ of $F$:
:$F = F_1 \cup F_2$
:$F_1 \subsetneq F$
:$F_2 \subsetneq F$
From Set Complement inverts Subsets and Equal Relative Complements iff Equal Subsets:
:$S \setminus F \subsetne... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U \in \tau$.
Let $F = S \setminus U$.
Let $F$ be a [[Definition:Closed Set (Topology)|closed]] [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspace]].
Then:
:$U$ is a [[Definition:Meet-Irredu... | By definition of [[Definition:Closed Set (Topology)|closed set]]:
:$F$ is a [[Definition:Closed Set (Topology)|closed set]]
By definition of [[Definition:Closed Set (Topology)|closed]] [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspace]] there exists [[Definition:Proper Subset|prop... | Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace/Necessary Condition | https://proofwiki.org/wiki/Meet-Irreducible_Open_Set_iff_Complement_is_Closed_Irreducible_Subspace/Necessary_Condition | https://proofwiki.org/wiki/Meet-Irreducible_Open_Set_iff_Complement_is_Closed_Irreducible_Subspace/Necessary_Condition | [
"Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace"
] | [
"Definition:Topological Space",
"Definition:Closed Set/Topology",
"Definition:Irreducible Space",
"Definition:Topological Subspace",
"Definition:Meet-Irreducible Open Set"
] | [
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Irreducible Space",
"Definition:Topological Subspace",
"Definition:Proper Subset",
"Definition:Closed Set/Topology",
"Set Complement inverts Subsets",
"Equal Relative Complements iff Equ... |
proofwiki-21819 | Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace/Sufficient Condition | Let $T = \struct{S, \tau}$ be a topological space.
Let $U \in \tau$ not be a meet-irreducible open set.
Let $F = S \setminus U$.
Then:
:$F$ is not a closed irreducible subspace | From Characterization of Meet-Irreducible Open Set there exists $V_1, V_2\in \tau$:
:$V_1 \nsubseteq U$
:$V_2 \nsubseteq U$
:$V_1 \cap V_2 \subseteq U$
Let:
:$U_1 = V_1 \cup U$
:$U_2 = V_2 \cup U$
From {{Open-set-axiom|1}}:
:$U_1, U_2 \in \tau$
From Set is Subset of Union:
:$U \subsetneq U_1$
:$U \subsetneq U_2$
We hav... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U \in \tau$ not be a [[Definition:Meet-Irreducible Open Set|meet-irreducible open set]].
Let $F = S \setminus U$.
Then:
:$F$ is not a [[Definition:Closed Set (Topology)|closed]] [[Definition:Irreducible Space|irreducible]] [[D... | From [[Characterization of Meet-Irreducible Open Set]] there exists $V_1, V_2\in \tau$:
:$V_1 \nsubseteq U$
:$V_2 \nsubseteq U$
:$V_1 \cap V_2 \subseteq U$
Let:
:$U_1 = V_1 \cup U$
:$U_2 = V_2 \cup U$
From {{Open-set-axiom|1}}:
:$U_1, U_2 \in \tau$
From [[Set is Subset of Union]]:
:$U \subsetneq U_1$
:$U \subsetn... | Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace/Sufficient Condition | https://proofwiki.org/wiki/Meet-Irreducible_Open_Set_iff_Complement_is_Closed_Irreducible_Subspace/Sufficient_Condition | https://proofwiki.org/wiki/Meet-Irreducible_Open_Set_iff_Complement_is_Closed_Irreducible_Subspace/Sufficient_Condition | [
"Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace"
] | [
"Definition:Topological Space",
"Definition:Meet-Irreducible Open Set",
"Definition:Closed Set/Topology",
"Definition:Irreducible Space",
"Definition:Topological Subspace"
] | [
"Characterization of Meet-Irreducible Open Set",
"Set is Subset of Union",
"Intersection Distributes over Union",
"Set Intersection is Idempotent",
"Union with Superset is Superset",
"Set Complement inverts Subsets",
"Equal Relative Complements iff Equal Subsets",
"De Morgan's Laws (Set Theory)",
"D... |
proofwiki-21820 | Equation of Rectangular Hyperbola in Reduced Form | Let $\KK$ be a rectangular hyperbola whose transverse axis and conjugate axis are of length $2 a$.
Let $\KK$ be aligned in a cartesian plane in reduced form.
$\KK$ can be expressed by the equation:
:$x^2 - y^2 = a^2$ | From Equation of Hyperbola in Reduced Form in Cartesian Frame, a hyperbola can be expressed by the equation:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
For a rectangular hyperbola:
:$a = b$
Hence $\KK$ can be expressed by the equation:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {a^2} = 1$
Multiplying both sides by $a^2$
:... | Let $\KK$ be a [[Definition:Rectangular Hyperbola|rectangular hyperbola]] whose [[Definition:Transverse Axis of Hyperbola|transverse axis]] and [[Definition:Conjugate Axis of Hyperbola|conjugate axis]] are of [[Definition:Length (Linear Measure)|length]] $2 a$.
Let $\KK$ be aligned in a [[Definition:Cartesian Plane|ca... | From [[Equation of Hyperbola in Reduced Form/Cartesian Frame|Equation of Hyperbola in Reduced Form in Cartesian Frame]], a [[Definition:Hyperbola|hyperbola]] can be expressed by the equation:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
For a [[Definition:Rectangular Hyperbola|rectangular hyperbola]]:
:$a = b$
Henc... | Equation of Rectangular Hyperbola in Reduced Form | https://proofwiki.org/wiki/Equation_of_Rectangular_Hyperbola_in_Reduced_Form | https://proofwiki.org/wiki/Equation_of_Rectangular_Hyperbola_in_Reduced_Form | [
"Rectangular Hyperbolas"
] | [
"Definition:Rectangular Hyperbola",
"Definition:Hyperbola/Transverse Axis",
"Definition:Hyperbola/Conjugate Axis",
"Definition:Linear Measure/Length",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Hyperbola"
] | [
"Equation of Hyperbola in Reduced Form/Cartesian Frame",
"Definition:Hyperbola",
"Definition:Rectangular Hyperbola",
"Definition:Multiplication/Real Numbers"
] |
proofwiki-21821 | Asymptotes to Rectangular Hyperbola | Let $\KK$ be a rectangular hyperbola.
The asymptotes of $\KK$ are perpendicular. | Let $\KK$ be embedded in a cartesian plane in reduced form with the equation:
:$x^2 - y^2 = a^2$
From Asymptotes to Hyperbola in Reduced Form, the asymptotes of $\KK$ can be expressed in the form:
:$y = \pm \dfrac b a x$
By definition of a rectangular hyperbola, $\KK$ is such that:
:$a = b$
Hence the asymptotes of $\KK... | Let $\KK$ be a [[Definition:Rectangular Hyperbola|rectangular hyperbola]].
The [[Definition:Asymptote of Hyperbola|asymptotes]] of $\KK$ are [[Definition:Perpendicular Lines|perpendicular]]. | Let $\KK$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] in [[Definition:Reduced Form of Hyperbola|reduced form]] with the equation:
:$x^2 - y^2 = a^2$
From [[Asymptotes to Hyperbola in Reduced Form]], the [[Definition:Asymptote of Hyperbola|asymptotes]] of $\KK$ can be expressed in the form:
:$y = \p... | Asymptotes to Rectangular Hyperbola | https://proofwiki.org/wiki/Asymptotes_to_Rectangular_Hyperbola | https://proofwiki.org/wiki/Asymptotes_to_Rectangular_Hyperbola | [
"Rectangular Hyperbolas"
] | [
"Definition:Rectangular Hyperbola",
"Definition:Hyperbola/Asymptote",
"Definition:Right Angle/Perpendicular"
] | [
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Hyperbola",
"Asymptotes to Hyperbola in Reduced Form",
"Definition:Hyperbola/Asymptote",
"Definition:Rectangular Hyperbola",
"Definition:Hyperbola/Asymptote",
"Definition:Slope/Straight Line",
"Definition:Slope/Straight Line",
"Con... |
proofwiki-21822 | Frame Homomorphism of Continuous Mapping is Frame Homomorphism | Let $T_1 = \struct{S_1, \tau_1}, T_2 = \struct{S_2, \tau_2}$ be topological spaces.
Let $f : T_1 \to T_2$ be a continuous mapping.
Let $\map \Omega {T_1} = \struct{\tau_1, \subseteq}$ and $\map \Omega {T_2} = \struct{\tau_2, \subseteq}$ denote the frames of $T_1$ and $T_2$ respectively.
Let $\map \Omega f : \map \Omega... | Recall the definition of continuous mapping:
:$f$ is continuous {{iff}} $U \in \tau_2 \implies f^{-1} \sqbrk U \in \tau_1$
By definition of frame homomorphism of $f$:
:$\map \Omega f$ is the inverse image mapping restricted to $\tau_2 \times \tau_1$
By definition of inverse image mapping:
:$\forall U \in \tau_2 : \map ... | Let $T_1 = \struct{S_1, \tau_1}, T_2 = \struct{S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f : T_1 \to T_2$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]].
Let $\map \Omega {T_1} = \struct{\tau_1, \subseteq}$ and $\map \Omega {T_2} = \struct{\tau_2, \su... | Recall the definition of [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]]:
:$f$ is [[Definition:Everywhere Continuous Mapping (Topology)|continuous]] {{iff}} $U \in \tau_2 \implies f^{-1} \sqbrk U \in \tau_1$
By definition of [[Definition:Frame Homomorphism of Continuous Mapping|frame homomo... | Frame Homomorphism of Continuous Mapping is Frame Homomorphism | https://proofwiki.org/wiki/Frame_Homomorphism_of_Continuous_Mapping_is_Frame_Homomorphism | https://proofwiki.org/wiki/Frame_Homomorphism_of_Continuous_Mapping_is_Frame_Homomorphism | [
"Continuous Mappings",
"Frame Homomorphisms"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Frame of Topological Space",
"Definition:Frame Homomorphism of Continuous Mapping",
"Definition:Frame Homomorphism"
] | [
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Frame Homomorphism of Continuous Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Restriction/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Inverse Image Ma... |
proofwiki-21823 | Rotation of Rectangular Hyperbola from Reduced to Standard Form | Let $\KK$ be a rectangular hyperbola embedded in a Cartesian plane in reduced form:
:$x^2 - y^2 = a^2$
Let $\KK$ be rotated $45 \degrees$ clockwise about the origin.
Then $\KK$ is in standard form:
:$x y = c^2$
where $c = \dfrac a {\sqrt 2}$. | Let $\tuple {x, y}$ denote an arbitrary point on $\KK$ before rotation.
Let $\tuple {x', y'}$ denote the image of $\tuple {x, y}$ after rotation.
Then:
{{begin-eqn}}
{{eqn | l = \begin {pmatrix} x' \\ y' \end {pmatrix}
| r = \begin {bmatrix} \map \cos {-45 \degrees} & -\map \sin {-45 \degrees} \\ \map \sin {-45 \... | Let $\KK$ be a [[Definition:Rectangular Hyperbola|rectangular hyperbola]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]] in [[Definition:Reduced Form of Hyperbola|reduced form]]:
:$x^2 - y^2 = a^2$
Let $\KK$ be [[Definition:Plane Rotation|rotated]] $45 \degrees$ [[Definition:Clockwise|clockwise]] about t... | Let $\tuple {x, y}$ denote an arbitrary [[Definition:Point|point]] on $\KK$ before [[Definition:Plane Rotation|rotation]].
Let $\tuple {x', y'}$ denote the [[Definition:Image of Element under Mapping|image]] of $\tuple {x, y}$ after [[Definition:Plane Rotation|rotation]].
Then:
{{begin-eqn}}
{{eqn | l = \begin {pma... | Rotation of Rectangular Hyperbola from Reduced to Standard Form | https://proofwiki.org/wiki/Rotation_of_Rectangular_Hyperbola_from_Reduced_to_Standard_Form | https://proofwiki.org/wiki/Rotation_of_Rectangular_Hyperbola_from_Reduced_to_Standard_Form | [
"Rectangular Hyperbolas"
] | [
"Definition:Rectangular Hyperbola",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Hyperbola",
"Definition:Rotation (Geometry)/Plane",
"Definition:Clockwise",
"Definition:Coordinate System/Origin",
"Definition:Rectangular Hyperbola/Standard Form"
] | [
"Definition:Point",
"Definition:Rotation (Geometry)/Plane",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Rotation (Geometry)/Plane",
"Matrix Equation of Plane Rotation",
"Definition:Clockwise",
"Definition:Negative/Real Number",
"Sine Function is Odd",
"Cosine Function is Even",
"S... |
proofwiki-21824 | Equation of Rectangular Hyperbola in Standard Form/Parametric Form | Let $\KK$ be a rectangular hyperbola in standard form.
$\KK$ can be expressed in parametric form as:
:$\begin {cases} x = c t \\ y = \dfrac c t \end {cases}$ | Let the point $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = c t
}}
{{eqn | l = y
| r = \dfrac c t
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = x y
| r = c t \times \dfrac c t
| c =
}}
{{eqn | r = c^2
| c =
}}
{{end-eqn}}
{{qed}} | Let $\KK$ be a [[Definition:Rectangular Hyperbola|rectangular hyperbola]] in [[Definition:Standard Form of Rectangular Hyperbola|standard form]].
$\KK$ can be expressed in [[Definition:Parametric Equation|parametric form]] as:
:$\begin {cases} x = c t \\ y = \dfrac c t \end {cases}$ | Let the point $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = c t
}}
{{eqn | l = y
| r = \dfrac c t
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = x y
| r = c t \times \dfrac c t
| c =
}}
{{eqn | r = c^2
| c =
}}
{{end-eqn}}
{{qed}} | Equation of Rectangular Hyperbola in Standard Form/Parametric Form | https://proofwiki.org/wiki/Equation_of_Rectangular_Hyperbola_in_Standard_Form/Parametric_Form | https://proofwiki.org/wiki/Equation_of_Rectangular_Hyperbola_in_Standard_Form/Parametric_Form | [
"Equation of Hyperbola in Reduced Form"
] | [
"Definition:Rectangular Hyperbola",
"Definition:Rectangular Hyperbola/Standard Form",
"Definition:Parametric Equation"
] | [] |
proofwiki-21825 | Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form 2 | $K$ can be expressed in parametric form as:
:$\begin {cases} x = a \sec \theta \\ y = b \tan \theta \end {cases}$ | Let the point $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = a \sec \theta
}}
{{eqn | l = y
| r = b \tan \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \frac {x^2} {a^2} - \frac {y^2} {b^2}
| r = \frac {\paren {a \sec \theta}^2} {a^2} - \frac {\paren {b \tan \theta}^2}... | $K$ can be expressed in [[Definition:Parametric Equation|parametric form]] as:
:$\begin {cases} x = a \sec \theta \\ y = b \tan \theta \end {cases}$ | Let the point $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = a \sec \theta
}}
{{eqn | l = y
| r = b \tan \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \frac {x^2} {a^2} - \frac {y^2} {b^2}
| r = \frac {\paren {a \sec \theta}^2} {a^2} - \frac {\paren {b \tan \theta... | Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form 2 | https://proofwiki.org/wiki/Equation_of_Hyperbola_in_Reduced_Form/Cartesian_Frame/Parametric_Form_2 | https://proofwiki.org/wiki/Equation_of_Hyperbola_in_Reduced_Form/Cartesian_Frame/Parametric_Form_2 | [
"Equation of Hyperbola in Reduced Form"
] | [
"Definition:Parametric Equation"
] | [
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-21826 | Frame of Open Sets Functor is Contravariant | Let $\mathbf{Top}$ denote the category of topological spaces.
Let $\mathbf{Frm}$ denote the category of frames.
Then:
:the open sets functor $\mathbf \Omega : \mathbf{Top} \to \mathbf{Frm}$ is a contravariant functor | Recall the open sets functor $\mathbf \Omega : \mathbf{Top} \to \mathbf{Frm}$ is defined by:
{{DefineFunctor
|ob = $\map \Omega T := $ the frame of topological space $T$
|mor = $\map \Omega f := $ the frame homomorphism of continuous mapping $f$
}} | Let $\mathbf{Top}$ denote the [[Definition:Category of Topological Spaces|category of topological spaces]].
Let $\mathbf{Frm}$ denote the [[Definition:Category of Frames|category of frames]].
Then:
:the [[Definition:Frame of Open Sets Functor|open sets functor]] $\mathbf \Omega : \mathbf{Top} \to \mathbf{Frm}$ is a ... | Recall the [[Definition:Frame of Open Sets Functor|open sets functor]] $\mathbf \Omega : \mathbf{Top} \to \mathbf{Frm}$ is defined by:
{{DefineFunctor
|ob = $\map \Omega T := $ the [[Definition:Frame of Topological Space|frame of topological space $T$]]
|mor = $\map \Omega f := $ the [[Definition:Frame Homomorphism of... | Frame of Open Sets Functor is Contravariant | https://proofwiki.org/wiki/Frame_of_Open_Sets_Functor_is_Contravariant | https://proofwiki.org/wiki/Frame_of_Open_Sets_Functor_is_Contravariant | [
"Functors"
] | [
"Definition:Category of Topological Spaces",
"Definition:Category of Frames",
"Definition:Frame of Open Sets Functor",
"Definition:Functor/Contravariant"
] | [
"Definition:Frame of Open Sets Functor",
"Definition:Frame of Topological Space",
"Definition:Frame Homomorphism of Continuous Mapping",
"Definition:Frame Homomorphism",
"Definition:Frame of Topological Space"
] |
proofwiki-21827 | Solution to Hypergeometric Differential Equation | The '''hypergeometric series''':
{{begin-eqn}}
{{eqn | l = \map F {a, b, c; z}
| r = \sum_{n \mathop = 0}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} \, n!} z^n
}}
{{end-eqn}}
defines a solution to the '''hypergeometric differential equation''':
:$x \paren {1 - x} \dfrac {\d^2 y} {\d x^2} + ... | Let:
{{begin-eqn}}
{{eqn | l = y
| r = 1 + \frac {a b} {1! c} x + \frac {a \paren {a + 1} b \paren {b + 1} } {2! c \paren {c + 1} } x^2 + \frac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {3! c \paren {c + 1} \paren {c + 2} } x^3 + \cdots
| c =
}}
{{eqn | r = \sum_{n \mathop = 0}^\in... | The '''[[Definition:Hypergeometric Series|hypergeometric series]]''':
{{begin-eqn}}
{{eqn | l = \map F {a, b, c; z}
| r = \sum_{n \mathop = 0}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} \, n!} z^n
}}
{{end-eqn}}
defines a [[Definition:Solution to Differential Equation|solution]] to the ''... | Let:
{{begin-eqn}}
{{eqn | l = y
| r = 1 + \frac {a b} {1! c} x + \frac {a \paren {a + 1} b \paren {b + 1} } {2! c \paren {c + 1} } x^2 + \frac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {3! c \paren {c + 1} \paren {c + 2} } x^3 + \cdots
| c =
}}
{{eqn | r = \sum_{n \mathop = 0}^\in... | Solution to Hypergeometric Differential Equation | https://proofwiki.org/wiki/Solution_to_Hypergeometric_Differential_Equation | https://proofwiki.org/wiki/Solution_to_Hypergeometric_Differential_Equation | [
"Gaussian Hypergeometric Function",
"Hypergeometric Differential Equations",
"Hypergeometric Series"
] | [
"Definition:Hypergeometric Series",
"Definition:Differential Equation/Solution",
"Definition:Hypergeometric Differential Equation"
] | [
"Derivative of Constant",
"Power Rule for Derivatives",
"Derivative of Constant",
"Power Rule for Derivatives",
"Sum of Indices of Rising Factorial",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Indices of Rising Factorial",
"Definition:Fraction/Numerator",
"Definiti... |
proofwiki-21828 | Frame Homomorphism is Lower Adjoint of Galois Connection | Let $L_1 = \struct{S_1, \preceq_1}, L_2 = \struct{S_2, \preceq_2}$ be frames.
Let $f : L_1 \to L_2$ be a frame homomorphism.
Then there exists a Galois connection $g = \tuple {\upperadjoint g, \loweradjoint g}$:
:the lower adjoint $\loweradjoint g = f$ | By definition of frame homomorphism:
:$f$ is arbitrary join preserving
From All Suprema Preserving Mapping is Lower Adjoint of Galois Connection:
:there exists a Galois connection $g = \tuple {\upperadjoint g, \loweradjoint g} : \loweradjoint g = f$
{{qed}}
Category:Frame Homomorphisms
Category:Galois Connections
nv22g... | Let $L_1 = \struct{S_1, \preceq_1}, L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $f : L_1 \to L_2$ be a [[Definition:Frame Homomorphism|frame homomorphism]].
Then there exists a [[Definition:Galois Connection|Galois connection]] $g = \tuple {\upperadjoint g, \loweradjoint g}$:
... | By definition of [[Definition:Frame Homomorphism|frame homomorphism]]:
:$f$ is [[Definition:Arbitrary Join Preserving Mapping|arbitrary join preserving]]
From [[All Suprema Preserving Mapping is Lower Adjoint of Galois Connection]]:
:there exists a [[Definition:Galois Connection|Galois connection]] $g = \tuple {\upper... | Frame Homomorphism is Lower Adjoint of Galois Connection | https://proofwiki.org/wiki/Frame_Homomorphism_is_Lower_Adjoint_of_Galois_Connection | https://proofwiki.org/wiki/Frame_Homomorphism_is_Lower_Adjoint_of_Galois_Connection | [
"Frame Homomorphisms",
"Galois Connections"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Frame Homomorphism",
"Definition:Galois Connection",
"Definition:Galois Connection/Lower Adjoint"
] | [
"Definition:Frame Homomorphism",
"Definition:Arbitrary Join Preserving Mapping",
"All Suprema Preserving Mapping is Lower Adjoint of Galois Connection",
"Definition:Galois Connection",
"Category:Frame Homomorphisms",
"Category:Galois Connections"
] |
proofwiki-21829 | Rational Root Theorem | Let $\map P x$ be a polynomial whose coefficients are all integers:
:$\map P x = a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0$
where $a_n \ne 0$ and $a_0 \ne 0$.
Let the polynomial equation $\map P x = 0$ have a root which is a rational number expressed in canonical form as $\dfrac p q$.
Then:
:the leading coef... | {{Recall|Canonical Form of Rational Number}}
{{:Definition:Canonical Form of Rational Number}}
We have {{hypothesis}} that $\dfrac p q$ is a root of $P$.
Hence:
{{begin-eqn}}
{{eqn | l = \map P {\dfrac p q}
| r = 0
| c = {{Defof|Root of Polynomial}}
}}
{{eqn | ll= \leadsto
| l = 0
| r = a_n \par... | Let $\map P x$ be a [[Definition:Polynomial|polynomial]] whose [[Definition:Polynomial Coefficient|coefficients]] are all [[Definition:Integer|integers]]:
:$\map P x = a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0$
where $a_n \ne 0$ and $a_0 \ne 0$.
Let the [[Definition:Polynomial Equation|polynomial equation... | {{Recall|Canonical Form of Rational Number}}
{{:Definition:Canonical Form of Rational Number}}
We have {{hypothesis}} that $\dfrac p q$ is a [[Definition:Root of Polynomial|root]] of $P$.
Hence:
{{begin-eqn}}
{{eqn | l = \map P {\dfrac p q}
| r = 0
| c = {{Defof|Root of Polynomial}}
}}
{{eqn | ll= \leads... | Rational Root Theorem | https://proofwiki.org/wiki/Rational_Root_Theorem | https://proofwiki.org/wiki/Rational_Root_Theorem | [
"Polynomial Equations",
"Named Theorems"
] | [
"Definition:Polynomial",
"Definition:Coefficient of Polynomial",
"Definition:Integer",
"Definition:Polynomial Equation",
"Definition:Root of Equation",
"Definition:Rational Number",
"Definition:Rational Number/Canonical Form",
"Definition:Leading Coefficient of Polynomial",
"Definition:Divisor (Alge... | [
"Definition:Root of Polynomial",
"Definition:Multiplication/Integers",
"Definition:Subtraction/Integers",
"Distributive Laws/Arithmetic",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Addition/Integers",
"Definition:Multiplication/Integers",
"Definition:Integer",
"Euclid's... |
proofwiki-21830 | Impossible Proposition Strictly Implies Every Proposition | Let $P$ be a proposition of modal logic.
Let $P$ be not possibly true.
Then:
:$\forall Q: P \implies Q$
where $Q$ is an arbitrary proposition in the universe of discourse. | {{ProofWanted|More background needed into Modal Logic}} | Let $P$ be a [[Definition:Proposition|proposition]] of [[Definition:Modal Logic|modal logic]].
Let $P$ be not [[Definition:Possibility (Modal Logic)|possibly true]].
Then:
:$\forall Q: P \implies Q$
where $Q$ is an arbitrary [[Definition:Proposition|proposition]] in the [[Definition:Universe of Discourse|universe of ... | {{ProofWanted|More background needed into Modal Logic}} | Impossible Proposition Strictly Implies Every Proposition | https://proofwiki.org/wiki/Impossible_Proposition_Strictly_Implies_Every_Proposition | https://proofwiki.org/wiki/Impossible_Proposition_Strictly_Implies_Every_Proposition | [
"Paradoxes of Strict Implication"
] | [
"Definition:Proposition",
"Definition:Modal Logic",
"Definition:Possibility (Modal Logic)",
"Definition:Proposition",
"Definition:Universe of Discourse"
] | [] |
proofwiki-21831 | Necessary Proposition is Strictly Implied by Every Proposition | Let $P$ be a proposition of modal logic.
Let $P$ be necessarily true.
Then:
:$\forall Q: Q \implies P$
where $Q$ is an arbitrary proposition in the universe of discourse. | {{ProofWanted|More background needed into Modal Logic}} | Let $P$ be a [[Definition:Proposition|proposition]] of [[Definition:Modal Logic|modal logic]].
Let $P$ be [[Definition:Necessary (Modal Logic)|necessarily true]].
Then:
:$\forall Q: Q \implies P$
where $Q$ is an arbitrary [[Definition:Proposition|proposition]] in the [[Definition:Universe of Discourse|universe of dis... | {{ProofWanted|More background needed into Modal Logic}} | Necessary Proposition is Strictly Implied by Every Proposition | https://proofwiki.org/wiki/Necessary_Proposition_is_Strictly_Implied_by_Every_Proposition | https://proofwiki.org/wiki/Necessary_Proposition_is_Strictly_Implied_by_Every_Proposition | [
"Paradoxes of Strict Implication"
] | [
"Definition:Proposition",
"Definition:Modal Logic",
"Definition:Necessary (Modal Logic)",
"Definition:Proposition",
"Definition:Universe of Discourse"
] | [] |
proofwiki-21832 | Impulse Imparted by Constant Force | Let a constant force $\mathbf F$ be applied to a particle $P$ from time $t_1$ to time $t_2$.
Then the impulse $\mathbf J$ imparted to $P$ is:
:$\mathbf J = \mathbf F \paren {t_2 - t_1}$ | {{begin-eqn}}
{{eqn | l = \mathbf J
| r = \int_{t_1}^{t_2} \mathbf F \rd t
| c = {{Defof|Impulse}}
}}
{{eqn | r = \bigintlimits {\mathbf F t} {t \mathop = t_1} {t \mathop = t_2}
| c = Definite Integral of Constant
}}
{{eqn | r = \mathbf F \paren {t_2 - t_1}
| c =
}}
{{end-eqn}}
{{qed}} | Let a [[Definition:Constant|constant]] [[Definition:Force|force]] $\mathbf F$ be applied to a [[Definition:Particle|particle]] $P$ from [[Definition:Time Instant|time]] $t_1$ to [[Definition:Time Instant|time]] $t_2$.
Then the [[Definition:Impulse|impulse]] $\mathbf J$ imparted to $P$ is:
:$\mathbf J = \mathbf F \pare... | {{begin-eqn}}
{{eqn | l = \mathbf J
| r = \int_{t_1}^{t_2} \mathbf F \rd t
| c = {{Defof|Impulse}}
}}
{{eqn | r = \bigintlimits {\mathbf F t} {t \mathop = t_1} {t \mathop = t_2}
| c = [[Definite Integral of Constant]]
}}
{{eqn | r = \mathbf F \paren {t_2 - t_1}
| c =
}}
{{end-eqn}}
{{qed}} | Impulse Imparted by Constant Force | https://proofwiki.org/wiki/Impulse_Imparted_by_Constant_Force | https://proofwiki.org/wiki/Impulse_Imparted_by_Constant_Force | [
"Impulse"
] | [
"Definition:Constant",
"Definition:Force",
"Definition:Particle",
"Definition:Instant of Time",
"Definition:Instant of Time",
"Definition:Impulse"
] | [
"Integral of Constant/Definite"
] |
proofwiki-21833 | Impulse equals Change in Momentum | Let a force $\mathbf F$ be applied to a particle $P$.
Then the impulse $\mathbf J$ imparted to $P$ is equal to the change of momentum of $P$. | {{begin-eqn}}
{{eqn | l = \mathbf F
| r = \dfrac {\d \mathbf p} {\d t}
| c = Newton's Second Law of Motion: $\mathbf p$ is the (linear) momentum of $P$
}}
{{eqn | ll= \leadsto
| l = \int_{t_1}^{t_2} \mathbf F \rd t
| r = \int_{t_1}^{t_2} \dfrac {\d \mathbf p} {\d t}
| c = integrating $\mat... | Let a [[Definition:Force|force]] $\mathbf F$ be applied to a [[Definition:Particle|particle]] $P$.
Then the [[Definition:Impulse|impulse]] $\mathbf J$ imparted to $P$ is equal to the change of [[Definition:Momentum|momentum]] of $P$. | {{begin-eqn}}
{{eqn | l = \mathbf F
| r = \dfrac {\d \mathbf p} {\d t}
| c = [[Newton's Second Law of Motion]]: $\mathbf p$ is the [[Definition:Linear Momentum|(linear) momentum]] of $P$
}}
{{eqn | ll= \leadsto
| l = \int_{t_1}^{t_2} \mathbf F \rd t
| r = \int_{t_1}^{t_2} \dfrac {\d \mathbf p} {... | Impulse equals Change in Momentum | https://proofwiki.org/wiki/Impulse_equals_Change_in_Momentum | https://proofwiki.org/wiki/Impulse_equals_Change_in_Momentum | [
"Impulse"
] | [
"Definition:Force",
"Definition:Particle",
"Definition:Impulse",
"Definition:Momentum"
] | [
"Newton's Laws of Motion/Second Law",
"Definition:Linear Momentum",
"Definition:Definite Integral",
"Definition:Time",
"Definition:Definite Integral/Limits of Integration",
"Fundamental Theorem of Calculus",
"Definition:Impulse",
"Definition:Linear Momentum"
] |
proofwiki-21834 | Category of Locales with Localic Mappings is Category | Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.
Then:
:$\mathbf{Loc_*}$ is a metacategory | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory.
For any two localic mappings their composition (in the usual set theoretic sense) is again a localic mapping by Composite Localic Mapping is Localic Mapping.
For any locale $L = \struct{S, \preceq}$, we have the identity mapping $\operatorna... | Let $\mathbf{Loc_*}$ denote the [[Definition:Category of Locales with Localic Mappings|category of locales with localic mappings]].
Then:
:$\mathbf{Loc_*}$ is a [[Definition:Metacategory|metacategory]] | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a [[Definition:Metacategory|metacategory]].
For any two [[Definition:Localic Mapping|localic mappings]] their [[Definition:Composition of Mappings|composition]] (in the usual [[Definition:Set Theory|set theoretic]] sense) is again a [[Definition:Localic M... | Category of Locales with Localic Mappings is Category | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Category | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Category | [
"Locales"
] | [
"Definition:Category of Locales with Localic Mappings",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Composition of Mappings",
"Definition:Set Theory",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Composite Localic Mapping is Localic Mapping",
"Definition:Locale (Lattice Theory)",
"Definition:Identi... |
proofwiki-21835 | Composite Localic Mapping is Localic Mapping | Let $L_1 = \struct{S_1, \preceq_1}, L_2 = \struct{S_2, \preceq_2}$ and $L_3 = \struct{S_3, \preceq_3}$ be locales.
Let $f_1 : L_1 \to L_2$ and $f_2 : L_2 \to L_3$ be localic mappings.
Then:
:the composite $f_2 \circ f_1 : L_1 \to L_3$ is a localic mapping. | By definition of a localic mapping:
:$f_1, f_2$ are the upper adjoint of Galois connections $\tuple{f_1, d_1}, \tuple{f_2, d_2}$ respectively
where $d_1 : L_2 \to L_1$ and $d_2 : L_3 \to L_2$ are frame homomorphsims.
From Composite Frame Homomorphism is Frame Homomorphism:
:$d_1 \circ d_2 : L_3 \to L_2$ is a frame homo... | Let $L_1 = \struct{S_1, \preceq_1}, L_2 = \struct{S_2, \preceq_2}$ and $L_3 = \struct{S_3, \preceq_3}$ be [[Definition:Locale (Lattice Theory)|locales]].
Let $f_1 : L_1 \to L_2$ and $f_2 : L_2 \to L_3$ be [[Definition:Localic Mapping|localic mappings]].
Then:
:the [[Definition:Composite Mapping|composite]] $f_2 \cir... | By definition of a [[Definition:Localic Mapping|localic mapping]]:
:$f_1, f_2$ are the [[Definition:Upper Adjoint|upper adjoint]] of [[Definition:Galois Connection|Galois connections]] $\tuple{f_1, d_1}, \tuple{f_2, d_2}$ respectively
where $d_1 : L_2 \to L_1$ and $d_2 : L_3 \to L_2$ are [[Definition:Frame Homomorphism... | Composite Localic Mapping is Localic Mapping | https://proofwiki.org/wiki/Composite_Localic_Mapping_is_Localic_Mapping | https://proofwiki.org/wiki/Composite_Localic_Mapping_is_Localic_Mapping | [
"Localic Mappings"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Composition of Mappings",
"Definition:Continuous Map (Locale)/Localic Mapping"
] | [
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Galois Connection/Upper Adjoint",
"Definition:Galois Connection",
"Definition:Frame Homomorphism",
"Composite Frame Homomorphism is Frame Homomorphism",
"Definition:Frame Homomorphism",
"Composition of Galois Connections is Galois Connect... |
proofwiki-21836 | Identity Mapping is Localic Mapping | Let $L = \struct {S, \preceq}$ be a locale.
Let $\operatorname{id}_S : S \to S$ be the identity mapping on $S$.
Then:
:$\operatorname{id}_S : L \to L$ is a localic mapping. | By definition of localic mapping, we need to show that $\operatorname{id}_S$ is the upper adjoint of a Galois connection where the lower adjoint is a frame homomorphism of $L$ to $L$.
From Identity Mapping forms Galois Connection:
:$\tuple{\operatorname{id}_S, \operatorname{id}_S}$ is a Galois connection
From Identity... | Let $L = \struct {S, \preceq}$ be a [[Definition:Locale (Lattice Theory)|locale]].
Let $\operatorname{id}_S : S \to S$ be the [[Definition:Identity Mapping|identity mapping]] on $S$.
Then:
:$\operatorname{id}_S : L \to L$ is a [[Definition:Localic Mapping|localic mapping]]. | By definition of [[Definition:Localic Mapping|localic mapping]], we need to show that $\operatorname{id}_S$ is the [[Definition:Upper Adjoint|upper adjoint]] of a [[Definition:Galois Connection|Galois connection]] where the [[Definition:Lower Adjoint|lower adjoint]] is a [[Definition:Frame Homomorphism|frame homomorph... | Identity Mapping is Localic Mapping | https://proofwiki.org/wiki/Identity_Mapping_is_Localic_Mapping | https://proofwiki.org/wiki/Identity_Mapping_is_Localic_Mapping | [
"Localic Mappings"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Identity Mapping",
"Definition:Continuous Map (Locale)/Localic Mapping"
] | [
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Galois Connection/Upper Adjoint",
"Definition:Galois Connection",
"Definition:Galois Connection/Lower Adjoint",
"Definition:Frame Homomorphism",
"Identity Mapping forms Galois Connection",
"Definition:Galois Connection",
"Identity Mappi... |
proofwiki-21837 | Category of Locales with Localic Mappings is Isomorphic to Category of Locales | Let $\mathbf{Loc}$ denote the category of locales.
Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.
Then:
:$\mathbf{Loc_*}$ is isomorphic to $\mathbf{Loc}$
with isomorphisms:
:$F : \mathbf{Loc} \to \mathbf{Loc_*}$ defined by:
::for each locale $L$ of $\mathbf{Loc} : \map F L = L$
::for each co... | By definitions of category of locales and category of locales with localic mappings:
:the objects of $\mathbf{Loc}$ and $\mathbf{Loc_*}$ are locales
:the morphisms of $\mathbf{Loc}$ are continuous maps
:the morphisms of $\mathbf{Loc_*}$ are localic mappings
By definition of continuous maps:
:$f : L_1 \to L_2$ is a cont... | Let $\mathbf{Loc}$ denote the [[Definition:Category of Locales|category of locales]].
Let $\mathbf{Loc_*}$ denote the [[Definition:Category of Locales with Localic Mappings|category of locales with localic mappings]].
Then:
:$\mathbf{Loc_*}$ is [[Definition:Isomorphic Categories|isomorphic]] to $\mathbf{Loc}$
with [... | By definitions of [[Definition:Category of Locales|category of locales]] and [[Definition:Category of Locales with Localic Mappings|category of locales with localic mappings]]:
:the [[Definition:Object (Category Theory)|objects]] of $\mathbf{Loc}$ and $\mathbf{Loc_*}$ are [[Definition:Locale (Lattice Theory)|locales]]
... | Category of Locales with Localic Mappings is Isomorphic to Category of Locales | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales | [
"Category of Locales",
"Category of Locales with Localic Mappings is Isomorphic to Category of Locales"
] | [
"Definition:Category of Locales",
"Definition:Category of Locales with Localic Mappings",
"Definition:Isomorphism of Categories/Isomorphic Categories",
"Definition:Isomorphism of Categories",
"Definition:Locale (Lattice Theory)",
"Definition:Continuous Map (Locale)",
"Definition:Frame Homomorphism",
"... | [
"Definition:Category of Locales",
"Definition:Category of Locales with Localic Mappings",
"Definition:Object (Category Theory)",
"Definition:Locale (Lattice Theory)",
"Definition:Morphism",
"Definition:Continuous Map (Locale)",
"Definition:Morphism",
"Definition:Continuous Map (Locale)/Localic Mapping... |
proofwiki-21838 | Category of Locales with Localic Mappings is Isomorphic to Category of Locales/Lemma 1 | :$F : \mathbf{Loc} \to \mathbf{Loc_*}$ is a well-defined covariant functor | === $F$ is Well-defined ===
By definition of $\mathbf{Loc}$ and $\mathbf {Loc_*}$ the objects of $\mathbf{Loc}$ are the objects of $\mathbf{Loc_*}$.
The object functor of $F$ is the identity object functor and so is well-defined.
Let $f : L_1 \to L_2$ be a continuous map of $\mathbf{Loc}$.
By definition of continuous m... | :$F : \mathbf{Loc} \to \mathbf{Loc_*}$ is a [[Definition:Well-Defined|well-defined]] [[Definition:Covariant Functor|covariant functor]] | === $F$ is Well-defined ===
By definition of [[Definition:Category of Locales|$\mathbf{Loc}$]] and [[Definition:Category of Locales with Localic Mappings|$\mathbf {Loc_*}$]] the [[Definition:Object|objects]] of $\mathbf{Loc}$ are the [[Definition:Object|objects]] of $\mathbf{Loc_*}$.
The [[Definition:Object Functor|o... | Category of Locales with Localic Mappings is Isomorphic to Category of Locales/Lemma 1 | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales/Lemma_1 | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales/Lemma_1 | [
"Category of Locales with Localic Mappings is Isomorphic to Category of Locales"
] | [
"Definition:Well-Defined",
"Definition:Functor/Covariant"
] | [
"Definition:Category of Locales",
"Definition:Category of Locales with Localic Mappings",
"Definition:Object",
"Definition:Object",
"Definition:Object Functor",
"Definition:Identity Functor",
"Definition:Object Functor",
"Definition:Well-Defined",
"Definition:Continuous Map (Locale)",
"Definition:... |
proofwiki-21839 | Category of Locales with Localic Mappings is Isomorphic to Category of Locales/Lemma 2 | :$G : \mathbf{Loc_*} \to \mathbf{Loc}$ is a well-defined covariant functor | === $G$ is Well-defined ===
By definition of $\mathbf{Loc_*}$ and $\mathbf {Loc}$ the objects of $\mathbf{Loc_*}$ are the objects of $\mathbf{Loc}$.
The object functor of $G$ is the identity object functor and so is well-defined.
Let $g : L_1 \to L_2$ be a localic mapping of $\mathbf{Loc_*}$.
By definition of localic m... | :$G : \mathbf{Loc_*} \to \mathbf{Loc}$ is a [[Definition:Well-Defined|well-defined]] [[Definition:Covariant Functor|covariant functor]] | === $G$ is Well-defined ===
By definition of [[Definition:Category of Locales with Localic Mappings|$\mathbf{Loc_*}$]] and [[Definition:Category of Locales|$\mathbf {Loc}$]] the [[Definition:Object|objects]] of $\mathbf{Loc_*}$ are the [[Definition:Object|objects]] of $\mathbf{Loc}$.
The [[Definition:Object Functor|o... | Category of Locales with Localic Mappings is Isomorphic to Category of Locales/Lemma 2 | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales/Lemma_2 | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales/Lemma_2 | [
"Category of Locales with Localic Mappings is Isomorphic to Category of Locales"
] | [
"Definition:Well-Defined",
"Definition:Functor/Covariant"
] | [
"Definition:Category of Locales with Localic Mappings",
"Definition:Category of Locales",
"Definition:Object",
"Definition:Object",
"Definition:Object Functor",
"Definition:Identity Functor",
"Definition:Object Functor",
"Definition:Well-Defined",
"Definition:Continuous Map (Locale)/Localic Mapping"... |
proofwiki-21840 | Category of Locales with Localic Mappings is Isomorphic to Category of Locales/Lemma 3 | :$GF = \operatorname{id}_{\mathbf {Loc}}$ | By definition of $\mathbf{Loc}$ and $\mathbf {Loc_*}$ the objects of $\mathbf{Loc}$ are the objects of $\mathbf{Loc_*}$.
The object functors of $F$ and $G$ are the identity object functor and so $GF$ is the identity object functor.
Let $f : L_1 \to L_2$ be a continuous map of $\mathbf{Loc}$.
By definition of continuous... | :$GF = \operatorname{id}_{\mathbf {Loc}}$ | By definition of [[Definition:Category of Locales|$\mathbf{Loc}$]] and [[Definition:Category of Locales with Localic Mappings|$\mathbf {Loc_*}$]] the [[Definition:Object|objects]] of $\mathbf{Loc}$ are the [[Definition:Object|objects]] of $\mathbf{Loc_*}$.
The [[Definition:Object Functor|object functors]] of $F$ and $... | Category of Locales with Localic Mappings is Isomorphic to Category of Locales/Lemma 3 | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales/Lemma_3 | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales/Lemma_3 | [
"Category of Locales with Localic Mappings is Isomorphic to Category of Locales"
] | [] | [
"Definition:Category of Locales",
"Definition:Category of Locales with Localic Mappings",
"Definition:Object",
"Definition:Object",
"Definition:Object Functor",
"Definition:Identity Functor",
"Definition:Object Functor",
"Definition:Identity Functor",
"Definition:Object Functor",
"Definition:Conti... |
proofwiki-21841 | Category of Locales with Localic Mappings is Isomorphic to Category of Locales/Lemma 4 | :$FG = \operatorname{id}_{\mathbf {Loc_*}}$ | By definition of $\mathbf{Loc_*}$ and $\mathbf {Loc}$ the objects of $\mathbf{Loc_*}$ are the objects of $\mathbf{Loc}$.
The object functors of $G$ and $F$ are the identity object functor and so $FG$ is the identity object functor.
Let $g : L_1 \to L_2$ be a localic mapping of $\mathbf{Loc_*}$.
By definition of localic... | :$FG = \operatorname{id}_{\mathbf {Loc_*}}$ | By definition of [[Definition:Category of Locales with Localic Mappings|$\mathbf{Loc_*}$]] and [[Definition:Category of Locales|$\mathbf {Loc}$]] the [[Definition:Object|objects]] of $\mathbf{Loc_*}$ are the [[Definition:Object|objects]] of $\mathbf{Loc}$.
The [[Definition:Object Functor|object functors]] of $G$ and $... | Category of Locales with Localic Mappings is Isomorphic to Category of Locales/Lemma 4 | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales/Lemma_4 | https://proofwiki.org/wiki/Category_of_Locales_with_Localic_Mappings_is_Isomorphic_to_Category_of_Locales/Lemma_4 | [
"Category of Locales with Localic Mappings is Isomorphic to Category of Locales"
] | [] | [
"Definition:Category of Locales with Localic Mappings",
"Definition:Category of Locales",
"Definition:Object",
"Definition:Object",
"Definition:Object Functor",
"Definition:Identity Functor",
"Definition:Object Functor",
"Definition:Identity Functor",
"Definition:Object Functor",
"Definition:Conti... |
proofwiki-21842 | Cube of n minus 23 Greater than Square of (4n-7) | Let $n \in \Z$ such that $n \ge 12$.
Then:
:$n^3 - 23 > \paren {4 n - 7}^2$ | The proof proceeds by induction.
For all $n \in \Z$ such that $n \ge 12$, let $\map P n$ be the proposition:
:$n^3 - 23 > \paren {4 n - 7}^2$ | Let $n \in \Z$ such that $n \ge 12$.
Then:
:$n^3 - 23 > \paren {4 n - 7}^2$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z$ such that $n \ge 12$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$n^3 - 23 > \paren {4 n - 7}^2$ | Cube of n minus 23 Greater than Square of (4n-7) | https://proofwiki.org/wiki/Cube_of_n_minus_23_Greater_than_Square_of_(4n-7) | https://proofwiki.org/wiki/Cube_of_n_minus_23_Greater_than_Square_of_(4n-7) | [
"Inequalities"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-21843 | Point with Zero Second Derivative is not necessarily Point of Inflection | Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.
Let
:$\map {f' '} \xi = 0$
where $\map {f' '} \xi$ denotes the second derivative of $f$ at $\xi \in \openint a b$.
Then it is not necessarily the case that $f$ has a point of inflection at $\xi$. | Consider the function:
:$f: \R \to \R: \forall x \in \R: \map f x = x^4$
This has a local minimum at $x = 0$ at which $\map {f' '} x = 0$.
But $x = 0$ is not a point of inflection of $f$.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice]] [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Let
:$\map {f' '} \xi = 0$
where $\map {f' '} \xi$ denotes the [[Definition:Second Deri... | Consider the [[Definition:Real Function|function]]:
:$f: \R \to \R: \forall x \in \R: \map f x = x^4$
This has a [[Definition:Local Minimum|local minimum]] at $x = 0$ at which $\map {f' '} x = 0$.
But $x = 0$ is not a [[Definition:Point of Inflection|point of inflection]] of $f$.
{{qed}} | Point with Zero Second Derivative is not necessarily Point of Inflection | https://proofwiki.org/wiki/Point_with_Zero_Second_Derivative_is_not_necessarily_Point_of_Inflection | https://proofwiki.org/wiki/Point_with_Zero_Second_Derivative_is_not_necessarily_Point_of_Inflection | [
"Points of Inflection",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Point of Inflection"
] | [
"Definition:Real Function",
"Definition:Minimum Value of Real Function/Local",
"Definition:Point of Inflection"
] |
proofwiki-21844 | Information Contained in Letter of Alphabet | Let $\psi$ be a letter of the English alphabet.
To a first degree of approximation, the quantity of information contained in $\psi$ is $4 \cdotp 7$.
That is, there is approximately $4 \cdotp 7$ times as much information conveyed by transmission of a single letter as a single bit. | Let it be assumed for the purposes of this exercise that each letter has an equal probability of occurring as an element of a message.
There are $26$ letters of the English alphabet.
The amount of information contained in $\psi$ is therefore:
:$\map I \psi = \dfrac {\lg 26} {\lg 2} \approx 4 \cdotp 7$
where $\lg$ denot... | Let $\psi$ be a [[Definition:Letter of Alphabet|letter]] of the [[Definition:Natural Language|English alphabet]].
To a first degree of approximation, the quantity of [[Definition:Information|information]] contained in $\psi$ is $4 \cdotp 7$.
That is, there is approximately $4 \cdotp 7$ times as much [[Definition:Info... | Let it be assumed for the purposes of this exercise that each [[Definition:Letter of Alphabet|letter]] has an equal [[Definition:Probability|probability]] of [[Definition:Occurrence|occurring]] as an [[Definition:Element|element]] of a [[Definition:Message|message]].
There are $26$ [[Definition:Letter of Alphabet|lett... | Information Contained in Letter of Alphabet | https://proofwiki.org/wiki/Information_Contained_in_Letter_of_Alphabet | https://proofwiki.org/wiki/Information_Contained_in_Letter_of_Alphabet | [
"Information Contained in Letter of Alphabet",
"Information",
"Information Theory"
] | [
"Definition:Letter of Alphabet",
"Definition:Natural Language",
"Definition:Information",
"Definition:Information",
"Definition:Transmission",
"Definition:Letter of Alphabet",
"Definition:Bit"
] | [
"Definition:Letter of Alphabet",
"Definition:Probability",
"Definition:Occurrence",
"Definition:Element",
"Definition:Message",
"Definition:Letter of Alphabet",
"Definition:Natural Language",
"Definition:Information",
"Definition:General Logarithm/Binary"
] |
proofwiki-21845 | Floor Function/Examples/Floor of 4.35 | :$\floor {4 \cdotp 35} = 4$ | We have that:
:$4 \le 4 \cdotp 35 < 5$
Hence $4$ is the floor of $4 \cdotp 35$ by definition.
{{qed}} | :$\floor {4 \cdotp 35} = 4$ | We have that:
:$4 \le 4 \cdotp 35 < 5$
Hence $4$ is the [[Definition:Floor Function|floor]] of $4 \cdotp 35$ by definition.
{{qed}} | Floor Function/Examples/Floor of 4.35 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_4.35 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_4.35 | [
"Examples of Floor Function"
] | [] | [
"Definition:Floor Function"
] |
proofwiki-21846 | Ceiling Function/Examples/Ceiling of 4.35 | :$\ceiling {4 \cdotp 35} = 5$ | We have that:
:$4 < 4 \cdotp 35 \le 5$
Hence $5$ is the ceiling of $4 \cdotp 35$ by definition.
{{qed}} | :$\ceiling {4 \cdotp 35} = 5$ | We have that:
:$4 < 4 \cdotp 35 \le 5$
Hence $5$ is the [[Definition:Ceiling Function|ceiling]] of $4 \cdotp 35$ by definition.
{{qed}} | Ceiling Function/Examples/Ceiling of 4.35 | https://proofwiki.org/wiki/Ceiling_Function/Examples/Ceiling_of_4.35 | https://proofwiki.org/wiki/Ceiling_Function/Examples/Ceiling_of_4.35 | [
"Examples of Ceiling Function"
] | [] | [
"Definition:Ceiling Function"
] |
proofwiki-21847 | Primitives which Differ by Constant/Corollary | Let $f$ be an integrable function on the closed interval $\closedint a b$.
Then there exist an uncountable number of primitives for $f$ on $\closedint a b$. | By definition of integrable function, $f$ has a primitive $F$ (at least one).
By Primitives which Differ by Constant, for every real number $C$, if $\map F x$ is a primitive of $f$, then so is $\map G x$ where:
:$\forall x \in \closedint a b: \map G x = \map F x + c$
The Real Numbers are Uncountable.
Hence the result.
... | Let $f$ be an [[Definition:Integrable Function|integrable function]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then there exist an [[Definition:Uncountable Set|uncountable number]] of [[Definition:Primitive (Calculus)|primitives]] for $f$ on $\closedint a b$. | By definition of [[Definition:Integrable Function|integrable function]], $f$ has a [[Definition:Primitive (Calculus)|primitive]] $F$ (at least one).
By [[Primitives which Differ by Constant]], for every [[Definition:Real Number|real number]] $C$, if $\map F x$ is a [[Definition:Primitive (Calculus)|primitive]] of $f$,... | Primitives which Differ by Constant/Corollary | https://proofwiki.org/wiki/Primitives_which_Differ_by_Constant/Corollary | https://proofwiki.org/wiki/Primitives_which_Differ_by_Constant/Corollary | [
"Primitives which Differ by Constant"
] | [
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Uncountable/Set",
"Definition:Primitive (Calculus)"
] | [
"Definition:Integrable Function",
"Definition:Primitive (Calculus)",
"Primitives which Differ by Constant",
"Definition:Real Number",
"Definition:Primitive (Calculus)",
"Real Numbers are Uncountably Infinite",
"Category:Primitives which Differ by Constant"
] |
proofwiki-21848 | Position of Interpolated Point under Linear Interpolation | Let $f$ be a real function.
Let $y_1, y_2, \ldots, y_n$ be known values of $f$ corresponding to $x_1, x_2, \ldots, x_n$ respectively.
Let $x'$ be in the domain of $f$ such that $x_i < x' < x_{i + 1}$.
Let $y' = \map f {x'}$ be determined according to linear interpolation.
Then:
:$y' = y_i + \dfrac {\paren {x' - x_i} \p... | By definition of linear interpolation, the three points $\tuple {x_i, y_i}$, $\tuple {x', y'}$ and $\tuple {x_{i + 1}, y_{i + 1} }$ are all collinear.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {y' - y_i} {x' - x_i}
| r = \dfrac {y_{i + 1} - y_i} {x_{i + 1} - x_i}
| c = Two-Point Form of Equation of Straight Li... | Let $f$ be a [[Definition:Real Function|real function]].
Let $y_1, y_2, \ldots, y_n$ be [[Definition:Known|known]] [[Definition:Value of Element under Mapping|values]] of $f$ corresponding to $x_1, x_2, \ldots, x_n$ respectively.
Let $x'$ be in the [[Definition:Domain of Mapping|domain]] of $f$ such that $x_i < x' < ... | By definition of [[Definition:Linear Interpolation|linear interpolation]], the three [[Definition:Point|points]] $\tuple {x_i, y_i}$, $\tuple {x', y'}$ and $\tuple {x_{i + 1}, y_{i + 1} }$ are all [[Definition:Collinear Points|collinear]].
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {y' - y_i} {x' - x_i}
| r = \dfra... | Position of Interpolated Point under Linear Interpolation | https://proofwiki.org/wiki/Position_of_Interpolated_Point_under_Linear_Interpolation | https://proofwiki.org/wiki/Position_of_Interpolated_Point_under_Linear_Interpolation | [
"Linear Interpolation"
] | [
"Definition:Real Function",
"Definition:Given",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Interpolation/Linear"
] | [
"Definition:Interpolation/Linear",
"Definition:Point",
"Definition:Collinear/Points",
"Equation of Straight Line in Plane/Two-Point Form",
"Definition:Multiplication/Real Numbers"
] |
proofwiki-21849 | Equation of Catenary/Whewell | The '''catenary''' can be described by the Whewell equation:
:$s = a \tan \psi$
where:
:$s$ is the arc length
:$\psi$ is the turning angle
:$a$ is a constant. | By definition, the '''catenary''' is the shape made by an ideally flexible chain hanging freely from two arbitrary points.
Let the '''catenary''' $\CC$ lie in a Cartesian plane.
Let the lowest point on $\CC$ be $P_0$.
Let $P$ be an arbitrary point on the chain.
Let $s$ be the length along the arc of the chain from the ... | The '''[[Definition:Catenary|catenary]]''' can be described by the [[Definition:Whewell Equation|Whewell equation]]:
:$s = a \tan \psi$
where:
:$s$ is the [[Definition:Arc Length|arc length]]
:$\psi$ is the [[Definition:Turning Angle|turning angle]]
:$a$ is a [[Definition:Constant|constant]]. | By definition, the '''[[Definition:Catenary|catenary]]''' is the shape made by an [[Definition:Ideal (Physics)|ideally]] [[Definition:Flexible Chain|flexible chain]] hanging freely from two arbitrary [[Definition:Point|points]].
Let the '''[[Definition:Catenary|catenary]]''' $\CC$ lie in a [[Definition:Cartesian Plan... | Equation of Catenary/Whewell | https://proofwiki.org/wiki/Equation_of_Catenary/Whewell | https://proofwiki.org/wiki/Equation_of_Catenary/Whewell | [
"Catenary",
"Whewell Equations"
] | [
"Definition:Catenary",
"Definition:Intrinsic Equation/Whewell Equation",
"Definition:Arc Length",
"Definition:Turning Angle",
"Definition:Constant"
] | [
"Definition:Catenary",
"Definition:Ideal (Physics)",
"Definition:Flexible Chain",
"Definition:Point",
"Definition:Catenary",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Point",
"Definition:Chain (Physics)",
"Definition:Linear Measure",
"Definition:Curve/Arc",
"Definition:Flex... |
proofwiki-21850 | Image of Element under Inverse Mapping/Corollary 2 | :$\forall y \in T: \map f {\map {f^{-1} } y} = y$ | {{begin-eqn}}
{{eqn | q = \forall x \in S, y \in T
| l = \map f x
| r = y
| c =
}}
{{eqn | lo= \iff
| l = \map {f^{-1} } y
| r = x
| c = Image of Element under Inverse Mapping
}}
{{eqn | q = \forall y \in T
| ll= \leadsto
| l = \map f {\map {f^{-1} } y}
| r = y
... | :$\forall y \in T: \map f {\map {f^{-1} } y} = y$ | {{begin-eqn}}
{{eqn | q = \forall x \in S, y \in T
| l = \map f x
| r = y
| c =
}}
{{eqn | lo= \iff
| l = \map {f^{-1} } y
| r = x
| c = [[Image of Element under Inverse Mapping]]
}}
{{eqn | q = \forall y \in T
| ll= \leadsto
| l = \map f {\map {f^{-1} } y}
| r = y... | Image of Element under Inverse Mapping/Corollary 2 | https://proofwiki.org/wiki/Image_of_Element_under_Inverse_Mapping/Corollary_2 | https://proofwiki.org/wiki/Image_of_Element_under_Inverse_Mapping/Corollary_2 | [
"Image of Element under Inverse Mapping"
] | [] | [
"Image of Element under Inverse Mapping"
] |
proofwiki-21851 | Image of Element under Inverse Mapping/Corollary 1 | :$\forall x \in S: \map {f^{-1} } {\map f x} = x$ | {{begin-eqn}}
{{eqn | q = \forall x \in S, y \in T
| l = \map f x
| r = y
| c =
}}
{{eqn | lo= \iff
| l = \map {f^{-1} } y
| r = x
| c = Image of Element under Inverse Mapping
}}
{{eqn | q = \forall x \in S
| ll= \leadsto
| l = \map {f^{-1} } {\map f x}
| r = x
... | :$\forall x \in S: \map {f^{-1} } {\map f x} = x$ | {{begin-eqn}}
{{eqn | q = \forall x \in S, y \in T
| l = \map f x
| r = y
| c =
}}
{{eqn | lo= \iff
| l = \map {f^{-1} } y
| r = x
| c = [[Image of Element under Inverse Mapping]]
}}
{{eqn | q = \forall x \in S
| ll= \leadsto
| l = \map {f^{-1} } {\map f x}
| r = x... | Image of Element under Inverse Mapping/Corollary 1 | https://proofwiki.org/wiki/Image_of_Element_under_Inverse_Mapping/Corollary_1 | https://proofwiki.org/wiki/Image_of_Element_under_Inverse_Mapping/Corollary_1 | [
"Image of Element under Inverse Mapping"
] | [] | [
"Image of Element under Inverse Mapping"
] |
proofwiki-21852 | Inverse of Strictly Monotone Continuous Real Function is Strictly Monotone and Continuous | Let $f$ be a continuous real function which is defined on the closed interval $I := \closedint a b$.
Let $f$ be strictly monotone on $I$.
Then $f$ has an inverse function $f^{-1}$ which is continuous and strictly monotone on $f \sqbrk I$. | The function $f$ is a bijection from Strictly Monotone Real Function is Bijective.
From Inverse of Strictly Monotone Function, $f^{-1}$ is strictly monotone on on $f \sqbrk I$.
From Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone, $f^{-1}$ is a continuous real function.
Hence the result.... | Let $f$ be a [[Definition:Continuous Real Function|continuous real function]] which is defined on the [[Definition:Closed Real Interval|closed interval]] $I := \closedint a b$.
Let $f$ be [[Definition:Strictly Monotone Real Function|strictly monotone]] on $I$.
Then $f$ has an [[Definition:Inverse Mapping|inverse fun... | The function $f$ is a [[Definition:Bijection|bijection]] from [[Strictly Monotone Real Function is Bijective]].
From [[Inverse of Strictly Monotone Function]], $f^{-1}$ is [[Definition:Strictly Monotone Real Function|strictly monotone]] on on $f \sqbrk I$.
From [[Continuous Real Function on Closed Interval is Biject... | Inverse of Strictly Monotone Continuous Real Function is Strictly Monotone and Continuous | https://proofwiki.org/wiki/Inverse_of_Strictly_Monotone_Continuous_Real_Function_is_Strictly_Monotone_and_Continuous | https://proofwiki.org/wiki/Inverse_of_Strictly_Monotone_Continuous_Real_Function_is_Strictly_Monotone_and_Continuous | [
"Inverse of Strictly Monotone Continuous Real Function is Strictly Monotone and Continuous",
"Strictly Monotone Real Functions",
"Continuous Real Functions",
"Inverse Mappings"
] | [
"Definition:Continuous Real Function",
"Definition:Real Interval/Closed",
"Definition:Strictly Monotone/Real Function",
"Definition:Inverse Mapping",
"Definition:Continuous Real Function",
"Definition:Strictly Monotone/Real Function"
] | [
"Definition:Bijection",
"Strictly Monotone Real Function is Bijective",
"Inverse of Strictly Monotone Function",
"Definition:Strictly Monotone/Real Function",
"Continuous Real Function on Closed Interval is Bijective iff Strictly Monotone",
"Definition:Continuous Real Function"
] |
proofwiki-21853 | Area under Arc of Sine Function | :$\ds \int_0^\pi \sin x \rd x = 2$ | {{begin-eqn}}
{{eqn | l = \int_0^\pi \sin x \rd x
| r = \bigintlimits {-\cos x} 0 \pi
| c = Primitive of Sine Function
}}
{{eqn | r = 2
| c = Cosine of $\pi$, {{cos|0}}
}}
{{end-eqn}}
{{qed}}
Category:Sine Function
Category:Definite Integrals involving Sine Function
scun1ir668xgx0evll4r7ss32kk3bvw | :$\ds \int_0^\pi \sin x \rd x = 2$ | {{begin-eqn}}
{{eqn | l = \int_0^\pi \sin x \rd x
| r = \bigintlimits {-\cos x} 0 \pi
| c = [[Primitive of Sine Function]]
}}
{{eqn | r = 2
| c = [[Cosine of Straight Angle|Cosine of $\pi$]], {{cos|0}}
}}
{{end-eqn}}
{{qed}}
[[Category:Sine Function]]
[[Category:Definite Integrals involving Sine Func... | Area under Arc of Sine Function | https://proofwiki.org/wiki/Area_under_Arc_of_Sine_Function | https://proofwiki.org/wiki/Area_under_Arc_of_Sine_Function | [
"Sine Function",
"Definite Integrals involving Sine Function"
] | [] | [
"Primitive of Sine Function",
"Cosine of Straight Angle",
"Category:Sine Function",
"Category:Definite Integrals involving Sine Function"
] |
proofwiki-21854 | Matroid Rank Function Iff Matroid 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 $M = \struct{S, \mathscr I}$.
{{iff}}:
:$\rho$ satisfies the rank axioms | Follows immediately from:
:* Formulation 1 Rank Axioms Implies Rank Function of Matroid
:* Rank Function of Matroid Satisfies Formulation 1 Rank Axioms | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Then:
:$\rho$ is the [[Definition:Rank Function (Matroid)|rank function]] of a [[Definition:Matroid|matroid... | Follows immediately from:
:* [[Formulation 1 Rank Axioms Implies Rank Function of Matroid]]
:* [[Rank Function of Matroid Satisfies Formulation 1 Rank Axioms]] | Matroid Rank Function Iff Matroid Rank Axioms | https://proofwiki.org/wiki/Matroid_Rank_Function_Iff_Matroid_Rank_Axioms | https://proofwiki.org/wiki/Matroid_Rank_Function_Iff_Matroid_Rank_Axioms | [
"Matroid Rank Functions"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Definition:Rank Function (Matroid)",
"Definition:Matroid",
"Axiom:Rank Axioms (Matroid)"
] | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid",
"Rank Function of Matroid Satisfies Formulation 1 Rank Axioms"
] |
proofwiki-21855 | Principle of Open Induction for Real Numbers | Let $a < b$ be real numbers.
Let $S$ be an open set of real numbers.
Suppose that, for every $x \in \closedint a b$ such that:
:$\hointr a x \subseteq S$
it also holds that:
:$x \in S$
Then, $\closedint a b \subseteq S$. | {{AimForCont}} there exists some $x \in \closedint a b$ such that:
:$x \notin S$
Let:
:$T := \closedint a b \setminus S$
be the set of all such $x$.
By Set Difference is Subset:
:$T \subseteq \closedint a b$
so by Subset of Bounded Below Set is Bounded Below:
:$T$ is bounded below.
Since $T$ is non-empty by assumption,... | Let $a < b$ be [[Definition:Real Number|real numbers]].
Let $S$ be an [[Definition:Open Set of Real Numbers|open set of real numbers]].
Suppose that, for every $x \in \closedint a b$ such that:
:$\hointr a x \subseteq S$
it also holds that:
:$x \in S$
Then, $\closedint a b \subseteq S$. | {{AimForCont}} there exists some $x \in \closedint a b$ such that:
:$x \notin S$
Let:
:$T := \closedint a b \setminus S$
be the [[Definition:Set|set]] of all such $x$.
By [[Set Difference is Subset]]:
:$T \subseteq \closedint a b$
so by [[Subset of Bounded Below Set is Bounded Below]]:
:$T$ is [[Definition:Bounded Be... | Principle of Open Induction for Real Numbers | https://proofwiki.org/wiki/Principle_of_Open_Induction_for_Real_Numbers | https://proofwiki.org/wiki/Principle_of_Open_Induction_for_Real_Numbers | [
"Mathematical Induction",
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Open Set/Real Analysis/Real Numbers"
] | [
"Definition:Set",
"Set Difference is Subset",
"Subset of Bounded Below Set is Bounded Below",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Non-Empty Set",
"Greatest Lower Bound Property",
"Definition:Infimum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Definition:Lowe... |
proofwiki-21856 | Equivalence of Definitions of Matroid Rank Axioms/Formulation 2 Implies Formulation 1 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 2 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 2}}
Then $\rho$ satisfies formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}} | ==== $\rho$ satisfies $(\text R 1)$ ====
We have:
{{begin-eqn}}
{{eqn | l = 0
| o = \le
| r = \map \rho \O
| c = Rank axiom $(\text R 4)$
}}
{{eqn | o = \le
| r = \card \O
| c = Rank axiom $(\text R 4)$
}}
{{eqn | r = 0
| c = Cardinality of Empty Set
}}
{{end-eqn}}
Hence:
:$\map \r... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 2|formulation 2]] of the [[Axiom:Rank Axioms (M... | ==== $\rho$ satisfies $(\text R 1)$ ====
We have:
{{begin-eqn}}
{{eqn | l = 0
| o = \le
| r = \map \rho \O
| c = [[Axiom:Rank Axioms (Matroid)/Definition 2|Rank axiom $(\text R 4)$]]
}}
{{eqn | o = \le
| r = \card \O
| c = [[Axiom:Rank Axioms (Matroid)/Definition 2|Rank axiom $(\text R ... | Equivalence of Definitions of Matroid Rank Axioms/Formulation 2 Implies Formulation 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Rank_Axioms/Formulation_2_Implies_Formulation_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Rank_Axioms/Formulation_2_Implies_Formulation_1 | [
"Equivalence of Definitions of Matroid Rank Axioms"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 2",
"Axiom:Rank Axioms (Matroid)",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)"
] | [
"Axiom:Rank Axioms (Matroid)/Definition 2",
"Axiom:Rank Axioms (Matroid)/Definition 2",
"Cardinality of Empty Set",
"Axiom:Rank Axioms (Matroid)/Definition 2",
"Axiom:Rank Axioms (Matroid)/Definition 2",
"Axiom:Rank Axioms (Matroid)/Definition 2",
"Cardinality of Singleton",
"Axiom:Rank Axioms (Matroi... |
proofwiki-21857 | Equivalence of Definitions of Matroid Rank Axioms/Formulation 1 Implies Formulation 2 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Then $\rho$ satisfies formulation 2 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 2}} | Follows immediately from:
:* Formulation 1 Rank Axioms Implies Rank Function of Matroid
:* Rank Function of Matroid Satisfies Formulation 2 Rank Axioms | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (M... | Follows immediately from:
:* [[Formulation 1 Rank Axioms Implies Rank Function of Matroid]]
:* [[Rank Function of Matroid Satisfies Formulation 2 Rank Axioms]] | Equivalence of Definitions of Matroid Rank Axioms/Formulation 1 Implies Formulation 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Rank_Axioms/Formulation_1_Implies_Formulation_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Rank_Axioms/Formulation_1_Implies_Formulation_2 | [
"Equivalence of Definitions of Matroid Rank Axioms"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Axiom:Rank Axioms (Matroid)/Definition 2",
"Axiom:Rank Axioms (Matroid)"
] | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid",
"Rank Function of Matroid Satisfies Formulation 2 Rank Axioms"
] |
proofwiki-21858 | Limit to Infinity of Binomial Coefficient over Power | Let $k \in \R \setminus \set {-1, -2, -3, \dotsc}$.
Then:
:$\ds \lim_{r \mathop \to \infty} \dfrac {\dbinom r k} {r^k} = \frac 1 {\map \Gamma {k + 1} }$ | {{begin-eqn}}
{{eqn | l = \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k}
| r = \lim_{r \mathop \to \infty} \frac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} r^k}
| c = Gamma Function Extends Factorial
}}
{{eqn | r = \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \f... | Let $k \in \R \setminus \set {-1, -2, -3, \dotsc}$.
Then:
:$\ds \lim_{r \mathop \to \infty} \dfrac {\dbinom r k} {r^k} = \frac 1 {\map \Gamma {k + 1} }$ | {{begin-eqn}}
{{eqn | l = \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k}
| r = \lim_{r \mathop \to \infty} \frac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} r^k}
| c = [[Gamma Function Extends Factorial]]
}}
{{eqn | r = \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} ... | Limit to Infinity of Binomial Coefficient over Power | https://proofwiki.org/wiki/Limit_to_Infinity_of_Binomial_Coefficient_over_Power | https://proofwiki.org/wiki/Limit_to_Infinity_of_Binomial_Coefficient_over_Power | [
"Binomial Coefficients",
"Limit to Infinity of Binomial Coefficient over Power"
] | [] | [
"Gamma Function Extends Factorial",
"Stirling's Formula for Gamma Function",
"Exponent Combination Laws/Quotient of Powers",
"Exponent Combination Laws/Product of Powers",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Exponent Combination Laws/Product of Powers"
] |
proofwiki-21859 | Limit to Infinity of Binomial Coefficient over Power/Integer | Let $k \in \N$.
Then:
:$\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k} = \frac 1 {k!}$ | {{begin-eqn}}
{{eqn | q = \forall r \in \R
| l = \frac {\dbinom r k} {r^k}
| r = \frac 1 {k !} \cdot \frac {r^{\underline k} } {r^k}
| c = {{Defof|Binomial Coefficient/Real Numbers|Binomial Coefficient}}
}}
{{eqn | r = \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \frac {r - j} r
| c = {{Defof|Fal... | Let $k \in \N$.
Then:
:$\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k} = \frac 1 {k!}$ | {{begin-eqn}}
{{eqn | q = \forall r \in \R
| l = \frac {\dbinom r k} {r^k}
| r = \frac 1 {k !} \cdot \frac {r^{\underline k} } {r^k}
| c = {{Defof|Binomial Coefficient/Real Numbers|Binomial Coefficient}}
}}
{{eqn | r = \frac 1 {k !} \prod_{j \mathop = 0}^{k - 1} \frac {r - j} r
| c = {{Defof|Fal... | Limit to Infinity of Binomial Coefficient over Power/Integer | https://proofwiki.org/wiki/Limit_to_Infinity_of_Binomial_Coefficient_over_Power/Integer | https://proofwiki.org/wiki/Limit_to_Infinity_of_Binomial_Coefficient_over_Power/Integer | [
"Limit to Infinity of Binomial Coefficient over Power"
] | [] | [
"Combination Theorem for Sequences/Real/Product Rule",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Sequence of Powers of Reciprocals is Null Sequence/Corollary",
"Category:Limit to Infinity of Binomial Coefficient over Power"
] |
proofwiki-21860 | Cover is Cover of Subset | Let $S$ be a set.
Let $\CC$ be a cover of $S$.
Let $T \subseteq S$ be a subset of $S$.
Then, $\CC$ is a cover of $T$. | By definition of a cover:
:$\ds S \subseteq \bigcup C$
But then, by Subset Relation is Transitive:
:$\ds T \subseteq \bigcup C$
Therefore, $C$ is a cover of $T$ by definition.
{{qed}}
Category:Covers
gwxoy47g2air474ny30lt1cw1t9uby5 | Let $S$ be a [[Definition:Set|set]].
Let $\CC$ be a [[Definition:Cover of Set|cover]] of $S$.
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Then, $\CC$ is a [[Definition:Cover of Set|cover]] of $T$. | By definition of a [[Definition:Cover of Set|cover]]:
:$\ds S \subseteq \bigcup C$
But then, by [[Subset Relation is Transitive]]:
:$\ds T \subseteq \bigcup C$
Therefore, $C$ is a [[Definition:Cover of Set|cover]] of $T$ by definition.
{{qed}}
[[Category:Covers]]
gwxoy47g2air474ny30lt1cw1t9uby5 | Cover is Cover of Subset | https://proofwiki.org/wiki/Cover_is_Cover_of_Subset | https://proofwiki.org/wiki/Cover_is_Cover_of_Subset | [
"Covers"
] | [
"Definition:Set",
"Definition:Cover of Set",
"Definition:Subset",
"Definition:Cover of Set"
] | [
"Definition:Cover of Set",
"Subset Relation is Transitive",
"Definition:Cover of Set",
"Category:Covers"
] |
proofwiki-21861 | Outer Jordan Content is Monotone | Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space.
Suppose that $A \subseteq B$.
Then:
:$\map {m^*} A \le \map {m^*} B$
where $m^*$ denotes the outer Jordan content. | Let $C$ be a finite covering of $B$ by closed rectangles.
Since $A \subseteq B$, by Cover is Cover of Subset:
:$C$ is a finite covering of $A$ by closed rectangles.
Therefore, by definition of the outer Jordan content:
:$\ds \map {m^*} A \le \sum_{R \mathop \in C} \map V R$
But, since $C$ was arbitrary:
:$\map {m^*} A$... | Let $A, B \subseteq \R^n$ be [[Definition:Bounded (Metric Space)|bounded]] [[Definition:Subspace|subspaces]] of [[Definition:Euclidean Space|Euclidean $n$-space]].
Suppose that $A \subseteq B$.
Then:
:$\map {m^*} A \le \map {m^*} B$
where $m^*$ denotes the [[Definition:Outer Jordan Content|outer Jordan content]]. | Let $C$ be a [[Definition:Finite Cover|finite covering]] of $B$ by [[Definition:Closed Rectangle|closed rectangles]].
Since $A \subseteq B$, by [[Cover is Cover of Subset]]:
:$C$ is a [[Definition:Finite Cover|finite covering]] of $A$ by [[Definition:Closed Rectangle|closed rectangles]].
Therefore, by definition of t... | Outer Jordan Content is Monotone | https://proofwiki.org/wiki/Outer_Jordan_Content_is_Monotone | https://proofwiki.org/wiki/Outer_Jordan_Content_is_Monotone | [
"Outer Jordan Content",
"Measure Theory"
] | [
"Definition:Bounded Metric Space",
"Definition:Subspace",
"Definition:Euclidean Space",
"Definition:Outer Jordan Content"
] | [
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Cover is Cover of Subset",
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Definition:Outer Jordan Content",
"Definition:Lower Bound of Set",
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Defini... |
proofwiki-21862 | Inner Jordan Content is Monotone | Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space.
Suppose that $A \subseteq B$.
Then:
:$\map {m_*} A \le \map {m_*} B$
where $m_*$ denotes the inner Jordan content. | Let $R \subseteq \R^n$ be a closed $n$-rectangle that contains $B$.
Then, by Subset Relation is Transitive, $R$ contains $A$ as well.
By Set Difference with Subset is Superset of Set Difference:
:$R \setminus B \subseteq R \setminus A$
So, by Outer Jordan Content is Monotone:
:$\map {m^*} {R \setminus B} \le \map {m^*}... | Let $A, B \subseteq \R^n$ be [[Definition:Bounded (Metric Space)|bounded]] [[Definition:Subspace|subspaces]] of [[Definition:Euclidean Space|Euclidean $n$-space]].
Suppose that $A \subseteq B$.
Then:
:$\map {m_*} A \le \map {m_*} B$
where $m_*$ denotes the [[Definition:Inner Jordan Content|inner Jordan content]]. | Let $R \subseteq \R^n$ be a [[Definition:Closed Rectangle|closed $n$-rectangle]] that [[Definition:Set Containment|contains]] $B$.
Then, by [[Subset Relation is Transitive]], $R$ [[Definition:Set Containment|contains]] $A$ as well.
By [[Set Difference with Subset is Superset of Set Difference]]:
:$R \setminus B \sub... | Inner Jordan Content is Monotone | https://proofwiki.org/wiki/Inner_Jordan_Content_is_Monotone | https://proofwiki.org/wiki/Inner_Jordan_Content_is_Monotone | [
"Jordan Content"
] | [
"Definition:Bounded Metric Space",
"Definition:Subspace",
"Definition:Euclidean Space",
"Definition:Inner Jordan Content"
] | [
"Definition:Closed Rectangle",
"Definition:Subset",
"Subset Relation is Transitive",
"Definition:Subset",
"Set Difference with Subset is Superset of Set Difference",
"Outer Jordan Content is Monotone",
"Definition:Inner Jordan Content",
"Category:Jordan Content"
] |
proofwiki-21863 | Union of Covers is Cover of Union | Let $\sequence {S_i}_{i \in I}$ be an indexed family of sets.
For each $i \in I$, let $\CC_i$ be a cover of $S_i$.
Then, $\ds \bigcup_{i \mathop \in I} \CC_i$ is a cover of $\ds \bigcup_{i \mathop \in I} S_i$. | Let $\ds x \in \bigcup_{i \mathop \in I} S_i$ be arbitrary.
By definition of union, there is some $i \in I$ such that:
:$x \in S_i$
Then, by definition of cover, there is some $C \in \CC_i$ such that:
:$x \in C$
But, by definition of union:
:$\ds C \in \bigcup_{i \mathop \in I} \CC_i$
Hence, there is some $\ds C \in \b... | Let $\sequence {S_i}_{i \in I}$ be an [[Definition:Indexed Family of Sets|indexed family of sets]].
For each $i \in I$, let $\CC_i$ be a [[Definition:Cover of Set|cover]] of $S_i$.
Then, $\ds \bigcup_{i \mathop \in I} \CC_i$ is a [[Definition:Cover of Set|cover]] of $\ds \bigcup_{i \mathop \in I} S_i$. | Let $\ds x \in \bigcup_{i \mathop \in I} S_i$ be arbitrary.
By definition of [[Definition:Union of Family|union]], there is some $i \in I$ such that:
:$x \in S_i$
Then, by definition of [[Definition:Cover of Set|cover]], there is some $C \in \CC_i$ such that:
:$x \in C$
But, by definition of [[Definition:Union of Fa... | Union of Covers is Cover of Union | https://proofwiki.org/wiki/Union_of_Covers_is_Cover_of_Union | https://proofwiki.org/wiki/Union_of_Covers_is_Cover_of_Union | [
"Covers"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Cover of Set",
"Definition:Cover of Set"
] | [
"Definition:Set Union/Family of Sets",
"Definition:Cover of Set",
"Definition:Set Union/Family of Sets",
"Definition:Cover of Set",
"Category:Covers"
] |
proofwiki-21864 | Outer Jordan Content is Subadditive | Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space.
Then:
:$\map {m^*} {A \cup B} \le \map {m^*} A + \map {m^*} B$
where $m^*$ denotes the outer Jordan content. | Let $\epsilon > 0$ be arbitrary.
By Characterizing Property of Infimum of Subset of Real Numbers, select:
:$C$ to be a finite covering of $A$ by closed $n$-rectangles such that:
::$\ds \sum_{R \mathop \in C} \map V R < \map {m^*} A + \frac \epsilon 2$
:$D$ to be a finite covering of $B$ by closed $n$-rectangles such th... | Let $A, B \subseteq \R^n$ be [[Definition:Bounded (Metric Space)|bounded]] [[Definition:Subspace|subspaces]] of [[Definition:Euclidean Space|Euclidean $n$-space]].
Then:
:$\map {m^*} {A \cup B} \le \map {m^*} A + \map {m^*} B$
where $m^*$ denotes the [[Definition:Outer Jordan Content|outer Jordan content]]. | Let $\epsilon > 0$ be arbitrary.
By [[Characterizing Property of Infimum of Subset of Real Numbers]], select:
:$C$ to be a [[Definition:Finite Cover|finite covering]] of $A$ by [[Definition:Closed Rectangle|closed $n$-rectangles]] such that:
::$\ds \sum_{R \mathop \in C} \map V R < \map {m^*} A + \frac \epsilon 2$
:$D... | Outer Jordan Content is Subadditive | https://proofwiki.org/wiki/Outer_Jordan_Content_is_Subadditive | https://proofwiki.org/wiki/Outer_Jordan_Content_is_Subadditive | [
"Outer Jordan Content"
] | [
"Definition:Bounded Metric Space",
"Definition:Subspace",
"Definition:Euclidean Space",
"Definition:Outer Jordan Content"
] | [
"Characterizing Property of Infimum of Subset of Real Numbers",
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Union of Finite Sets is Finite",
"Definition:Finite Set",
"Definition:Closed Rectangle",
"Union of Covers... |
proofwiki-21865 | Jordan Content is Monotone | Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space.
Suppose that $A \subseteq B$.
Further suppose that the Jordan content of both $A$ and $B$ exists.
Then:
:$\map m A \le \map m B$
where $m$ denotes the Jordan content. | By definition of the Jordan content:
:$\map m A = \map {m^*} A$
:$\map m B = \map {m^*} B$
where $m^*$ denotes the outer Jordan content.
The result follows from Outer Jordan Content is Monotone.
{{qed}}
Category:Jordan Content
qssstvz29iq77ap9a1s53n4kcsaml15 | Let $A, B \subseteq \R^n$ be [[Definition:Bounded (Metric Space)|bounded]] [[Definition:Subspace|subspaces]] of [[Definition:Euclidean Space|Euclidean $n$-space]].
Suppose that $A \subseteq B$.
Further suppose that the [[Definition:Jordan Content|Jordan content]] of both $A$ and $B$ exists.
Then:
:$\map m A \le \ma... | By definition of the [[Definition:Jordan Content|Jordan content]]:
:$\map m A = \map {m^*} A$
:$\map m B = \map {m^*} B$
where $m^*$ denotes the [[Definition:Outer Jordan Content|outer Jordan content]].
The result follows from [[Outer Jordan Content is Monotone]].
{{qed}}
[[Category:Jordan Content]]
qssstvz29iq77ap9a... | Jordan Content is Monotone | https://proofwiki.org/wiki/Jordan_Content_is_Monotone | https://proofwiki.org/wiki/Jordan_Content_is_Monotone | [
"Jordan Content"
] | [
"Definition:Bounded Metric Space",
"Definition:Subspace",
"Definition:Euclidean Space",
"Definition:Jordan Content",
"Definition:Jordan Content"
] | [
"Definition:Jordan Content",
"Definition:Outer Jordan Content",
"Outer Jordan Content is Monotone",
"Category:Jordan Content"
] |
proofwiki-21866 | Set of Intersections with Superset is Cover | Let $S$ be a set.
Let $\CC$ be a cover of $S$.
Let $T \supseteq S$ be a superset of $S$.
Then:
:$\set {C \cap T : C \in \CC}$
is a cover of $S$. | Let $x \in S$ be arbitrary.
By definition of cover, there is some $C \in \CC$ such that:
:$x \in C$
By definition of superset:
:$x \in T$
Therefore, by definition of intersection:
:$x \in C \cap T$
As $x \in S$ was arbitrary, the result follows.
{{qed}}
Category:Covers
Category:Set Intersection
srrtqp72vfb03dmcu68n0idn... | Let $S$ be a [[Definition:Set|set]].
Let $\CC$ be a [[Definition:Cover of Set|cover]] of $S$.
Let $T \supseteq S$ be a [[Definition:Superset|superset]] of $S$.
Then:
:$\set {C \cap T : C \in \CC}$
is a [[Definition:Cover of Set|cover]] of $S$. | Let $x \in S$ be arbitrary.
By definition of [[Definition:Cover of Set|cover]], there is some $C \in \CC$ such that:
:$x \in C$
By definition of [[Definition:Superset|superset]]:
:$x \in T$
Therefore, by definition of [[Definition:Set Intersection|intersection]]:
:$x \in C \cap T$
As $x \in S$ was arbitrary, the re... | Set of Intersections with Superset is Cover | https://proofwiki.org/wiki/Set_of_Intersections_with_Superset_is_Cover | https://proofwiki.org/wiki/Set_of_Intersections_with_Superset_is_Cover | [
"Covers",
"Set Intersection"
] | [
"Definition:Set",
"Definition:Cover of Set",
"Definition:Subset/Superset",
"Definition:Cover of Set"
] | [
"Definition:Cover of Set",
"Definition:Subset/Superset",
"Definition:Set Intersection",
"Category:Covers",
"Category:Set Intersection"
] |
proofwiki-21867 | Outer Jordan Content of Scaled Set | Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space.
Let $c_1, c_2, \dotsc, c_n \in \R_{\ge 0}$ be non-negative real numbers.
Let $M' \subseteq \R^n$ be defined as:
:$M' = \set {\tuple {c_1 x_1, c_2 x_2, \dotsc, c_n x_n} : \tuple {x_1, x_2, \dotsc, x_n} \in \R^n}$
Then:
:$\map {m^*} {M'} = c_1 c_2 \dots... | Let $\epsilon > 0$ be arbitrary.
By Characterizing Property of Infimum of Subset of Real Numbers, let $C$ be a finite covering of $M$ by closed $n$-rectangles such that:
:$\ds \sum_{R \mathop \in C} \map V C < \map {m^*} M + \frac \epsilon {c_1 c_2 \dotsm c_n + 1}$
Let $C'$ be defined as:
:$\ds C' = \set {\closedint {c... | Let $M \subseteq \R^n$ be a [[Definition:Bounded (Metric Space)|bounded]] [[Definition:Subspace|subspace]] of [[Definition:Euclidean Space|Euclidean $n$-space]].
Let $c_1, c_2, \dotsc, c_n \in \R_{\ge 0}$ be [[Definition:Non-Negative Real Number|non-negative real numbers]].
Let $M' \subseteq \R^n$ be defined as:
:$M'... | Let $\epsilon > 0$ be arbitrary.
By [[Characterizing Property of Infimum of Subset of Real Numbers]], let $C$ be a [[Definition:Finite Cover|finite covering]] of $M$ by [[Definition:Closed Rectangle|closed $n$-rectangles]] such that:
:$\ds \sum_{R \mathop \in C} \map V C < \map {m^*} M + \frac \epsilon {c_1 c_2 \dotsm... | Outer Jordan Content of Scaled Set | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Scaled_Set | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Scaled_Set | [
"Outer Jordan Content"
] | [
"Definition:Bounded Metric Space",
"Definition:Subspace",
"Definition:Euclidean Space",
"Definition:Positive/Real Number",
"Definition:Outer Jordan Content"
] | [
"Characterizing Property of Infimum of Subset of Real Numbers",
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Definition:Cover of Set",
"Real Number Ordering is Compatible with Multiplication",
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Real Plus Epsilon",
... |
proofwiki-21868 | Inner Jordan Content is Well-Defined | Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space.
Let:
{{begin-eqn}}
{{eqn | l = R
| r = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}
}}
{{eqn | l = R'
| r = \prod_{i \mathop = 1}^n \closedint {a'_i} {b'_i}
}}
{{end-eqn}}
be closed $n$-rectangles that contain $M$.
Then:
:$\map V R - \ma... | {{WLOG}}, suppose that $R$ and $R'$ differ in only a single bound.
We will give the proof for the case where $a_k < a'_k$, and all other bounds are equivalent; the other cases are symmetric.
Under this assumption:
:$\ds R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$
:$\ds R' = \prod_{i \mathop = 1}^{k - 1} \closed... | Let $M \subseteq \R^n$ be a [[Definition:Bounded (Metric Space)|bounded]] [[Definition:Subspace|subspace]] of [[Definition:Euclidean Space|Euclidean $n$-space]].
Let:
{{begin-eqn}}
{{eqn | l = R
| r = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}
}}
{{eqn | l = R'
| r = \prod_{i \mathop = 1}^n \closedint ... | {{WLOG}}, suppose that $R$ and $R'$ differ in only a single bound.
We will give the proof for the case where $a_k < a'_k$, and all other bounds are equivalent; the other cases are symmetric.
Under this assumption:
:$\ds R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$
:$\ds R' = \prod_{i \mathop = 1}^{k - 1} \clos... | Inner Jordan Content is Well-Defined | https://proofwiki.org/wiki/Inner_Jordan_Content_is_Well-Defined | https://proofwiki.org/wiki/Inner_Jordan_Content_is_Well-Defined | [
"Jordan Content"
] | [
"Definition:Bounded Metric Space",
"Definition:Subspace",
"Definition:Euclidean Space",
"Definition:Closed Rectangle",
"Definition:Subset",
"Definition:Content of Rectangle",
"Definition:Inner Jordan Content",
"Definition:Closed Rectangle"
] | [
"Outer Jordan Content of Subdivision",
"Definition:Subdivision of Interval/Rectangle",
"Outer Jordan Content of Open Rectangle",
"Outer Jordan Content of Subdivision",
"Category:Jordan Content"
] |
proofwiki-21869 | Euler Phi Function of 5 | :$\map \phi 5 = 4$ | From Euler Phi Function of Prime:
:$\map \phi p = p - 1$
As $5$ is a prime number it follows that:
:$\map \phi 5 = 5 - 1 = 4$
{{qed}}
Category:Examples of Euler Phi Function
Category:5
aiv01ec8b4bv1w67p6ve471m97yt5fr | :$\map \phi 5 = 4$ | From [[Euler Phi Function of Prime]]:
:$\map \phi p = p - 1$
As $5$ is a [[Definition:Prime Number|prime number]] it follows that:
:$\map \phi 5 = 5 - 1 = 4$
{{qed}}
[[Category:Examples of Euler Phi Function]]
[[Category:5]]
aiv01ec8b4bv1w67p6ve471m97yt5fr | Euler Phi Function of 5 | https://proofwiki.org/wiki/Euler_Phi_Function_of_5 | https://proofwiki.org/wiki/Euler_Phi_Function_of_5 | [
"Examples of Euler Phi Function",
"5"
] | [] | [
"Euler Phi Function of Prime",
"Definition:Prime Number",
"Category:Examples of Euler Phi Function",
"Category:5"
] |
proofwiki-21870 | Group of Units Ring of Integers Modulo p^2 is Cyclic | Let $p$ be a prime.
Let $\struct {\Z / p^2 \Z, +, \times}$ be the ring of integers modulo $p^2$.
Let $U = \struct {\paren {\Z / p^2 \Z}^\times, \times}$ denote the group of units of $\struct {\Z / p^2 \Z, +, \times}$.
Then $U$ is cyclic. | The case $p = 2$ follows from Isomorphism between Group of Units Ring of Integers Modulo $2^n$ and $C_2 \times C_{2^{n - 2} }$.
Next, we suppose $p > 2$.
From Ring of Integers Modulo Prime is Field and Group of Units of Finite Field is Cyclic, $\paren {\Z / p \Z}^\times$ is cyclic.
Let $\eqclass g p$ be a primitive ele... | Let $p$ be a [[Definition:Prime Number|prime]].
Let $\struct {\Z / p^2 \Z, +, \times}$ be the [[Definition:Ring (Abstract Algebra)|ring]] of [[Definition:Integers Modulo m|integers modulo $p^2$]].
Let $U = \struct {\paren {\Z / p^2 \Z}^\times, \times}$ denote the [[Definition:Group of Units of Ring|group of units]] o... | The case $p = 2$ follows from [[Isomorphism between Group of Units Ring of Integers Modulo 2^n and C2 x C2^(n-2)|Isomorphism between Group of Units Ring of Integers Modulo $2^n$ and $C_2 \times C_{2^{n - 2} }$]].
Next, we suppose $p > 2$.
From [[Ring of Integers Modulo Prime is Field]] and [[Group of Units of Finite ... | Group of Units Ring of Integers Modulo p^2 is Cyclic | https://proofwiki.org/wiki/Group_of_Units_Ring_of_Integers_Modulo_p^2_is_Cyclic | https://proofwiki.org/wiki/Group_of_Units_Ring_of_Integers_Modulo_p^2_is_Cyclic | [
"Ring of Integers Modulo m"
] | [
"Definition:Prime Number",
"Definition:Ring (Abstract Algebra)",
"Definition:Integers Modulo m",
"Definition:Group of Units/Ring",
"Definition:Cyclic Group"
] | [
"Isomorphism between Group of Units Ring of Integers Modulo 2^n and C2 x C2^(n-2)",
"Ring of Integers Modulo Prime is Field",
"Group of Units of Finite Field is Cyclic",
"Definition:Cyclic Group",
"Definition:Primitive Element of Cyclic Modulo Group",
"Definition:Order of Group Element",
"Congruence by ... |
proofwiki-21871 | Axes of Symmetry of Kappa Curve | Let $\KK$ be a kappa curve expressed in Cartesian coordinates as:
:$x^4 + x^2 y^2 = a^2 y^2$
$\KK$ has two axes of bilateral symmetry:
:the $x$-axis
:the $y$-axis. | We have:
:$(1): \quad x^4 + x^2 y^2 = a^2 y^2$
;Symmetry about the $y$-axis
Substituting $-x$ for $x$ in $(1)$:
{{begin-eqn}}
{{eqn | l = \paren {-x}^4 + \paren {-x}^2 y^2
| r = x^4 + x^2 y^2
| c =
}}
{{eqn | r = a^2 y^2
| c =
}}
{{end-eqn}}
Thus if $\tuple {x, y}$ satisfies $(1)$, then $\tuple {-x,... | Let $\KK$ be a [[Definition:Kappa Curve|kappa curve]] expressed in [[Definition:Cartesian Coordinates|Cartesian coordinates]] as:
:$x^4 + x^2 y^2 = a^2 y^2$
$\KK$ has two [[Definition:Axis of Bilateral Symmetry|axes of bilateral symmetry]]:
:the [[Definition:X-Axis|$x$-axis]]
:the [[Definition:Y-Axis|$y$-axis]]. | We have:
:$(1): \quad x^4 + x^2 y^2 = a^2 y^2$
;Symmetry about the [[Definition:Y-Axis|$y$-axis]]
Substituting $-x$ for $x$ in $(1)$:
{{begin-eqn}}
{{eqn | l = \paren {-x}^4 + \paren {-x}^2 y^2
| r = x^4 + x^2 y^2
| c =
}}
{{eqn | r = a^2 y^2
| c =
}}
{{end-eqn}}
Thus if $\tuple {x, y}$ satisf... | Axes of Symmetry of Kappa Curve | https://proofwiki.org/wiki/Axes_of_Symmetry_of_Kappa_Curve | https://proofwiki.org/wiki/Axes_of_Symmetry_of_Kappa_Curve | [
"Kappa Curve",
"Bilateral Symmetry"
] | [
"Definition:Kappa Curve",
"Definition:Cartesian Coordinate System",
"Definition:Bilateral Symmetry/Axis",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis"
] | [
"Definition:Axis/Y-Axis",
"Definition:Axis/X-Axis",
"Definition:Invariant",
"Definition:Transformation",
"Definition:Bilateral Symmetry",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis"
] |
proofwiki-21872 | Asymptotes of Kappa Curve | Let $\KK$ be a kappa curve expressed in Cartesian coordinates as:
:$x^4 + x^2 y^2 = a^2 y^2$
$\KK$ has two asymptotes:
:the line $x = a$
:the line $x = -a$. | We have:
{{begin-eqn}}
{{eqn | l = x^4 + x^2 y^2
| r = a^2 y^2
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = x^4
| r = y^2 \paren {a^2 - x^2}
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = y^2
| r = \dfrac {x^4} {a^2 - x^2}
| c = dividing top and bottom by $a^2 - x^... | Let $\KK$ be a [[Definition:Kappa Curve|kappa curve]] expressed in [[Definition:Cartesian Coordinates|Cartesian coordinates]] as:
:$x^4 + x^2 y^2 = a^2 y^2$
$\KK$ has two [[Definition:Asymptote|asymptotes]]:
:the [[Definition:Straight Line|line]] $x = a$
:the [[Definition:Straight Line|line]] $x = -a$. | We have:
{{begin-eqn}}
{{eqn | l = x^4 + x^2 y^2
| r = a^2 y^2
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = x^4
| r = y^2 \paren {a^2 - x^2}
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = y^2
| r = \dfrac {x^4} {a^2 - x^2}
| c = [[Definition:Real Division|dividing... | Asymptotes of Kappa Curve | https://proofwiki.org/wiki/Asymptotes_of_Kappa_Curve | https://proofwiki.org/wiki/Asymptotes_of_Kappa_Curve | [
"Kappa Curve",
"Examples of Asymptotes"
] | [
"Definition:Kappa Curve",
"Definition:Cartesian Coordinate System",
"Definition:Asymptote",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line"
] | [
"Definition:Division/Field/Real Numbers",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line",
"Definition:Kappa Curve",
"Definition:Asymptote"
] |
proofwiki-21873 | Kappa Curve has Double Cusp at Origin | Let $\KK$ be a kappa curve expressed in Cartesian coordinates as:
:$x^4 + x^2 y^2 = a^2 y^2$
$\KK$ has a double cusp at the origin.
This double cusp is a double cusp of the first kind. | We have {{hypothesis}} that $\KK$ is given by:
:$x^4 + x^2 y^2 = a^2 y^2$
Let us define the real-valued function $F: \R^2 \to \R$:
:$\forall \tuple {x, y} \in \R^2: \map F {x, y} = x^4 + x^2 y^2 - a^2 y^2$
We have:
:$\map F {0, 0} = 0$
so the origin lies on $\KK$.
Computing the partial derivatives of $F$ {{WRT}} $x$ an... | Let $\KK$ be a [[Definition:Kappa Curve|kappa curve]] expressed in [[Definition:Cartesian Coordinates|Cartesian coordinates]] as:
:$x^4 + x^2 y^2 = a^2 y^2$
$\KK$ has a [[Definition:Double Cusp|double cusp]] at the [[Definition:Origin|origin]].
This [[Definition:Double Cusp|double cusp]] is a [[Definition:Double Cusp... | We have {{hypothesis}} that $\KK$ is given by:
:$x^4 + x^2 y^2 = a^2 y^2$
Let us define the [[Definition:Real-Valued Function|real-valued function]] $F: \R^2 \to \R$:
:$\forall \tuple {x, y} \in \R^2: \map F {x, y} = x^4 + x^2 y^2 - a^2 y^2$
We have:
:$\map F {0, 0} = 0$
so the [[Definition:Origin|origin]] lies on... | Kappa Curve has Double Cusp at Origin | https://proofwiki.org/wiki/Kappa_Curve_has_Double_Cusp_at_Origin | https://proofwiki.org/wiki/Kappa_Curve_has_Double_Cusp_at_Origin | [
"Kappa Curve",
"Examples of Double Cusps"
] | [
"Definition:Kappa Curve",
"Definition:Cartesian Coordinate System",
"Definition:Double Cusp",
"Definition:Coordinate System/Origin",
"Definition:Double Cusp",
"Definition:Double Cusp/First Kind"
] | [
"Definition:Real-Valued Function",
"Definition:Coordinate System/Origin",
"Definition:Partial Derivative",
"Definition:Coordinate System/Origin",
"Definition:Singular Point",
"Definition:Double Line",
"Definition:Coordinate System/Origin",
"Definition:Double Cusp/First Kind",
"Definition:Coordinate ... |
proofwiki-21874 | Formulation 1 Rank Axioms Implies Rank Function of Matroid | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Then $\rho$ is the rank function of a matroid on $S$. | Let
:$X \in \mathscr I$
{{AimForCont}}
:$\exists Y \subseteq X : Y \notin \mathscr I$
Let:
:$Y_0 \subseteq X : \card {Y_0} = \max \set{\card Z : Z \subseteq X \land Z \notin \mathscr I}$
By definition of $\mathscr I$:
:$Y_0 \notin \mathscr I \leadsto \map \rho {Y_0} \ne \card {Y_0}$
From Lemma 2:
:$\map \rho {Y_0} < \c... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (Ma... | Let
:$X \in \mathscr I$
{{AimForCont}}
:$\exists Y \subseteq X : Y \notin \mathscr I$
Let:
:$Y_0 \subseteq X : \card {Y_0} = \max \set{\card Z : Z \subseteq X \land Z \notin \mathscr I}$
By definition of $\mathscr I$:
:$Y_0 \notin \mathscr I \leadsto \map \rho {Y_0} \ne \card {Y_0}$
From [[Formulation 1 Rank Axio... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 5/Proof 1 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_5/Proof_1 | [
"Matroid Theory",
"Matroid Rank Functions",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Definition:Rank Function (Matroid)",
"Definition:Matroid"
] | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2",
"Definition:Proper Subset",
"Set Difference with Proper Subset",
"Cardinality of Set Union/Corollary",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Definition:Contradiction",
"Axiom:Matroid Axioms"
] |
proofwiki-21875 | Formulation 1 Rank Axioms Implies Rank Function of Matroid | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Then $\rho$ is the rank function of a matroid on $S$. | We prove the contrapositive statement:
:$\forall X, Y \subseteq S: Y \notin \mathscr I \land Y \subseteq X \implies X \notin \mathscr I$
Let $X, Y \subseteq S : Y \notin \mathscr I$ and $Y \subseteq X$.
====== Case 1: $Y = X$ ======
Let $Y = X$.
Then $X \notin \mathscr I$.
{{qed|lemma}}
====== Case 2: $Y \subset X$ ===... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (Ma... | We prove the [[Definition:Contrapositive Statement|contrapositive statement]]:
:$\forall X, Y \subseteq S: Y \notin \mathscr I \land Y \subseteq X \implies X \notin \mathscr I$
Let $X, Y \subseteq S : Y \notin \mathscr I$ and $Y \subseteq X$.
====== Case 1: $Y = X$ ======
Let $Y = X$.
Then $X \notin \mathscr I$.
{{... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 5/Proof 2 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_5/Proof_2 | [
"Matroid Theory",
"Matroid Rank Functions",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Definition:Rank Function (Matroid)",
"Definition:Matroid"
] | [
"Definition:Contrapositive Statement",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2",
"Cardinality of Set Union/Corollary",
"Definition:Contrapositive Statement",
"Axi... |
proofwiki-21876 | Formulation 1 Rank Axioms Implies Rank Function of Matroid | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Then $\rho$ is the rank function of a matroid on $S$. | Let:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \card X}$
It is to be shown that:
:* $\quad \mathscr I$ satisfies the matroid axioms
and
:* $\quad \rho$ is the rank function of the matroid $M = \struct{S, \mathscr I}$
==== Matroid Axioms ====
===== Matroid Axiom $(\text I 1)$ =====
{{:Formulation 1 Rank Axioms I... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (Ma... | Let:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \card X}$
It is to be shown that:
:* $\quad \mathscr I$ satisfies the [[Axiom:Matroid Axioms|matroid axioms]]
and
:* $\quad \rho$ is the [[Definition:Rank Function (Matroid)|rank function]] of the [[Definition:Matroid|matroid]] $M = \struct{S, \mathscr I}$
==== ... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Proof 1 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Proof_1 | [
"Matroid Theory",
"Matroid Rank Functions",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Definition:Rank Function (Matroid)",
"Definition:Matroid"
] | [
"Axiom:Matroid Axioms",
"Definition:Rank Function (Matroid)",
"Definition:Matroid",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 4",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 5",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 6",
"Formulati... |
proofwiki-21877 | Formulation 1 Rank Axioms Implies Rank Function of Matroid | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Then $\rho$ is the rank function of a matroid on $S$. | === Lemma 1 ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1}}
=== Lemma 2 ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2}}
=== Lemma 3 ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 3}}
Let:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (Ma... | === [[Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1|Lemma 1]] ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1}}
=== [[Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2|Lemma 2]] ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2}}
... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Proof 2 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Proof_2 | [
"Matroid Theory",
"Matroid Rank Functions",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Definition:Rank Function (Matroid)",
"Definition:Matroid"
] | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 3",
"Axiom:Matroid Axioms",
"Definition:Rank Function (Matroid)",
"Definition:Matroid",
"Formulati... |
proofwiki-21878 | Rank Function of Matroid Satisfies Formulation 2 Rank Axioms | Let $S$ be a finite set.
Let $\rho$ is the rank function of a matroid $M = \struct{S, \mathscr I}$.
Then $\rho$ satisfies formulation 2 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 2}} | ==== $\rho$ satisfies $(\text R 4)$ ====
This follows immediately from Bounds for Rank of Subset.
{{qed|lemma}}
==== $\rho$ satisfies $(\text R 5)$ ====
This follows immediately from Rank Function is Increasing.
{{qed|lemma}}
==== $\rho$ satisfies $(\text R 6)$ ====
Let $X, Y \subseteq S$.
Let $A$ be a maximal independ... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho$ is the [[Definition:Rank Function (Matroid)|rank function]] of a [[Definition:Matroid|matroid]] $M = \struct{S, \mathscr I}$.
Then $\rho$ satisfies [[Axiom:Rank Axioms (Matroid)/Definition 2|formulation 2]] of the [[Axiom:Rank Axioms (Matroid)|rank axioms... | ==== $\rho$ satisfies $(\text R 4)$ ====
This follows immediately from [[Bounds for Rank of Subset]].
{{qed|lemma}}
==== $\rho$ satisfies $(\text R 5)$ ====
This follows immediately from [[Rank Function is Increasing]].
{{qed|lemma}}
==== $\rho$ satisfies $(\text R 6)$ ====
Let $X, Y \subseteq S$.
Let $A$ be a... | Rank Function of Matroid Satisfies Formulation 2 Rank Axioms | https://proofwiki.org/wiki/Rank_Function_of_Matroid_Satisfies_Formulation_2_Rank_Axioms | https://proofwiki.org/wiki/Rank_Function_of_Matroid_Satisfies_Formulation_2_Rank_Axioms | [
"Matroid Rank Functions"
] | [
"Definition:Finite Set",
"Definition:Rank Function (Matroid)",
"Definition:Matroid",
"Axiom:Rank Axioms (Matroid)/Definition 2",
"Axiom:Rank Axioms (Matroid)"
] | [
"Bounds for Rank of Subset",
"Rank Function is Increasing",
"Definition:Maximal/Set",
"Definition:Matroid/Independent Set",
"Cardinality of Maximal Independent Subset Equals Rank of Set",
"Independent Subset is Contained in Maximal Independent Subset",
"Definition:Maximal/Set",
"Definition:Matroid/Ind... |
proofwiki-21879 | Rank Function of Matroid Satisfies Formulation 1 Rank Axioms | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be the rank function of a matroid on $S$.
Then $\rho$ satisfies formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}} | Let $\rho$ be the rank function of a matroid $M = \struct{S, \mathscr I}$ on $S$
==== $\rho$ satisfies $(\text R 1)$ ====
$\rho$ satisfies $(\text R 1)$ follows immediately from Rank of Empty Set is Zero.
{{qed|lemma}}
==== $\rho$ satisfies $(\text R 2)$ ====
Let:
:$X \subseteq S$
:$y \in S$
From Rank Function is Incre... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be the [[Definition:Rank Function (Matroid)|rank function]] of a [[Definition:Matroid|matroid]] on $S$.
Then $\rho$ satisfies [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (Matroid)|rank axioms]... | Let $\rho$ be the [[Definition:Rank Function (Matroid)|rank function]] of a [[Definition:Matroid|matroid]] $M = \struct{S, \mathscr I}$ on $S$
==== $\rho$ satisfies $(\text R 1)$ ====
$\rho$ satisfies $(\text R 1)$ follows immediately from [[Rank of Empty Set is Zero]].
{{qed|lemma}}
==== $\rho$ satisfies $(\text ... | Rank Function of Matroid Satisfies Formulation 1 Rank Axioms | https://proofwiki.org/wiki/Rank_Function_of_Matroid_Satisfies_Formulation_1_Rank_Axioms | https://proofwiki.org/wiki/Rank_Function_of_Matroid_Satisfies_Formulation_1_Rank_Axioms | [
"Matroid Rank Functions"
] | [
"Definition:Finite Set",
"Definition:Rank Function (Matroid)",
"Definition:Matroid",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)"
] | [
"Definition:Rank Function (Matroid)",
"Definition:Matroid",
"Rank of Empty Set is Zero",
"Rank Function is Increasing",
"Definition:Maximal/Set",
"Definition:Matroid/Independent Set",
"Cardinality of Maximal Independent Subset Equals Rank of Set",
"Axiom:Matroid Axioms",
"Cardinality of Set Differen... |
proofwiki-21880 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Proof 1 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Then $\rho$ is the rank function of a matroid on $S$. | Let:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \card X}$
It is to be shown that:
:* $\quad \mathscr I$ satisfies the matroid axioms
and
:* $\quad \rho$ is the rank function of the matroid $M = \struct{S, \mathscr I}$
==== Matroid Axioms ====
===== Matroid Axiom $(\text I 1)$ =====
{{:Formulation 1 Rank Axioms I... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (M... | Let:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \card X}$
It is to be shown that:
:* $\quad \mathscr I$ satisfies the [[Axiom:Matroid Axioms|matroid axioms]]
and
:* $\quad \rho$ is the [[Definition:Rank Function (Matroid)|rank function]] of the [[Definition:Matroid|matroid]] $M = \struct{S, \mathscr I}$
==== ... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Proof 1 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Proof_1 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Proof_1 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Definition:Rank Function (Matroid)",
"Definition:Matroid"
] | [
"Axiom:Matroid Axioms",
"Definition:Rank Function (Matroid)",
"Definition:Matroid",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 4",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 5",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 6",
"Formulati... |
proofwiki-21881 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Proof 2 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Then $\rho$ is the rank function of a matroid on $S$. | === Lemma 1 ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1}}
=== Lemma 2 ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2}}
=== Lemma 3 ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 3}}
Let:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (M... | === [[Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1|Lemma 1]] ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1}}
=== [[Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2|Lemma 2]] ===
{{:Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2}}
... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Proof 2 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Proof_2 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Proof_2 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Definition:Rank Function (Matroid)",
"Definition:Matroid"
] | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 3",
"Axiom:Matroid Axioms",
"Definition:Rank Function (Matroid)",
"Definition:Matroid",
"Formulati... |
proofwiki-21882 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1 | :$\forall A, B \subseteq S: A \subseteq B \implies \map \rho A \le \map \rho B$ | {{AimForCont}}:
:$\exists A, B \subseteq S : A \subseteq B$ and $\map \rho A > \map \rho B$
Let $B \subseteq S$:
:$\exists A \subseteq B : \map rho A > \map \rho B$
Let $A_0 \subseteq B$:
:$\card {A_0} = \max \set{\card A : A \subseteq B \land \map \rho A > \map \rho B}$
As $\map \rho {A_0} > \map \rho B$:
:$A_0 \ne B$... | :$\forall A, B \subseteq S: A \subseteq B \implies \map \rho A \le \map \rho B$ | {{AimForCont}}:
:$\exists A, B \subseteq S : A \subseteq B$ and $\map \rho A > \map \rho B$
Let $B \subseteq S$:
:$\exists A \subseteq B : \map rho A > \map \rho B$
Let $A_0 \subseteq B$:
:$\card {A_0} = \max \set{\card A : A \subseteq B \land \map \rho A > \map \rho B}$
As $\map \rho {A_0} > \map \rho B$:
:$A_0 \... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 1 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_1 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_1 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [] | [
"Set Difference with Proper Subset",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Definition:Contradiction",
"Category:Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] |
proofwiki-21883 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2 | :$\forall A \subseteq S: \map \rho A \le \card A$ | {{AimForCont}}:
:$\exists A \subseteq S : \map \rho A > \card A$
Let:
:$A_0 \subseteq S : \card{A_0} = \min \set{\card A : \map \rho A > \card A}$
We have:
{{begin-eqn}}
{{eqn | l = \map \rho \O
| r = 0
| c = Rank axiom $(\text R 1)$
}}
{{eqn | r = \card \O
| c = Cardinality of Empty Set
}}
{{end-eqn}... | :$\forall A \subseteq S: \map \rho A \le \card A$ | {{AimForCont}}:
:$\exists A \subseteq S : \map \rho A > \card A$
Let:
:$A_0 \subseteq S : \card{A_0} = \min \set{\card A : \map \rho A > \card A}$
We have:
{{begin-eqn}}
{{eqn | l = \map \rho \O
| r = 0
| c = [[Axiom:Rank Axioms (Matroid)/Definition 1|Rank axiom $(\text R 1)$]]
}}
{{eqn | r = \card \O
... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 2 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_2 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_2 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [] | [
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Cardinality of Empty Set",
"Cardinality of Set Difference",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Cardinality of Set Difference with Subset",
"Cardinality of Singleton",
"Definition:Contradiction",
"Category:Formulation 1 Rank Axioms Implies Rank Fun... |
proofwiki-21884 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 3 | Let:
:$A \subseteq S : \map \rho A = \card A$
Let:
:$B \subseteq S : \forall b \in B \setminus A : \map \rho {A \cup \set b} \ne \card{A \cup \set b}$
Then:
:$\map \rho {A \cup B} = \map \rho A$ | === Case 1 : $B \setminus A = \O$ ===
Let $B \setminus A = \O$.
From Set Difference with Superset is Empty Set:
:$B \subseteq A$
From Union with Superset is Superset:
:$A \cup B = A$
It follows that:
:$\map \rho {A \cup B} = \map \rho A$
{{qed|lemma}} | Let:
:$A \subseteq S : \map \rho A = \card A$
Let:
:$B \subseteq S : \forall b \in B \setminus A : \map \rho {A \cup \set b} \ne \card{A \cup \set b}$
Then:
:$\map \rho {A \cup B} = \map \rho A$ | === Case 1 : $B \setminus A = \O$ ===
Let $B \setminus A = \O$.
From [[Set Difference with Superset is Empty Set]]:
:$B \subseteq A$
From [[Union with Superset is Superset]]:
:$A \cup B = A$
It follows that:
:$\map \rho {A \cup B} = \map \rho A$
{{qed|lemma}} | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 3 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_3 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_3 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [] | [
"Set Difference with Superset is Empty Set",
"Union with Superset is Superset"
] |
proofwiki-21885 | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 8 | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
Let $\rho$ satisfy formulation 1 of the rank axioms:
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
Let $M = \struct{S, \mathscr I}$ be the matroid where:
:$\mathscr I = \set{X \subseteq S : \map \rho X = \ca... | Let $X \subseteq S$.
By definition of the rank function:
:$\map {\rho_M} X = \max \set{\card Y : Y \subseteq X, Y \in \mathscr I}$
Let $Y_0 \subseteq X$:
:$\card {Y_0} = \max \set{\card Y : Y \subseteq X, Y \in \mathscr I}$
Then:
{{begin-eqn}}
{{eqn | l = \map {\rho_M} X
| r = \card {Y_0}
| c = By choice of... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
Let $\rho$ satisfy [[Axiom:Rank Axioms (Matroid)/Definition 1|formulation 1]] of the [[Axiom:Rank Axioms (M... | Let $X \subseteq S$.
By definition of the [[Definition:Rank Function (Matroid)|rank function]]:
:$\map {\rho_M} X = \max \set{\card Y : Y \subseteq X, Y \in \mathscr I}$
Let $Y_0 \subseteq X$:
:$\card {Y_0} = \max \set{\card Y : Y \subseteq X, Y \in \mathscr I}$
Then:
{{begin-eqn}}
{{eqn | l = \map {\rho_M} X
... | Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 8 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_8 | https://proofwiki.org/wiki/Formulation_1_Rank_Axioms_Implies_Rank_Function_of_Matroid/Lemma_8 | [
"Formulation 1 Rank Axioms Implies Rank Function of Matroid"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Definition:Matroid",
"Definition:Rank Function (Matroid)",
"Definition:Matroid"
] | [
"Definition:Rank Function (Matroid)",
"Set Difference with Proper Subset",
"Formulation 1 Rank Axioms Implies Rank Function of Matroid/Lemma 3",
"Definition:Rank Function (Matroid)",
"Definition:Matroid"
] |
proofwiki-21886 | Equivalence of Definitions of Purely Inseparable Extension | Let $E / F$ be an algebraic field extension.
{{TFAE|def = Purely Inseparable Field Extension}} | {{tidy}}
{{MissingLinks}} | Let $E / F$ be an [[Definition:Algebraic Field Extension|algebraic field extension]].
{{TFAE|def = Purely Inseparable Field Extension}} | {{tidy}}
{{MissingLinks}} | Equivalence of Definitions of Purely Inseparable Extension | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Purely_Inseparable_Extension | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Purely_Inseparable_Extension | [
"Purely Inseparable Field Extensions"
] | [
"Definition:Algebraic Field Extension"
] | [] |
proofwiki-21887 | Transitivity of Separable Field Extensions | Let $E / K / F$ be a tower of fields.
Let $E / K$ and $K / F$ be separable.
Then $E / F$ is separable. | {{ProofWanted|Using Chapter V, $\S4$, Corollary 4.2 of Lang's Algebra}} | Let $E / K / F$ be a [[Definition:Tower of Fields|tower of fields]].
Let $E / K$ and $K / F$ be [[Definition:Separable Field Extension|separable]].
Then $E / F$ is [[Definition:Separable Field Extension|separable]]. | {{ProofWanted|Using Chapter V, $\S4$, Corollary 4.2 of Lang's Algebra}} | Transitivity of Separable Field Extensions | https://proofwiki.org/wiki/Transitivity_of_Separable_Field_Extensions | https://proofwiki.org/wiki/Transitivity_of_Separable_Field_Extensions | [
"Separable Field Extensions"
] | [
"Definition:Tower of Fields",
"Definition:Separable Extension",
"Definition:Separable Extension"
] | [] |
proofwiki-21888 | Separable Degree is At Most Equal To Degree | Let $E / F$ be a finite field extension.
Then $\index E F_s$ is finite, and:
:$\index E F_s \le \index E F$
where:
:$\index E F$ denotes the degree of $E / F$
:$\index E F_s$ denotes the separable degree of $E / F$. | {{ProofWanted|Theorem $4.1$ of Lang's Algebra}} | Let $E / F$ be a [[Definition:Finite Field Extension|finite field extension]].
Then $\index E F_s$ is finite, and:
:$\index E F_s \le \index E F$
where:
:$\index E F$ denotes the [[Definition:Degree of Field Extension|degree]] of $E / F$
:$\index E F_s$ denotes the [[Definition:Separable Degree|separable degree]] o... | {{ProofWanted|Theorem $4.1$ of Lang's Algebra}} | Separable Degree is At Most Equal To Degree | https://proofwiki.org/wiki/Separable_Degree_is_At_Most_Equal_To_Degree | https://proofwiki.org/wiki/Separable_Degree_is_At_Most_Equal_To_Degree | [
"Separable Field Extensions"
] | [
"Definition:Field Extension/Degree/Finite",
"Definition:Field Extension/Degree",
"Definition:Separable Degree"
] | [] |
proofwiki-21889 | Separable Degree of Field Extensions is Multiplicative | Let $E / F / k$ be a tower of fields. Then
:$\index E k_s = \index E F_s \index F k_s$
where $\index E F_s$ denotes the separable degree of $E / F$. | If $E / k$ is infinite, both sides are infinite.
Now assume $E / k$ is finite.
Let $L$ be the algebraic closure of $k$.
Let $\set{\sigma_i}$ be the family of distinct embedding of $F$ to $L$ fixing $k$,
By definition 2 of separable degree, the set $\set{\sigma_i}$ has $\index F k_s$ elements.
For each $i$, let $\set{\... | Let $E / F / k$ be a [[Definition:Tower of Fields|tower of fields]]. Then
:$\index E k_s = \index E F_s \index F k_s$
where $\index E F_s$ denotes the [[Definition:Separable Degree|separable degree]] of $E / F$. | If $E / k$ is [[Definition:Infinite Field Extension|infinite]], both sides are infinite.
Now assume $E / k$ is [[Definition:Finite Field Extension|finite]].
Let $L$ be the [[Definition:Algebraic Closure|algebraic closure]] of $k$.
Let $\set{\sigma_i}$ be the family of distinct [[Definition:Embedding (Galois Theory)... | Separable Degree of Field Extensions is Multiplicative | https://proofwiki.org/wiki/Separable_Degree_of_Field_Extensions_is_Multiplicative | https://proofwiki.org/wiki/Separable_Degree_of_Field_Extensions_is_Multiplicative | [
"Separable Field Extensions"
] | [
"Definition:Tower of Fields",
"Definition:Separable Degree"
] | [
"Definition:Field Extension/Degree/Infinite",
"Definition:Field Extension/Degree/Finite",
"Definition:Algebraic Closure",
"Definition:Embedding (Galois Theory)",
"Definition:Separable Degree",
"Definition:Separable Degree"
] |
proofwiki-21890 | Steinitz's Theorem | Let $E / F$ be a finite field extension.
{{TFAE}}
:$(1): \quad E / F$ is simple: there exists $\alpha \in E$ such that $E = \map F \alpha$
:$(2): \quad$ there are only finitely many intermediate fields between $E$ and $F$. | === $1$ implies $2$ ===
Suppose $E = F \sqbrk \alpha$ for some $\alpha \in F$.
Let $M$ be any intermediate field between $E$ and $F$.
Let $g$ be the minimal polynomial of $\alpha$ over $M$. Write
:$\ds g = x^ d + \sum _{i \mathop = 0}^{d \mathop - 1} a_ i x^ i$
Let $M'$ be the field extension of $F$ generated by all th... | Let $E / F$ be a [[Definition:Finite Field Extension|finite field extension]].
{{TFAE}}
:$(1): \quad E / F$ is [[Definition:Simple Field Extension|simple]]: there exists $\alpha \in E$ such that $E = \map F \alpha$
:$(2): \quad$ there are only finitely many [[Definition:Intermediate Field|intermediate fields]] between... | === $1$ implies $2$ ===
Suppose $E = F \sqbrk \alpha$ for some $\alpha \in F$.
Let $M$ be any [[Definition:Intermediate Field|intermediate field]] between $E$ and $F$.
Let $g$ be the [[Definition:Minimal Polynomial|minimal polynomial]] of $\alpha$ over $M$. Write
:$\ds g = x^ d + \sum _{i \mathop = 0}^{d \mathop - 1... | Steinitz's Theorem | https://proofwiki.org/wiki/Steinitz's_Theorem | https://proofwiki.org/wiki/Steinitz's_Theorem | [
"Field Extensions"
] | [
"Definition:Field Extension/Degree/Finite",
"Definition:Simple Field Extension",
"Definition:Intermediate Field"
] | [
"Definition:Intermediate Field",
"Definition:Minimal Polynomial",
"Definition:Minimal Polynomial",
"Degree Equation"
] |
proofwiki-21891 | Equivalence of Definitions of Separable Degree | Let $E / F$ be a field extension.
{{TFAE|def = Separable Degree}} | {{Refactor|level = medium|separate the lemma into its own page for neatness}} | Let $E / F$ be a [[Definition:Field Extension|field extension]].
{{TFAE|def = Separable Degree}} | {{Refactor|level = medium|separate the lemma into its own page for neatness}} | Equivalence of Definitions of Separable Degree | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Separable_Degree | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Separable_Degree | [
"Field Extensions"
] | [
"Definition:Field Extension"
] | [] |
proofwiki-21892 | Vector Space over an Infinite Field is not equal to the Union of Proper Subspaces | Let $U_1, U_2, \dots, U_n$ be proper subspaces of a vector space $V$ over an infinite field $F$,
Then $V$ is not equal to the union of $U_1, U_2, \dots, U_n$. | {{MissingLinks}}
{{AimForCont}} $V = V_1 \cup \cdots \cup V_n$.
We can assume that $n \geq 2$, and that $m$ is minimal $n$ with this property.
Choose and fix some $y \in V \setminus V_1$.
Let $x \in V_1$.
Since $F$ is infinite field, we can choose a subset $S \subset F$ of size $m+1$.
For each $\alpha \in S$, we can fi... | Let $U_1, U_2, \dots, U_n$ be [[Definition:Proper Vector Subspace|proper subspaces]] of a [[Definition:Vector Space|vector space]] $V$ over an [[Definition:Infinite Field|infinite field]] $F$,
Then $V$ is not equal to the [[Definition:Union|union]] of $U_1, U_2, \dots, U_n$. | {{MissingLinks}}
{{AimForCont}} $V = V_1 \cup \cdots \cup V_n$.
We can assume that $n \geq 2$, and that $m$ is minimal $n$ with this property.
Choose and fix some $y \in V \setminus V_1$.
Let $x \in V_1$.
Since $F$ is [[Definition:Infinite Field|infinite field]], we can choose a subset $S \subset F$ of size $m+1$.... | Vector Space over an Infinite Field is not equal to the Union of Proper Subspaces | https://proofwiki.org/wiki/Vector_Space_over_an_Infinite_Field_is_not_equal_to_the_Union_of_Proper_Subspaces | https://proofwiki.org/wiki/Vector_Space_over_an_Infinite_Field_is_not_equal_to_the_Union_of_Proper_Subspaces | [
"Vector Subspaces"
] | [
"Definition:Vector Subspace/Proper Subspace",
"Definition:Vector Space",
"Definition:Infinite Field",
"Definition:Union"
] | [
"Definition:Infinite Field",
"Definition:Injection"
] |
proofwiki-21893 | Kinetic Energy of Body at Constant Angular Speed | Let $B$ be a body rotating at an angular speed $\omega$ about some axis of rotation $\LL$.
Let $I$ denote the moment of inertia of $B$ about $\LL$.
Then the kinetic energy $T$ of $B$ brought about by this rotation is given by:
:$T = \dfrac {I \omega^2} 2$ | === Discrete Case ===
{{Recall|Moment of Inertia|Discrete Moment of Inertia|subdef = Discrete}}
{{:Definition:Moment of Inertia/Discrete}}
The total kinetic energy of $B$ is equal to the sum of the kinetic energies of every $P_i$.
{{begin-eqn}}
{{eqn | l = K
| r = \sum_i K_i
}}
{{eqn | r = \sum_i \frac {m_i v_i^2... | Let $B$ be a [[Definition:Body|body]] [[Definition:Space Rotation|rotating]] at an [[Definition:Angular Speed|angular speed]] $\omega$ about some [[Definition:Axis of Rotation|axis of rotation]] $\LL$.
Let $I$ denote the [[Definition:Moment of Inertia|moment of inertia]] of $B$ about $\LL$.
Then the [[Definition:Kin... | === Discrete Case ===
{{Recall|Moment of Inertia|Discrete Moment of Inertia|subdef = Discrete}}
{{:Definition:Moment of Inertia/Discrete}}
The total [[Definition:Kinetic Energy|kinetic energy]] of $B$ is equal to the [[Definition:Sum (Addition)|sum]] of the [[Definition:Kinetic Energy|kinetic energies]] of every $P_i... | Kinetic Energy of Body at Constant Angular Speed | https://proofwiki.org/wiki/Kinetic_Energy_of_Body_at_Constant_Angular_Speed | https://proofwiki.org/wiki/Kinetic_Energy_of_Body_at_Constant_Angular_Speed | [
"Kinetic Energy"
] | [
"Definition:Body",
"Definition:Rotation (Geometry)/Space",
"Definition:Angular Speed",
"Definition:Rotation (Geometry)/Axis",
"Definition:Moment of Inertia",
"Definition:Kinetic Energy",
"Definition:Rotation (Geometry)/Space"
] | [
"Definition:Kinetic Energy",
"Definition:Addition/Sum",
"Definition:Kinetic Energy",
"Kinetic Energy of Motion",
"Angular Speed of Particle in Circular Motion at Constant Speed",
"Indexed Summation of Multiple of Mapping",
"Definition:Kinetic Energy",
"Definition:Addition/Sum",
"Definition:Kinetic E... |
proofwiki-21894 | Outer Jordan Content of Right Triangle | Let $T \subseteq \R^2$ be defined as:
:$T = \set {\tuple {x, y} \in \R^2 : x \ge 0 \land y \ge 0 \land x + y \le 1}$
Then:
:$\map {m^*} T = \dfrac 1 2$
{{explain|notation}} | Let $\epsilon > 0$ be arbitrary.
By the Axiom of Archimedes, let $n \in \N$ such that:
:$n > 2 \epsilon$
Define $C \subseteq \powerset {\R^2}$ as:
:$C = \set {\closedint {\dfrac p n} {\dfrac {p + 1} n} \times \closedint {\dfrac q n} {\dfrac {q + 1} n} : p, q \in \set {0, 1, \dotsc, n - 1} \land p + q < n}$
By construct... | Let $T \subseteq \R^2$ be defined as:
:$T = \set {\tuple {x, y} \in \R^2 : x \ge 0 \land y \ge 0 \land x + y \le 1}$
Then:
:$\map {m^*} T = \dfrac 1 2$
{{explain|notation}} | Let $\epsilon > 0$ be arbitrary.
By the [[Axiom of Archimedes]], let $n \in \N$ such that:
:$n > 2 \epsilon$
Define $C \subseteq \powerset {\R^2}$ as:
:$C = \set {\closedint {\dfrac p n} {\dfrac {p + 1} n} \times \closedint {\dfrac q n} {\dfrac {q + 1} n} : p, q \in \set {0, 1, \dotsc, n - 1} \land p + q < n}$
By co... | Outer Jordan Content of Right Triangle | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Right_Triangle | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Right_Triangle | [] | [] | [
"Axiom of Archimedes",
"Definition:Finite Set",
"Definition:Closed Rectangle",
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Definition:Cover of Set"
] |
proofwiki-21895 | Set Difference Then Union Equals Union Then Set Difference | Let $S, A, B$ be sets.
Let $A \cap B = \O$.
Then:
:$\paren {S \setminus A} \cup B = \paren {S \cup B} \setminus A$ | We have:
{{begin-eqn}}
{{eqn | l = \paren {S \cup B} \setminus A
| r = \paren {S \setminus A} \cup \paren {B \setminus A}
| c = Set Difference is Right Distributive over Union
}}
{{eqn | r = \paren {S \setminus A} \cup B
| c = Set Difference with Disjoint Set
}}
{{end-eqn}}
{{qed}}
Category:Set Differ... | Let $S, A, B$ be [[Definition:Finite Set|sets]].
Let $A \cap B = \O$.
Then:
:$\paren {S \setminus A} \cup B = \paren {S \cup B} \setminus A$ | We have:
{{begin-eqn}}
{{eqn | l = \paren {S \cup B} \setminus A
| r = \paren {S \setminus A} \cup \paren {B \setminus A}
| c = [[Set Difference is Right Distributive over Union]]
}}
{{eqn | r = \paren {S \setminus A} \cup B
| c = [[Set Difference with Disjoint Set]]
}}
{{end-eqn}}
{{qed}}
[[Category... | Set Difference Then Union Equals Union Then Set Difference | https://proofwiki.org/wiki/Set_Difference_Then_Union_Equals_Union_Then_Set_Difference | https://proofwiki.org/wiki/Set_Difference_Then_Union_Equals_Union_Then_Set_Difference | [
"Set Difference",
"Set Union"
] | [
"Definition:Finite Set"
] | [
"Set Difference is Right Distributive over Union",
"Set Difference with Disjoint Set",
"Category:Set Difference",
"Category:Set Union"
] |
proofwiki-21896 | Lagrange Interpolation Formula/Formulation 1 | Let $\tuple {x_0, \ldots, x_n}$ and $\tuple {a_0, \ldots, a_n}$ be ordered tuples of real numbers such that $x_i \ne x_j$ for $i \ne j$.
Then there exists a unique polynomial $P \in \R \sqbrk X$ of degree at most $n$ such that:
:$\map P {x_i} = a_i$ for all $i \in \set {0, 1, \ldots, n}$
Moreover $P$ is given by the fo... | Recall the definition:
:$\ds \map {L_j} X = \prod_{\substack {0 \mathop \le i \mathop \le n \\ i \mathop \ne j}} \frac {X - x_i} {x_j - x_i} \in \R \sqbrk X$
{{mistake|Not sure if it's a mistake or a different way of defining it, but {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan ... | Let $\tuple {x_0, \ldots, x_n}$ and $\tuple {a_0, \ldots, a_n}$ be [[Definition:Ordered Tuple|ordered tuples]] of [[Definition:Real Number|real numbers]] such that $x_i \ne x_j$ for $i \ne j$.
Then there exists a [[Definition:Unique|unique]] [[Definition:Polynomial over Real Numbers|polynomial]] $P \in \R \sqbrk X$ of... | Recall the definition:
:$\ds \map {L_j} X = \prod_{\substack {0 \mathop \le i \mathop \le n \\ i \mathop \ne j}} \frac {X - x_i} {x_j - x_i} \in \R \sqbrk X$
{{mistake|Not sure if it's a mistake or a different way of defining it, but {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan... | Lagrange Interpolation Formula/Formulation 1 | https://proofwiki.org/wiki/Lagrange_Interpolation_Formula/Formulation_1 | https://proofwiki.org/wiki/Lagrange_Interpolation_Formula/Formulation_1 | [
"Lagrange Interpolation Formula"
] | [
"Definition:Ordered Tuple",
"Definition:Real Number",
"Definition:Unique",
"Definition:Polynomial/Real Numbers",
"Definition:Degree of Polynomial",
"Definition:Lagrange Basis Polynomial"
] | [
"Degree of Product of Polynomials over Ring/Corollary 2",
"Degree of Sum of Polynomials",
"Degree of Sum of Polynomials",
"Definition:Contradiction"
] |
proofwiki-21897 | N over 2 times Reciprocal of 1 Plus n Squared x Squared to the Power of 3 over 2 Delta Sequence | thumb600pxThe graph of the $\ds \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2}}$ delta sequence. As $n$ grows, the graph becomes thinner and taller. The area under each graph is equal to $1$.
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
:$\ds \map {\delta_n} x := \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{... | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x
| r = \int_0^\infty \frac 1 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd \paren {n x}
}}
{{eqn | r = \int_0^\infty \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y
| c = $n x = y$, Integration by Substitution
}}... | [[File:LorDeltaSequence2.png|thumb|600px|The graph of the $\ds \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2}}$ delta sequence. As $n$ grows, the graph becomes thinner and taller. The area under each graph is equal to $1$.]]
Let $\sequence {\map {\delta_n} x}$ be a [[Definition:Sequence|sequence]] such that:
:$\ds \ma... | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd x
| r = \int_0^\infty \frac 1 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} } \rd \paren {n x}
}}
{{eqn | r = \int_0^\infty \frac 1 2 \frac 1 {\paren{1 + y^2}^{3 / 2} } \rd y
| c = $n x = y$, [[Integration by Substitution/... | N over 2 times Reciprocal of 1 Plus n Squared x Squared to the Power of 3 over 2 Delta Sequence | https://proofwiki.org/wiki/N_over_2_times_Reciprocal_of_1_Plus_n_Squared_x_Squared_to_the_Power_of_3_over_2_Delta_Sequence | https://proofwiki.org/wiki/N_over_2_times_Reciprocal_of_1_Plus_n_Squared_x_Squared_to_the_Power_of_3_over_2_Delta_Sequence | [
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:LorDeltaSequence2.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Integration by Substitution/Definite Integral",
"Definite Integral of Even Function",
"Integration by Substitution/Definite Integral",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Mean Value Theorem for Integrals/Ge... |
proofwiki-21898 | Equivalence of Definitions of Matroid Base Axioms/Formulation 1 Iff Formulation 5 | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
Then:
:$\mathscr B$ satisfies formulation $1$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}}
{{iff}}
:$\mathscr B$ satisfies formulation $5$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 5}} | === Necessary Condition ===
From:
:* Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom
:* Matroid Bases Satisfy Formulation 4 Base Axiom
formulation $4$ follows from formulation $1$.
We have formulation $5$ follows from formulation $4$.
{{qed|lemma}}
=== Sufficient Condition ===
Let $\mathscr B$ satisfy f... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Then:
:$\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 1|formulation $1$ of base axiom]]:
{{:Axiom:Base Axiom (Matroid)/Formulati... | === Necessary Condition ===
From:
:* [[Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom]]
:* [[Matroid Bases Satisfy Formulation 4 Base Axiom]]
[[Axiom:Base Axiom (Matroid)/Formulation 4|formulation $4$]] follows from [[Axiom:Base Axiom (Matroid)/Formulation 1|formulation $1$]].
We have [[Axiom:Bas... | Equivalence of Definitions of Matroid Base Axioms/Formulation 1 Iff Formulation 5 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms/Formulation_1_Iff_Formulation_5 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms/Formulation_1_Iff_Formulation_5 | [
"Equivalence of Definitions of Matroid Base Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty",
"Definition:Set",
"Definition:Subset",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Axiom:Base Axiom (Matroid)/Formulation 5"
] | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom",
"Matroid Bases Satisfy Formulation 4 Base Axiom",
"Axiom:Base Axiom (Matroid)/Formulation 4",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Axiom:Base Axiom (Matroid)/Formulation 5",
"Axiom:Base Axiom (Matroid)/Formulation 4",
"Axiom:Base... |
proofwiki-21899 | Matroid Bases Iff Satisfies Matroid Base Axiom | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
Then $\mathscr B$ is the set of bases of a matroid on $S$ {{iff}} $\mathscr B$ satisfies any one of the base axioms:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}}{{:Axiom:Base Axiom (Matroid)/Formulation 2}}{{:Axiom:Base Axiom (Matroid)/For... | Follows immediately from:
:* Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom
:* Equivalence of Definitions of Matroid Base Axioms
{{qed}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Then $\mathscr B$ is the set of [[Definition:Base of Matroid|bases]] of a [[Definition:Matroid|matroid]] on $S$ {{iff}} $\mathscr B$ satisfie... | Follows immediately from:
:* [[Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom]]
:* [[Equivalence of Definitions of Matroid Base Axioms]]
{{qed}} | Matroid Bases Iff Satisfies Matroid Base Axiom | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Matroid_Base_Axiom | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Matroid_Base_Axiom | [
"Matroid Bases"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Definition:Base of Matroid",
"Definition:Matroid",
"Axiom:Base Axiom (Matroid)",
"Axiom:Base Axiom (Matroid)"
] | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom",
"Equivalence of Definitions of Matroid Base Axioms"
] |
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