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-2900 | Quotient Group is Group | Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Then the quotient group $G / N$ is indeed a group. | By Subgroup is Normal iff Left Cosets are Right Cosets, the set of left cosets for $N$ equals the set of right cosets.
It follows that $G / N$ does not depend on whether left cosets are used to define it or right cosets.
{{WLOG}}, we will work with the left cosets.
By definition of quotient group, the elements of $G / ... | Let $G$ be a [[Definition:Group|group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Then the [[Definition:Quotient Group|quotient group]] $G / N$ is indeed a [[Definition:Group|group]]. | By [[Subgroup is Normal iff Left Cosets are Right Cosets]], the [[Definition:Set|set]] of [[Definition:Left Coset|left cosets]] for $N$ equals the [[Definition:Set|set]] of [[Definition:Right Coset|right cosets]].
It follows that $G / N$ does not depend on whether [[Definition:Left Coset|left cosets]] are used to defi... | Quotient Group is Group | https://proofwiki.org/wiki/Quotient_Group_is_Group | https://proofwiki.org/wiki/Quotient_Group_is_Group | [
"Quotient Groups"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Group"
] | [
"Subgroup is Normal iff Left Cosets are Right Cosets",
"Definition:Set",
"Definition:Coset/Left Coset",
"Definition:Set",
"Definition:Coset/Right Coset",
"Definition:Coset/Left Coset",
"Definition:Coset/Right Coset",
"Definition:Coset/Left Coset",
"Definition:Quotient Group",
"Definition:Coset",
... |
proofwiki-2901 | Principal Ideal is Ideal | Let $\struct {R, +, \circ}$ be a ring with unity.
Let $a \in R$.
Let $\ideal a$ be the principal ideal of $R$ generated by $a$.
Then $\ideal a$ is an ideal of $R$. | First we establish that $\ideal a$ is an ideal of $R$, by verifying the conditions of Test for Ideal.
$\ideal a \ne \O$, as $1_R \circ a = a \in \ideal a$.
Let $x, y \in \ideal a$.
Then:
{{begin-eqn}}
{{eqn | q = \exists r, s \in R
| l = x
| r = r \circ a, y = s \circ a
| c =
}}
{{eqn | ll= \leadsto
... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $a \in R$.
Let $\ideal a$ be the [[Definition:Principal Ideal of Ring|principal ideal]] of $R$ generated by $a$.
Then $\ideal a$ is an [[Definition:Ideal of Ring|ideal]] of $R$. | First we establish that $\ideal a$ is an [[Definition:Ideal of Ring|ideal]] of $R$, by verifying the conditions of [[Test for Ideal]].
$\ideal a \ne \O$, as $1_R \circ a = a \in \ideal a$.
Let $x, y \in \ideal a$.
Then:
{{begin-eqn}}
{{eqn | q = \exists r, s \in R
| l = x
| r = r \circ a, y = s \circ... | Principal Ideal is Ideal | https://proofwiki.org/wiki/Principal_Ideal_is_Ideal | https://proofwiki.org/wiki/Principal_Ideal_is_Ideal | [
"Ideal Theory"
] | [
"Definition:Ring with Unity",
"Definition:Principal Ideal of Ring",
"Definition:Ideal of Ring"
] | [
"Definition:Ideal of Ring",
"Test for Ideal",
"Test for Ideal",
"Definition:Ideal of Ring"
] |
proofwiki-2902 | Bézout's Theorem | Let $X$ and $Y$ be two plane projective curves defined over a field $F$ that do not have a common component.
Then the total number of intersection points of $X$ and $Y$ with coordinates in an algebraically closed field $E$ which contains $F$, counted with their multiplicities, is equal to the product of the degrees of ... | The condition that $X$ and $Y$ have no common component is true if both $X$ and $Y$ are defined by different irreducible polynomials.
In particular, it holds for a pair of "generic" curves.
{{ExtractTheorem|Link to a proof of the above}}
{{ProofWanted}} | Let $X$ and $Y$ be two [[Definition:Plane Projective Curve|plane projective curves]] defined over a [[Definition:Field (Abstract Algebra)|field]] $F$ that do not have a common component.
Then the total number of [[Definition:Intersection (Geometry)|intersection points]] of $X$ and $Y$ with [[Definition:Cartesian Coord... | The condition that $X$ and $Y$ have no common component is [[Definition:True|true]] if both $X$ and $Y$ are defined by different [[Definition:Irreducible Polynomial|irreducible polynomials]].
In particular, it holds for a pair of "generic" curves.
{{ExtractTheorem|Link to a proof of the above}}
{{ProofWanted}} | Bézout's Theorem | https://proofwiki.org/wiki/Bézout's_Theorem | https://proofwiki.org/wiki/Bézout's_Theorem | [
"Bézout's Theorem",
"Polynomial Theory"
] | [
"Definition:Plane Projective Curve",
"Definition:Field (Abstract Algebra)",
"Definition:Intersection (Geometry)",
"Definition:Cartesian Coordinate System",
"Definition:Algebraically Closed Field",
"Definition:Multiplicity (Real Analysis)",
"Definition:Degree of Polynomial"
] | [
"Definition:True",
"Definition:Irreducible Polynomial"
] |
proofwiki-2903 | Forward Difference of Power | :$\Delta c^x = \paren {c - 1} c^x$
where $\Delta$ denotes the forward difference operator. | From the definitions:
{{begin-eqn}}
{{eqn | l = \Delta c^x
| r = c^{x + 1} - c^x
| c = {{Defof|Forward Difference Operator}}
}}
{{eqn | r = c \cdot c^x - c^x
}}
{{eqn | r = \paren {c - 1} c^x
}}
{{end-eqn}}
{{qed}}
Category:Finite Calculus
m0tf2vwc49nhv5xfxhzhmznqqd9sidk | :$\Delta c^x = \paren {c - 1} c^x$
where $\Delta$ denotes the [[Definition:Forward Difference Operator|forward difference operator]]. | From the definitions:
{{begin-eqn}}
{{eqn | l = \Delta c^x
| r = c^{x + 1} - c^x
| c = {{Defof|Forward Difference Operator}}
}}
{{eqn | r = c \cdot c^x - c^x
}}
{{eqn | r = \paren {c - 1} c^x
}}
{{end-eqn}}
{{qed}}
[[Category:Finite Calculus]]
m0tf2vwc49nhv5xfxhzhmznqqd9sidk | Forward Difference of Power | https://proofwiki.org/wiki/Forward_Difference_of_Power | https://proofwiki.org/wiki/Forward_Difference_of_Power | [
"Finite Calculus"
] | [
"Definition:Finite Difference Operator/Forward Difference"
] | [
"Category:Finite Calculus"
] |
proofwiki-2904 | Forward Difference of Harmonic Number Function | Let $H_x$ denote the harmonic number function.
Then:
:$\Delta H_x = \dfrac 1 {x + 1}$
where $\Delta H_x$ denotes the forward difference operator. | From the definitions:
{{begin-eqn}}
{{eqn | l = \Delta H_x
| r = H_{x + 1} - H_x
| c = {{Defof|Forward Difference Operator}}
}}
{{eqn | r = \sum_{k \mathop = 1}^{x + 1} \frac 1 k - \sum_{k \mathop = 1}^x \frac 1 k
| c = {{Defof|Harmonic Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^x \frac 1 k + \frac ... | Let $H_x$ denote the [[Definition:Harmonic Number|harmonic number function]].
Then:
:$\Delta H_x = \dfrac 1 {x + 1}$
where $\Delta H_x$ denotes the [[Definition:Forward Difference Operator|forward difference operator]]. | From the definitions:
{{begin-eqn}}
{{eqn | l = \Delta H_x
| r = H_{x + 1} - H_x
| c = {{Defof|Forward Difference Operator}}
}}
{{eqn | r = \sum_{k \mathop = 1}^{x + 1} \frac 1 k - \sum_{k \mathop = 1}^x \frac 1 k
| c = {{Defof|Harmonic Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^x \frac 1 k + \frac... | Forward Difference of Harmonic Number Function | https://proofwiki.org/wiki/Forward_Difference_of_Harmonic_Number_Function | https://proofwiki.org/wiki/Forward_Difference_of_Harmonic_Number_Function | [
"Finite Calculus",
"Harmonic Numbers"
] | [
"Definition:Harmonic Numbers",
"Definition:Finite Difference Operator/Forward Difference"
] | [
"Category:Finite Calculus",
"Category:Harmonic Numbers"
] |
proofwiki-2905 | Equality of Ordered Tuples | Let $a = \tuple {a_1, a_2, \ldots, a_n}$ and $b = \tuple {b_1, b_2, \ldots, b_n}$ be ordered tuples.
Then:
:$a = b \iff \forall i: 1 \le i \le n: a_i = b_i$
That is, for two ordered tuples to be equal, all the corresponding elements have to be equal. | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\tuple {a_1, a_2, \ldots, a_n} = \tuple {b_1, b_2, \ldots, b_n} \iff \forall i: 1 \le i \le n: a_i = b_i$
$\map P 1$ is true, as this just says $\tuple {a_1} = \tuple {b_1} \iff a_1 = b_1$ which is trivial. | Let $a = \tuple {a_1, a_2, \ldots, a_n}$ and $b = \tuple {b_1, b_2, \ldots, b_n}$ be [[Definition:Ordered Tuple as Ordered Set|ordered tuples]].
Then:
:$a = b \iff \forall i: 1 \le i \le n: a_i = b_i$
That is, for two [[Definition:Ordered Tuple as Ordered Set|ordered tuples]] to be [[Definition:Equal|equal]], all the... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\tuple {a_1, a_2, \ldots, a_n} = \tuple {b_1, b_2, \ldots, b_n} \iff \forall i: 1 \le i \le n: a_i = b_i$
$\map P 1$ is true, as this just says $\tuple {a_1} = \tuple {... | Equality of Ordered Tuples | https://proofwiki.org/wiki/Equality_of_Ordered_Tuples | https://proofwiki.org/wiki/Equality_of_Ordered_Tuples | [
"Cartesian Product",
"Equality of Ordered Tuples",
"Equality"
] | [
"Definition:Ordered Tuple as Ordered Set",
"Definition:Ordered Tuple as Ordered Set",
"Definition:Equals",
"Definition:Element",
"Definition:Equals"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-2906 | Zero of Power Set with Union | Let $S$ be a set and let $\powerset S$ be its power set.
Consider the algebraic structure $\struct {\powerset S, \cup}$, where $\cup$ denotes set union.
Then $S$ serves as the zero element for $\struct {\powerset S, \cup}$. | We note that by Set is Subset of Itself, $S \subseteq S$ and so $S \in \powerset S$ from the definition of the power set.
From Union with Superset is Superset, we have:
:$A \subseteq S \iff A \cup S = S = S \cup A$.
By definition of power set:
:$A \subseteq S \iff A \in \powerset S$
So:
:$\forall A \in \powerset S: A \... | Let $S$ be a [[Definition:Set|set]] and let $\powerset S$ be its [[Definition:Power Set|power set]].
Consider the [[Definition:Algebraic Structure|algebraic structure]] $\struct {\powerset S, \cup}$, where $\cup$ denotes [[Definition:Set Union|set union]].
Then $S$ serves as the [[Definition:Zero Element|zero elemen... | We note that by [[Set is Subset of Itself]], $S \subseteq S$ and so $S \in \powerset S$ from the definition of the [[Definition:Power Set|power set]].
From [[Union with Superset is Superset]], we have:
:$A \subseteq S \iff A \cup S = S = S \cup A$.
By definition of [[Definition:Power Set|power set]]:
:$A \subseteq S \... | Zero of Power Set with Union | https://proofwiki.org/wiki/Zero_of_Power_Set_with_Union | https://proofwiki.org/wiki/Zero_of_Power_Set_with_Union | [
"Set Union",
"Power Set",
"Zero Elements"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Algebraic Structure",
"Definition:Set Union",
"Definition:Zero Element"
] | [
"Set is Subset of Itself",
"Definition:Power Set",
"Union with Superset is Superset",
"Definition:Power Set"
] |
proofwiki-2907 | Zero of Power Set with Intersection | Let $S$ be a set and let $\powerset S$ be its power set.
Consider the algebraic structure $\struct {\powerset S, \cap}$, where $\cap$ denotes set intersection.
Then the empty set $\O$ serves as the zero element for $\struct {\powerset S, \cap}$. | From Empty Set is Element of Power Set:
:$\O \in \powerset S$
From Intersection with Empty Set:
:$\forall A \subseteq S: A \cap \O = \O = \O \cap A$
By definition of power set:
:$A \subseteq S \iff A \in \powerset S$
So:
:$\forall A \in \powerset S: A \cap \O = \O = \O \cap A$
Thus we see that $\O$ acts as the zero ele... | Let $S$ be a [[Definition:Set|set]] and let $\powerset S$ be its [[Definition:Power Set|power set]].
Consider the [[Definition:Algebraic Structure|algebraic structure]] $\struct {\powerset S, \cap}$, where $\cap$ denotes [[Definition:Set Intersection|set intersection]].
Then the [[Definition:Empty Set|empty set]] $\... | From [[Empty Set is Element of Power Set]]:
:$\O \in \powerset S$
From [[Intersection with Empty Set]]:
:$\forall A \subseteq S: A \cap \O = \O = \O \cap A$
By definition of [[Definition:Power Set|power set]]:
:$A \subseteq S \iff A \in \powerset S$
So:
:$\forall A \in \powerset S: A \cap \O = \O = \O \cap A$
Thus ... | Zero of Power Set with Intersection | https://proofwiki.org/wiki/Zero_of_Power_Set_with_Intersection | https://proofwiki.org/wiki/Zero_of_Power_Set_with_Intersection | [
"Set Intersection",
"Power Set",
"Empty Set",
"Zero Elements"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Algebraic Structure",
"Definition:Set Intersection",
"Definition:Empty Set",
"Definition:Zero Element"
] | [
"Empty Set is Element of Power Set",
"Intersection with Empty Set",
"Definition:Power Set",
"Definition:Zero Element"
] |
proofwiki-2908 | Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group | Let $\R_{>0}$ be the set of strictly positive real numbers:
:$\R_{>0} = \set {x \in \R: x > 0}$
The structure $\struct {\R_{>0}, \times}$ is an uncountable abelian group. | From Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers we have that $\struct {\R_{>0}, \times}$ is a subgroup of $\struct {\R_{\ne 0}, \times}$, where $\R_{\ne 0}$ is the set of real numbers without zero:
:$\R_{\ne 0} = \R \setminus \set 0$
From Subgroup of Abelian Group is Abel... | Let $\R_{>0}$ be the set of [[Definition:Strictly Positive Real Number|strictly positive real numbers]]:
:$\R_{>0} = \set {x \in \R: x > 0}$
The [[Definition:Algebraic Structure with One Operation|structure]] $\struct {\R_{>0}, \times}$ is an [[Definition:Uncountable Group|uncountable]] [[Definition:Abelian Group|abel... | From [[Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers]] we have that $\struct {\R_{>0}, \times}$ is a [[Definition:Subgroup|subgroup]] of $\struct {\R_{\ne 0}, \times}$, where $\R_{\ne 0}$ is the set of [[Definition:Real Number|real numbers]] without [[Definition:Zero (Number... | Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group | https://proofwiki.org/wiki/Strictly_Positive_Real_Numbers_under_Multiplication_form_Uncountable_Abelian_Group | https://proofwiki.org/wiki/Strictly_Positive_Real_Numbers_under_Multiplication_form_Uncountable_Abelian_Group | [
"Real Multiplication",
"Examples of Abelian Groups",
"Examples of Infinite Groups"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Algebraic Structure/One Operation",
"Definition:Infinite Group/Uncountable",
"Definition:Abelian Group"
] | [
"Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers",
"Definition:Subgroup",
"Definition:Real Number",
"Definition:Zero (Number)",
"Subgroup of Abelian Group is Abelian",
"Definition:Abelian Group"
] |
proofwiki-2909 | Solution to Legendre's Differential Equation | The solution of Legendre's differential equation:
{{:Definition:Legendre's Differential Equation}}
is:
:$\ds A \paren {\sum_{n \mathop = 0}^\infty \prod_{k \mathop = 1}^n \paren {\paren {2 n - 2 k} \paren {2 n - \paren {2 k - 1} } - p \paren {p + 1} } a_0 \frac {x^{2 n} } {2 n!} } + B \paren {\sum_{n \mathop = 0}^\inft... | Begin with the ansatz:
:$\ds y = \sum_{n \mathop = 0}^\infty a_n x^n$
Differentating {{WRT|Differentiation}} $x$:
:$\ds \dot y = \sum_{n \mathop = 0}^\infty a_n n x^{n - 1}$
:$\ds \ddot y = \sum_{n \mathop = 0}^\infty a_n n \paren {n - 1} x^{n - 2}$
Substituting in the original equation:
{{begin-eqn}}
{{eqn | l = \pare... | The solution of [[Definition:Legendre's Differential Equation|Legendre's differential equation]]:
{{:Definition:Legendre's Differential Equation}}
is:
:$\ds A \paren {\sum_{n \mathop = 0}^\infty \prod_{k \mathop = 1}^n \paren {\paren {2 n - 2 k} \paren {2 n - \paren {2 k - 1} } - p \paren {p + 1} } a_0 \frac {x^{2 n} ... | Begin with the [[Definition:Ansatz|ansatz]]:
:$\ds y = \sum_{n \mathop = 0}^\infty a_n x^n$
[[Definition:Differentiation|Differentating]] {{WRT|Differentiation}} $x$:
:$\ds \dot y = \sum_{n \mathop = 0}^\infty a_n n x^{n - 1}$
:$\ds \ddot y = \sum_{n \mathop = 0}^\infty a_n n \paren {n - 1} x^{n - 2}$
Substitutin... | Solution to Legendre's Differential Equation | https://proofwiki.org/wiki/Solution_to_Legendre's_Differential_Equation | https://proofwiki.org/wiki/Solution_to_Legendre's_Differential_Equation | [
"Legendre's Differential Equation",
"Second Order ODEs"
] | [
"Definition:Legendre's Differential Equation"
] | [
"Definition:Ansatz",
"Definition:Differentiation",
"Definition:Summation",
"Definition:Summation",
"Definition:Summation",
"Definition:Summation",
"Definition:Coefficient",
"Definition:Term",
"Definition:Recursive Sequence/Recurrence Relation",
"Definition:Legendre's Differential Equation",
"Def... |
proofwiki-2910 | Law of Mass Action | Let $\AA$ and $\BB$ be two chemical substances in a solution $C$ which are involved in a second-order reaction.
Let $x$ grams of $\CC$ contain $a x$ grams of $\AA$ and $b x$ grams of $\BB$, where $a + b = 1$.
Let there be $a A$ grams of $\AA$ and $b B$ grams of $\BB$ at time $t = t_0$, at which time $x = 0$.
Then:
:<no... | By the definition of a second-order reaction:
:The rate of formation of $\CC$ is jointly proportional to the quantities of $\AA$ and $\BB$ which have not yet transformed.
By definition of joint proportion:
:$\dfrac {\d x} {\d t} \propto \paren {A - x} a \paren {B - x} b$
or:
:$\dfrac {\d x} {\d t} = k a b \paren {A - x... | Let $\AA$ and $\BB$ be two [[Definition:Substance|chemical substances]] in a solution $C$ which are involved in a [[Definition:Second-Order Reaction|second-order reaction]].
Let $x$ [[Definition:Gram|grams]] of $\CC$ contain $a x$ [[Definition:Gram|grams]] of $\AA$ and $b x$ [[Definition:Gram|grams]] of $\BB$, where ... | By the definition of a [[Definition:Second-Order Reaction|second-order reaction]]:
:The [[Definition:Rate|rate]] of formation of $\CC$ is [[Definition:Joint Proportion|jointly proportional]] to the quantities of $\AA$ and $\BB$ which have not yet transformed.
By definition of [[Definition:Joint Proportion|joint prop... | Law of Mass Action | https://proofwiki.org/wiki/Law_of_Mass_Action | https://proofwiki.org/wiki/Law_of_Mass_Action | [
"Chemistry",
"First Order ODEs",
"Named Theorems"
] | [
"Definition:Substance",
"Definition:Second-Order Reaction",
"Definition:Metric System/Mass/Gram",
"Definition:Metric System/Mass/Gram",
"Definition:Metric System/Mass/Gram",
"Definition:Metric System/Mass/Gram",
"Definition:Metric System/Mass/Gram",
"Definition:Positive/Real Number",
"Definition:Con... | [
"Definition:Second-Order Reaction",
"Definition:Rate",
"Definition:Proportion/Joint",
"Definition:Proportion/Joint",
"Definition:Positive/Real Number",
"Definition:Constant",
"Solution to Separable Differential Equation",
"Definition:Partial Fractions Expansion",
"Definition:Primitive (Calculus)/Con... |
proofwiki-2911 | Motion of Simple Pendulum | Consider a simple pendulum consisting of a bob whose mass is $m$, at the end of a rod of negligible mass of length $a$.
Let the bob be pulled to one side so that the rod is at an angle $\alpha$ from the vertical and then released.
Let $T$ be the time period of the simple pendulum, that is, the time through which the bo... | At a time $t$, let:
:the rod be at an angle $\theta$ to the vertical
:the bob be travelling at a speed $v$
:the displacement of the bob from where it is when the rod is vertical, along its line of travel, be $s$.
:350px
At its maximum displacement, the speed of the bob is $0$, so its kinetic energy is $0$.
By the Princ... | Consider a [[Definition:Simple Pendulum|simple pendulum]] consisting of a [[Definition:Pendulum Bob|bob]] whose [[Definition:Mass|mass]] is $m$, at the end of a [[Definition:Rod|rod]] of negligible [[Definition:Mass|mass]] of [[Definition:Length (Linear Measure)|length]] $a$.
Let the [[Definition:Pendulum Bob|bob]] be... | At a time $t$, let:
:the [[Definition:Rod|rod]] be at an [[Definition:Angle|angle]] $\theta$ to [[Definition:Vertical|the vertical]]
:the [[Definition:Pendulum Bob|bob]] be travelling at a [[Definition:Speed|speed]] $v$
:the [[Definition:Displacement|displacement]] of the [[Definition:Pendulum Bob|bob]] from where it i... | Motion of Simple Pendulum | https://proofwiki.org/wiki/Motion_of_Simple_Pendulum | https://proofwiki.org/wiki/Motion_of_Simple_Pendulum | [
"Mathematical Physics",
"Simple Pendulums",
"Complete Elliptic Integral of the First Kind"
] | [
"Definition:Pendulum/Simple",
"Definition:Simple Pendulum/Bob",
"Definition:Mass",
"Definition:Rod",
"Definition:Mass",
"Definition:Linear Measure/Length",
"Definition:Simple Pendulum/Bob",
"Definition:Rod",
"Definition:Angle",
"Definition:Vertical",
"Definition:Time Period",
"Definition:Pendu... | [
"Definition:Rod",
"Definition:Angle",
"Definition:Vertical",
"Definition:Simple Pendulum/Bob",
"Definition:Speed",
"Definition:Displacement",
"Definition:Simple Pendulum/Bob",
"Definition:Rod",
"Definition:Vertical Line",
"File:MotionOfPendulum.png",
"Definition:Speed",
"Definition:Simple Pend... |
proofwiki-2912 | One-Parameter Family of Curves for First Order ODE | Every one-parameter family of curves is the general solution of some first order ordinary differential equation.
Conversely, every first order ordinary differential equation has as its general solution some one-parameter family of curves. | From Picard's Existence Theorem, every point in a given rectangle is passed through by some curve which is the solution of a given integral curve of a differential equation.
The equation of this family can be written as:
:$y = \map y {x, c}$
where different values of $c$ give different curves.
The integral curve which ... | Every [[Definition:One-Parameter Family of Curves|one-parameter family of curves]] is the [[Definition:General Solution to Differential Equation|general solution]] of some [[Definition:First Order Ordinary Differential Equation|first order ordinary differential equation]].
Conversely, every [[Definition:First Order Or... | From [[Picard's Existence Theorem]], every point in a given rectangle is passed through by some curve which is the solution of a given integral curve of a differential equation.
The equation of this family can be written as:
:$y = \map y {x, c}$
where different values of $c$ give different curves.
The integral curve ... | One-Parameter Family of Curves for First Order ODE | https://proofwiki.org/wiki/One-Parameter_Family_of_Curves_for_First_Order_ODE | https://proofwiki.org/wiki/One-Parameter_Family_of_Curves_for_First_Order_ODE | [
"First Order ODEs",
"One-Parameter Families"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Differential Equation/Solution/General Solution",
"Definition:First Order Ordinary Differential Equation",
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Family of C... | [
"Picard's Existence Theorem",
"Definition:Family of Curves/One-Parameter",
"Definition:First Order Ordinary Differential Equation"
] |
proofwiki-2913 | Orthogonal Trajectories of One-Parameter Family of Curves | Every one-parameter family of curves has a unique family of orthogonal trajectories. | Let $\map f {x, y, z}$ define a one-parameter family of curves $\FF$.
From One-Parameter Family of Curves for First Order ODE, there is a corresponding first order ODE:
:$\map F {x, y, \dfrac {\d y} {\d x} }$
whose solution is $\FF$.
From Slope of Orthogonal Curves, the slope of one curve is the negative reciprocal of ... | Every [[Definition:One-Parameter Family of Curves|one-parameter family of curves]] has a unique [[Definition:Orthogonal Trajectories|family of orthogonal trajectories]]. | Let $\map f {x, y, z}$ define a [[Definition:One-Parameter Family of Curves|one-parameter family of curves]] $\FF$.
From [[One-Parameter Family of Curves for First Order ODE]], there is a corresponding [[Definition:First Order Ordinary Differential Equation|first order ODE]]:
:$\map F {x, y, \dfrac {\d y} {\d x} }$
wh... | Orthogonal Trajectories of One-Parameter Family of Curves | https://proofwiki.org/wiki/Orthogonal_Trajectories_of_One-Parameter_Family_of_Curves | https://proofwiki.org/wiki/Orthogonal_Trajectories_of_One-Parameter_Family_of_Curves | [
"Orthogonal Trajectories",
"One-Parameter Families"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Orthogonal Trajectories"
] | [
"Definition:Family of Curves/One-Parameter",
"One-Parameter Family of Curves for First Order ODE",
"Definition:First Order Ordinary Differential Equation",
"Slope of Orthogonal Curves",
"Definition:Slope",
"Definition:Line/Curve",
"Definition:Reciprocal",
"Definition:Orthogonal Curves",
"Definition:... |
proofwiki-2914 | Brachistochrone is Cycloid | The shape of the brachistochrone is a cycloid. | :500px
Recall from the Snell-Descartes Law:
:$\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$
Here, we invoke a generalization of the Snell-Descartes Law.
This is justified, as we are attempting to demonstrate the curve that takes the smallest time.
Thus we have $\dfrac {\sin \alpha} v = k$, where $k$ is s... | The [[Definition:Geometric Figure|shape]] of the [[Definition:Brachistochrone|brachistochrone]] is a [[Definition:Cycloid|cycloid]]. | :[[File:Brachistochrone.png|500px]]
Recall from the [[Snell-Descartes Law]]:
:$\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$
Here, we invoke a generalization of the [[Snell-Descartes Law]].
This is justified, as we are attempting to demonstrate the [[Definition:Curve|curve]] that takes the smallest [[... | Brachistochrone is Cycloid/Proof 1 | https://proofwiki.org/wiki/Brachistochrone_is_Cycloid | https://proofwiki.org/wiki/Brachistochrone_is_Cycloid/Proof_1 | [
"Brachistochrone is Cycloid",
"Cycloids"
] | [
"Definition:Geometric Figure",
"Definition:Brachistochrone",
"Definition:Cycloid"
] | [
"File:Brachistochrone.png",
"Snell-Descartes Law",
"Snell-Descartes Law",
"Definition:Line/Curve",
"Definition:Time",
"Definition:Constant",
"Principle of Conservation of Energy",
"Definition:Speed",
"Definition:Potential Energy",
"Definition:Point",
"Principle of Conservation of Energy",
"Def... |
proofwiki-2915 | Brachistochrone is Cycloid | The shape of the brachistochrone is a cycloid. | Throughout this proof, we use the standard alignment of coordinate axes:
:$X$-axis pointing rightwards
:$Y$-axis is pointing upwards.
Suppose that the curve passes through the point $\tuple {x, y}$ for some value of variable $t$.
Due to smoothness of the curve, one can define velocity $v$ at a point $\tuple {\map x t, ... | The [[Definition:Geometric Figure|shape]] of the [[Definition:Brachistochrone|brachistochrone]] is a [[Definition:Cycloid|cycloid]]. | Throughout this [[Definition:Proof|proof]], we use the standard alignment of [[Definition:Coordinate Axis|coordinate axes]]:
:[[Definition:X-Axis|$X$-axis]] pointing rightwards
:[[Definition:Y-Axis|$Y$-axis]] is pointing upwards.
Suppose that the [[Definition:Curve|curve]] passes through the [[Definition:Point|point]]... | Brachistochrone is Cycloid/Proof 2 | https://proofwiki.org/wiki/Brachistochrone_is_Cycloid | https://proofwiki.org/wiki/Brachistochrone_is_Cycloid/Proof_2 | [
"Brachistochrone is Cycloid",
"Cycloids"
] | [
"Definition:Geometric Figure",
"Definition:Brachistochrone",
"Definition:Cycloid"
] | [
"Definition:Proof",
"Definition:Axis/Coordinate Axes",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Line/Curve",
"Definition:Point",
"Definition:Variable/Real",
"Definition:Smooth Real Function",
"Definition:Line/Curve",
"Definition:Velocity",
"Definition:Point",
"Definition... |
proofwiki-2916 | Equation of Cycloid | Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane.
Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.
Consider the cycloid traced out by the point $P$.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travel... | Let the circle have rolled so that the radius to the point $P = \tuple {x, y}$ is at angle $\theta$ to the vertical.
:700px
The center of the circle is at $\tuple {a \theta, a}$.
From the diagram above, we see:
:The $x$-coordinate is to the left of the center of the circle by $a \sin \theta$
:The $y$-coordinate is belo... | Consider a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $a$ rolling without slipping along the [[Definition:X-Axis|x-axis]] of a [[Definition:Cartesian Plane|cartesian plane]].
Consider the [[Definition:Point|point]] $P$ on the [[Definition:Circumference of Circle|circumference]] of this [[De... | Let the circle have rolled so that the [[Definition:Radius of Circle|radius]] to the point $P = \tuple {x, y}$ is at [[Definition:Angle|angle]] $\theta$ to [[Definition:Vertical|the vertical]].
:[[File:Cycloid.png|700px]]
The [[Definition:Center of Circle|center]] of the [[Definition:Circle|circle]] is at $\tuple {... | Equation of Cycloid | https://proofwiki.org/wiki/Equation_of_Cycloid | https://proofwiki.org/wiki/Equation_of_Cycloid | [
"Cycloids"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Axis/X-Axis",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center",
"Definition:Axis/Y-Axis",
"Definition:Cycloid",
... | [
"Definition:Circle/Radius",
"Definition:Angle",
"Definition:Vertical",
"File:Cycloid.png",
"Definition:Circle/Center",
"Definition:Circle",
"Definition:Diagram (Graphical Technique)",
"Definition:Cartesian Coordinate System/X Coordinate",
"Definition:Circle/Center",
"Definition:Circle",
"Definit... |
proofwiki-2917 | Length of Arc of Cycloid | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the length of one arc of the cycloid is $8 a$. | Let $L$ be the length of one arc of the cycloid.
From Arc Length for Parametric Equations:
:$\ds L = \int_0^{2 \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
where, from Equation of Cycloid:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
we... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the length of one [[Definition:Arc of Cycloid|arc]] of the [[Definition:Cycloid|cycloid]] is $8 a$. | Let $L$ be the [[Definition:Arc Length|length]] of one [[Definition:Arc of Cycloid|arc]] of the [[Definition:Cycloid|cycloid]].
From [[Arc Length for Parametric Equations]]:
:$\ds L = \int_0^{2 \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
where, from [[Equation... | Length of Arc of Cycloid/Proof 1 | https://proofwiki.org/wiki/Length_of_Arc_of_Cycloid | https://proofwiki.org/wiki/Length_of_Arc_of_Cycloid/Proof_1 | [
"Arc Length",
"Length of Arc of Cycloid",
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Cycloid/Arc",
"Definition:Cycloid"
] | [
"Definition:Arc Length",
"Definition:Cycloid/Arc",
"Definition:Cycloid",
"Arc Length for Parametric Equations",
"Equation of Cycloid",
"Derivative of Sine Function",
"Derivative of Cosine Function",
"Sum of Squares of Sine and Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denomina... |
proofwiki-2918 | Length of Arc of Cycloid | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the length of one arc of the cycloid is $8 a$. | Consider the tangent line $PQ$ to the cycloid generated by a generating circle moving at '''constant''' angular speed of $\omega = \dfrac \theta t$.
:750px
From Angular Speed of Particle in Circular Motion at Constant Speed, point $P$ is moving at a speed of $\dfrac {\d s} {\d t} = r \omega = 2 a \map \sin {\dfrac \the... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the length of one [[Definition:Arc of Cycloid|arc]] of the [[Definition:Cycloid|cycloid]] is $8 a$. | Consider the [[Definition:Tangent Line|tangent line]] $PQ$ to the [[Definition:Cycloid|cycloid]] generated by a [[Definition:Generating Circle of Cycloid|generating circle]] [[Definition:Motion|moving]] at '''[[Definition:Constant|constant]]''' [[Definition:Angular Speed|angular speed]] of $\omega = \dfrac \theta t$.
... | Length of Arc of Cycloid/Proof 2 | https://proofwiki.org/wiki/Length_of_Arc_of_Cycloid | https://proofwiki.org/wiki/Length_of_Arc_of_Cycloid/Proof_2 | [
"Arc Length",
"Length of Arc of Cycloid",
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Cycloid/Arc",
"Definition:Cycloid"
] | [
"Definition:Tangent Line",
"Definition:Cycloid",
"Definition:Cycloid/Generating Circle",
"Definition:Motion",
"Definition:Constant",
"Definition:Angular Speed",
"File:ArcLengthOfCycloid2.png",
"Angular Speed of Particle in Circular Motion at Constant Speed",
"Definition:Point",
"Definition:Motion"... |
proofwiki-2919 | Length of Arc of Cycloid | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the length of one arc of the cycloid is $8 a$. | {{AuthorRef|René Descartes|Descartes}} approximated the cycloid by substituting a polygon for the generating circle.
For example, a hexagon (where $n = 6$) produces $5$ arches (that is, $n - 1$).
Each arch is generated by turning through the external angle of the polygon.
For the hexagon, this is $\dfrac {2 \pi} ... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the length of one [[Definition:Arc of Cycloid|arc]] of the [[Definition:Cycloid|cycloid]] is $8 a$. | {{AuthorRef|René Descartes|Descartes}} approximated the [[Definition:Cycloid|cycloid]] by substituting a [[Definition:Polygon|polygon]] for the [[Definition:Generating Circle of Cycloid|generating circle]].
For example, a [[Definition:Hexagon|hexagon]] (where $n = 6$) produces $5$ [[Definition:Arc of Cycloid|arches]... | Length of Arc of Cycloid/Proof 3 | https://proofwiki.org/wiki/Length_of_Arc_of_Cycloid | https://proofwiki.org/wiki/Length_of_Arc_of_Cycloid/Proof_3 | [
"Arc Length",
"Length of Arc of Cycloid",
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Cycloid/Arc",
"Definition:Cycloid"
] | [
"Definition:Cycloid",
"Definition:Polygon",
"Definition:Cycloid/Generating Circle",
"Definition:Hexagon",
"Definition:Cycloid/Arc",
"Definition:Cycloid/Arc",
"Definition:Polygon/External Angle",
"Definition:Polygon",
"Definition:Hexagon",
"File:Cycloid of hexagon.png",
"Definition:Cycloid/Arc",
... |
proofwiki-2920 | Area under Arc of Cycloid | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the area under one arc of the cycloid is $3 \pi a^2$.
That is, the area under one arc of the cycloid is three times the area of the generating circle. | Let $A$ be the area under of one arc of the cycloid.
From Area under Curve, $A$ is defined by:
{{begin-eqn}}
{{eqn | l = A
| r = \int_0^{2 \pi a} y \rd x
| c =
}}
{{eqn | r = \int_0^{2 \pi} a \paren {1 - \cos \theta} \frac {\d x} {\d \theta} \rd \theta
| c =
}}
{{end-eqn}}
But:
:$\dfrac {\d x} {\d ... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
Then the [[Definition:Area|area]] under one [[Definition:Arc of Cycloid|arc of the cycloid]] is $3 \pi a^2$.
That is, the [[Definition:Area|area]] under one [[Definition:A... | Let $A$ be the [[Definition:Area|area]] under of one [[Definition:Arc of Cycloid|arc of the cycloid]].
From [[Area under Curve]], $A$ is defined by:
{{begin-eqn}}
{{eqn | l = A
| r = \int_0^{2 \pi a} y \rd x
| c =
}}
{{eqn | r = \int_0^{2 \pi} a \paren {1 - \cos \theta} \frac {\d x} {\d \theta} \rd \the... | Area under Arc of Cycloid | https://proofwiki.org/wiki/Area_under_Arc_of_Cycloid | https://proofwiki.org/wiki/Area_under_Arc_of_Cycloid | [
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Area",
"Definition:Cycloid/Arc",
"Definition:Area",
"Definition:Cycloid/Arc",
"Definition:Area",
"Definition:Cycloid/Generating Circle"
] | [
"Definition:Area",
"Definition:Cycloid/Arc",
"Area under Curve"
] |
proofwiki-2921 | Goldbach Conjecture implies Weak Goldbach Conjecture | The Goldbach Conjecture:
: Every even integer greater than $2$ is the sum of two primes
implies Goldbach's Weak Conjecture:
: Every odd integer greater than $7$ is the sum of three odd primes. | Take any odd integer $n$ such that $n > 7$.
Then $m = n - 3$ is an even integer $n$ such that $m > 4$.
If the Goldbach Conjecture holds, then $m$ is the sum of two primes: $m = p_1 + p_2$.
If one of them were $2$, then $m - 2$ would have to be even, which if it is prime it can not be.
So if $m > 4$, both $p_1$ and $p_2... | The [[Goldbach Conjecture]]:
: Every [[Definition:Even Integer|even integer]] greater than $2$ is the sum of two [[Definition:Prime Number|primes]]
implies [[Goldbach's Weak Conjecture]]:
: Every [[Definition:Odd Integer|odd integer]] greater than $7$ is the sum of three [[Definition:Odd Prime|odd primes]]. | Take any [[Definition:Odd Integer|odd integer]] $n$ such that $n > 7$.
Then $m = n - 3$ is an [[Definition:Even Integer|even integer]] $n$ such that $m > 4$.
If the [[Goldbach Conjecture]] holds, then $m$ is the sum of two [[Definition:Prime Number|primes]]: $m = p_1 + p_2$.
If one of them were $2$, then $m - 2$ wo... | Goldbach Conjecture implies Weak Goldbach Conjecture | https://proofwiki.org/wiki/Goldbach_Conjecture_implies_Weak_Goldbach_Conjecture | https://proofwiki.org/wiki/Goldbach_Conjecture_implies_Weak_Goldbach_Conjecture | [
"Number Theory",
"Goldbach Conjecture"
] | [
"Goldbach Conjecture",
"Definition:Even Integer",
"Definition:Prime Number",
"Goldbach's Weak Conjecture",
"Definition:Odd Integer",
"Definition:Odd Prime"
] | [
"Definition:Odd Integer",
"Definition:Even Integer",
"Goldbach Conjecture",
"Definition:Prime Number",
"Definition:Even Integer",
"Goldbach Conjecture",
"Goldbach's Weak Conjecture",
"Category:Number Theory",
"Category:Goldbach Conjecture"
] |
proofwiki-2922 | Euclidean Domain is UFD | Let $\struct {D, +, \times}$ be a Euclidean domain.
Then $\struct {D, +, \times}$ is a unique factorization domain. | By the definition of unique factorization domain, we need to show that:
For all $x \in D$ such that $x$ is non-zero and not a unit of $D$:
:$(1): \quad x$ has a complete factorization in $D$
:$(2): \quad$ Any two complete factorizations of $x$ in $D$ are equivalent. | Let $\struct {D, +, \times}$ be a [[Definition:Euclidean Domain|Euclidean domain]].
Then $\struct {D, +, \times}$ is a [[Definition:Unique Factorization Domain|unique factorization domain]]. | By the definition of [[Definition:Unique Factorization Domain|unique factorization domain]], we need to show that:
For all $x \in D$ such that $x$ is non-[[Definition:Ring Zero|zero]] and not a [[Definition:Unit of Ring|unit]] of $D$:
:$(1): \quad x$ has a [[Definition:Complete Factorization|complete factorization]]... | Euclidean Domain is UFD | https://proofwiki.org/wiki/Euclidean_Domain_is_UFD | https://proofwiki.org/wiki/Euclidean_Domain_is_UFD | [
"Euclidean Domains",
"Unique Factorization Domains",
"Factorization"
] | [
"Definition:Euclidean Domain",
"Definition:Unique Factorization Domain"
] | [
"Definition:Unique Factorization Domain",
"Definition:Ring Zero",
"Definition:Unit of Ring",
"Definition:Complete Factorization",
"Definition:Complete Factorization",
"Definition:Equivalent Factorizations",
"Definition:Unit of Ring",
"Definition:Complete Factorization",
"Definition:Unit of Ring",
... |
proofwiki-2923 | Quadrature of Parabola | Let $T$ be a parabola.
Consider the parabolic segment bounded by an arbitrary chord $AB$.
Let $C$ be the point on $T$ where the tangent to $T$ is parallel to $AB$.
Then the area $S$ of the parabolic segment $ABC$ of $T$ is given by:
:$S = \dfrac 4 3 \triangle ABC$ | {{WLOG}}, consider the parabola $y = a x^2$.
Let $A, B, C$ be the points:
{{begin-eqn}}
{{eqn | l = A
| r = \tuple {x_0, a {x_0}^2}
}}
{{eqn | l = B
| r = \tuple {x_2, a {x_2}^2}
}}
{{eqn | l = C
| r = \tuple {x_1, a {x_1}^2}
}}
{{end-eqn}}
:500px
The slope of the tangent at $C$ is given by using:
:$\... | Let $T$ be a [[Definition:Parabola|parabola]].
Consider the [[Definition:Parabolic Segment|parabolic segment]] bounded by an arbitrary [[Definition:Chord of Parabola|chord]] $AB$.
Let $C$ be the point on $T$ where the [[Definition:Tangent Line|tangent]] to $T$ is [[Definition:Parallel Lines|parallel]] to $AB$.
Then ... | {{WLOG}}, consider the [[Definition:Parabola|parabola]] $y = a x^2$.
Let $A, B, C$ be the points:
{{begin-eqn}}
{{eqn | l = A
| r = \tuple {x_0, a {x_0}^2}
}}
{{eqn | l = B
| r = \tuple {x_2, a {x_2}^2}
}}
{{eqn | l = C
| r = \tuple {x_1, a {x_1}^2}
}}
{{end-eqn}}
:[[File:ParabolaQuadrature2.png|5... | Quadrature of Parabola/Proof 1 | https://proofwiki.org/wiki/Quadrature_of_Parabola | https://proofwiki.org/wiki/Quadrature_of_Parabola/Proof_1 | [
"Quadrature of Parabola",
"Parabolas",
"Area Formulas"
] | [
"Definition:Parabola",
"Definition:Parabolic Segment",
"Definition:Chord of Conic Section/Parabola",
"Definition:Tangent Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Area",
"Definition:Parabolic Segment"
] | [
"Definition:Parabola",
"File:ParabolaQuadrature2.png",
"Definition:Tangent Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Vertical",
"Definition:Line/Straight Line",
"Definition:Bisection",
"Definition:Point",
"Definition:Quadrilateral/Parallelogram",
"Definition:Point",
"Definition:... |
proofwiki-2924 | Quadrature of Parabola | Let $T$ be a parabola.
Consider the parabolic segment bounded by an arbitrary chord $AB$.
Let $C$ be the point on $T$ where the tangent to $T$ is parallel to $AB$.
Then the area $S$ of the parabolic segment $ABC$ of $T$ is given by:
:$S = \dfrac 4 3 \triangle ABC$ | Let $T$ be the parabola which is the locus of points $\tuple {x, y}$ satisfying $y = x^2$.
By Area of Triangle Inscribed in Parabola:
:the point $C$ where the tangent to $T$ at $C$ is parallel to $AB$ has $x$-coordinate $\dfrac 1 2 \paren {x_0 + x_2}$.
Let $d$ be the horizontal distance between $A$ and $B$.
By Area of ... | Let $T$ be a [[Definition:Parabola|parabola]].
Consider the [[Definition:Parabolic Segment|parabolic segment]] bounded by an arbitrary [[Definition:Chord of Parabola|chord]] $AB$.
Let $C$ be the point on $T$ where the [[Definition:Tangent Line|tangent]] to $T$ is [[Definition:Parallel Lines|parallel]] to $AB$.
Then ... | Let $T$ be the [[Definition:Parabola|parabola]] which is the [[Definition:Locus|locus]] of [[Definition:Point|points]] $\tuple {x, y}$ satisfying $y = x^2$.
By [[Area of Triangle Inscribed in Parabola]]:
:the point $C$ where the [[Definition:Tangent Line|tangent]] to $T$ at $C$ is [[Definition:Parallel Lines|parallel]... | Quadrature of Parabola/Proof 2 | https://proofwiki.org/wiki/Quadrature_of_Parabola | https://proofwiki.org/wiki/Quadrature_of_Parabola/Proof_2 | [
"Quadrature of Parabola",
"Parabolas",
"Area Formulas"
] | [
"Definition:Parabola",
"Definition:Parabolic Segment",
"Definition:Chord of Conic Section/Parabola",
"Definition:Tangent Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Area",
"Definition:Parabolic Segment"
] | [
"Definition:Parabola",
"Definition:Locus",
"Definition:Point",
"Area of Triangle Inscribed in Parabola",
"Definition:Tangent Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Cartesian Coordinate System",
"Definition:Distance between Points",
"Area of Triangle Inscribed in Parabola",
"Def... |
proofwiki-2925 | Lune of Hippocrates | Take the circle whose center is $A$ and whose radius is $AB = AC = AD = AE$.
Let $C$ be the center of a circle whose radius is $CD = CF = CE$.
:400px
Consider the lune $DFEB$.
Its area is equal to that of the square $AEGC$. | :400px
The chords $DB$ and $EB$ are tangent to the arc $DFE$. They divide the lune into three regions: yellow, green and blue.
From Pythagoras's Theorem, $CD = \sqrt 2 AD$.
The green and blue areas are of equal area as each subtend a right angle.
The orange area also subtends a right angle.
So the area of the orange ar... | Take the [[Definition:Circle|circle]] whose [[Definition:Center of Circle|center]] is $A$ and whose [[Definition:Radius of Circle|radius]] is $AB = AC = AD = AE$.
Let $C$ be the [[Definition:Center of Circle|center]] of a circle whose [[Definition:Radius of Circle|radius]] is $CD = CF = CE$.
:[[File:LuneOfHippocrates... | :[[File:LuneOfHippocratesProof.png|400px]]
The [[Definition:Chord of Circle|chords]] $DB$ and $EB$ are [[Definition:Tangent to Circle|tangent]] to the [[Definition:Arc of Circle|arc]] $DFE$. They divide the [[Definition:Lune (Plane Geometry)|lune]] into three regions: yellow, green and blue.
From [[Pythagoras's Theor... | Lune of Hippocrates | https://proofwiki.org/wiki/Lune_of_Hippocrates | https://proofwiki.org/wiki/Lune_of_Hippocrates | [
"Lunes (Plane Geometry)",
"Plane Geometry"
] | [
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"File:LuneOfHippocrates.png",
"Definition:Lune (Plane Geometry)",
"Definition:Area",
"Definition:Quadrilateral/Square"
] | [
"File:LuneOfHippocratesProof.png",
"Definition:Circle/Chord",
"Definition:Tangent Line/Circle",
"Definition:Circle/Arc",
"Definition:Lune (Plane Geometry)",
"Pythagoras's Theorem",
"Definition:Subtend",
"Definition:Right Angle",
"Definition:Subtend",
"Definition:Right Angle",
"Definition:Lune (P... |
proofwiki-2926 | Integral of Power | :$\ds \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | From the Fundamental Theorem of Calculus:
:$(1): \quad \ds \int_0^b x^n \rd x = \bigintlimits {\map F x} 0 b = \map F b - \map F 0$
where $\map F x$ is a primitive of $x^n$.
By Primitive of Power, $\dfrac {x^{n + 1} } {n + 1}$ is a primitive of $x^n$.
Then:
{{begin-eqn}}
{{eqn | l = \int_0^b x^n \rd x
| r = \intl... | :$\ds \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | From the [[Fundamental Theorem of Calculus]]:
:$(1): \quad \ds \int_0^b x^n \rd x = \bigintlimits {\map F x} 0 b = \map F b - \map F 0$
where $\map F x$ is a [[Definition:Primitive (Calculus)|primitive]] of $x^n$.
By [[Primitive of Power]], $\dfrac {x^{n + 1} } {n + 1}$ is a [[Definition:Primitive (Calculus)|primiti... | Integral of Power/Conventional Proof | https://proofwiki.org/wiki/Integral_of_Power | https://proofwiki.org/wiki/Integral_of_Power/Conventional_Proof | [
"Integral Calculus"
] | [] | [
"Fundamental Theorem of Calculus",
"Definition:Primitive (Calculus)",
"Primitive of Power",
"Definition:Primitive (Calculus)"
] |
proofwiki-2927 | Integral of Power | :$\ds \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | First let $n$ be a positive integer.
Take a real number $r \in \R$ such that $0 < r < 1$ but reasonably close to $1$.
Consider a subdivision $S$ of the closed interval $\closedint 0 b$ defined as:
:$S = \set {0, \ldots, r^2 b, r b, b}$
that is, by taking as the points of subdivision successive powers of $r$.
Now we tak... | :$\ds \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | First let $n$ be a [[Definition:Positive Integer|positive integer]].
Take a [[Definition:Real Number|real number]] $r \in \R$ such that $0 < r < 1$ but reasonably close to $1$.
Consider a [[Definition:Subdivision of Interval|subdivision]] $S$ of the [[Definition:Closed Real Interval|closed interval]] $\closedint 0 b$... | Integral of Power/Fermat's Proof | https://proofwiki.org/wiki/Integral_of_Power | https://proofwiki.org/wiki/Integral_of_Power/Fermat's_Proof | [
"Integral Calculus"
] | [] | [
"Definition:Positive/Integer",
"Definition:Real Number",
"Definition:Subdivision of Interval",
"Definition:Real Interval/Closed",
"Definition:Upper Darboux Sum",
"Sum of Geometric Sequence",
"Definition:Positive/Integer",
"Definition:Strictly Positive",
"Definition:Rational Number",
"Definition:Ra... |
proofwiki-2928 | Integral of Power | :$\ds \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | Let $n \ge 2$.
Let:
:$\ds I_n := \int \sin^n x \rd x$
Then:
{{begin-eqn}}
{{eqn | l = I_n
| r = \int \sin^n x \rd x
| c =
}}
{{eqn | r = \int \sin^{n - 1} x \sin x \rd x
| c =
}}
{{eqn | r = \int \sin^{n - 1} x \map \rd {-\cos x}
| c = Derivative of Cosine Function
}}
{{eqn | r = - \sin^{n - 1... | :$\ds \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | Let $n \ge 2$.
Let:
:$\ds I_n := \int \sin^n x \rd x$
Then:
{{begin-eqn}}
{{eqn | l = I_n
| r = \int \sin^n x \rd x
| c =
}}
{{eqn | r = \int \sin^{n - 1} x \sin x \rd x
| c =
}}
{{eqn | r = \int \sin^{n - 1} x \map \rd {-\cos x}
| c = [[Derivative of Cosine Function]]
}}
{{eqn | r = - \sin... | Reduction Formula for Integral of Power of Sine/Proof 1 | https://proofwiki.org/wiki/Integral_of_Power | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Sine/Proof_1 | [
"Integral Calculus"
] | [] | [
"Derivative of Cosine Function",
"Integration by Parts",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Linear Combination of Integrals/Indefinite",
"Sum of Squares of Sine and Cosine",
"Primitive of Sine Function"
] |
proofwiki-2929 | Integral of Power | :$\ds \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin^{n - 1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {n - 1} \sin ^{n - 2} x \cos x
| c... | :$\ds \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin^{n - 1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {n - 1} \sin ^{n - 2} x \cos x
| ... | Reduction Formula for Integral of Power of Sine/Proof 2 | https://proofwiki.org/wiki/Integral_of_Power | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Sine/Proof_2 | [
"Integral Calculus"
] | [] | [
"Derivative of Composite Function",
"Derivative of Sine Function",
"Power Rule for Derivatives",
"Primitive of Sine Function",
"Integration by Parts",
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-2930 | Volume of Sphere | The volume $V$ of a sphere of radius $r$ is given by:
:$V = \dfrac {4 \pi r^3} 3$ | Consider the circle in the cartesian plane whose center is at $\tuple {a, 0}$ and whose radius is $a$.
From Equation of Circle, its equation is:
:$(1): \quad x^2 + y^2 = 2 a x$
Consider this circle as the cross-section through the center of a sphere which has the x-axis passing through its center, which is at $\tuple {... | The [[Definition:Volume|volume]] $V$ of a [[Definition:Sphere (Geometry)|sphere]] of [[Definition:Radius of Sphere|radius]] $r$ is given by:
:$V = \dfrac {4 \pi r^3} 3$ | Consider the [[Definition:Circle|circle]] in the [[Definition:Cartesian Plane|cartesian plane]] whose [[Definition:Center of Circle|center]] is at $\tuple {a, 0}$ and whose [[Definition:Radius of Circle|radius]] is $a$.
From [[Equation of Circle]], its equation is:
:$(1): \quad x^2 + y^2 = 2 a x$
Consider this circle... | Volume of Sphere/Proof by Archimedes | https://proofwiki.org/wiki/Volume_of_Sphere | https://proofwiki.org/wiki/Volume_of_Sphere/Proof_by_Archimedes | [
"Volume of Sphere",
"Spheres",
"Volume Formulas"
] | [
"Definition:Volume",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius"
] | [
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Equation of Circle",
"Definition:Cross-Section",
"Definition:Sphere/Geometry",
"Definition:Axis/X-Axis",
"Definition:Sphere/Geometry/Center",
"Definition:Sphere/Geometry",
"Definition:Cone... |
proofwiki-2931 | Volume of Sphere | The volume $V$ of a sphere of radius $r$ is given by:
:$V = \dfrac {4 \pi r^3} 3$ | :600px
Consider a sphere $S$ of radius $r$.
Consider a right circular cylinder $C$ whose bases are circles of radius $r$ and whose height is $2 r$.
Let $K$ be a double napped cone each of whose nappes has bases which coincide with the bases of $C$.
Let $K$ be removed from $C$ to leave a solid figure $C'$ described as a... | The [[Definition:Volume|volume]] $V$ of a [[Definition:Sphere (Geometry)|sphere]] of [[Definition:Radius of Sphere|radius]] $r$ is given by:
:$V = \dfrac {4 \pi r^3} 3$ | :[[File:VolumeOfSphereCavalieri.png|600px]]
Consider a [[Definition:Sphere (Geometry)|sphere]] $S$ of [[Definition:Radius of Sphere|radius]] $r$.
Consider a [[Definition:Right Circular Cylinder|right circular cylinder]] $C$ whose [[Definition:Base of Right Circular Cylinder|bases]] are [[Definition:Circle|circles]] o... | Volume of Sphere/Proof by Cavalieri | https://proofwiki.org/wiki/Volume_of_Sphere | https://proofwiki.org/wiki/Volume_of_Sphere/Proof_by_Cavalieri | [
"Volume of Sphere",
"Spheres",
"Volume Formulas"
] | [
"Definition:Volume",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius"
] | [
"File:VolumeOfSphereCavalieri.png",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius",
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder/Base",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cylinder/Height",
"Definition:Cone (Geometry)/Double Na... |
proofwiki-2932 | Volume of Sphere | The volume $V$ of a sphere of radius $r$ is given by:
:$V = \dfrac {4 \pi r^3} 3$ | === Construction ===
Describe a circle on the $x y$-plane.
Let its center be the origin.
By Equation of Circle, this circle is the locus of:
:$x^2 + y^2 = r^2$
where $r$ is a constant radius.
Solving for $y$:
:$y = \pm \sqrt {r^2 - x^2}$
Considering only the upper half of the circle:
:$y = \sqrt {r^2 - x^2}$
:300px
Thi... | The [[Definition:Volume|volume]] $V$ of a [[Definition:Sphere (Geometry)|sphere]] of [[Definition:Radius of Sphere|radius]] $r$ is given by:
:$V = \dfrac {4 \pi r^3} 3$ | === Construction ===
Describe a [[Definition:Circle|circle]] on the [[Definition:Cartesian Plane|$x y$-plane]].
Let its [[Definition:Center of Circle|center]] be the [[Definition:Origin|origin]].
By [[Equation of Circle]], this circle is the [[Definition:Locus|locus]] of:
:$x^2 + y^2 = r^2$
where $r$ is a constant [... | Volume of Sphere/Proof by Method of Disks | https://proofwiki.org/wiki/Volume_of_Sphere | https://proofwiki.org/wiki/Volume_of_Sphere/Proof_by_Method_of_Disks | [
"Volume of Sphere",
"Spheres",
"Volume Formulas"
] | [
"Definition:Volume",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius"
] | [
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Equation of Circle",
"Definition:Locus",
"Definition:Circle/Radius",
"File:Semicircle.png",
"Definition:Geometric Figure/Plane Figure",
"Definition:Circle/Semicircle",
"Definiti... |
proofwiki-2933 | Einstein's Mass-Energy Equation | The energy imparted to a body to cause that body to move causes the body to increase in mass by a value $M$ as given by the equation:
:$E = M c^2$
where $c$ is the speed of light. | From Einstein's Law of Motion, we have:
:$\mathbf F = \dfrac {m_0 \mathbf a} {\paren {1 - \dfrac {v^2} {c^2} }^{\tfrac 3 2} }$
where:
:$\mathbf F$ is the force on the body
:$\mathbf a$ is the acceleration induced on the body
:$v$ is the magnitude of the velocity of the body
:$c$ is the speed of light
:$m_0$ is the rest... | The [[Definition:Energy|energy]] imparted to a [[Definition:Body|body]] to cause that [[Definition:Body|body]] to move causes the [[Definition:Body|body]] to increase in [[Definition:Mass|mass]] by a value $M$ as given by the equation:
:$E = M c^2$
where $c$ is the [[Definition:Speed of Light|speed of light]]. | From [[Einstein's Law of Motion]], we have:
:$\mathbf F = \dfrac {m_0 \mathbf a} {\paren {1 - \dfrac {v^2} {c^2} }^{\tfrac 3 2} }$
where:
:$\mathbf F$ is the [[Definition:Force|force]] on the [[Definition:Body|body]]
:$\mathbf a$ is the [[Definition:Acceleration|acceleration]] induced on the [[Definition:Body|body]]
... | Einstein's Mass-Energy Equation | https://proofwiki.org/wiki/Einstein's_Mass-Energy_Equation | https://proofwiki.org/wiki/Einstein's_Mass-Energy_Equation | [
"Einstein's Mass-Energy Equation",
"Mass",
"Energy",
"Relativistic Mechanics",
"Physics"
] | [
"Definition:Energy",
"Definition:Body",
"Definition:Body",
"Definition:Body",
"Definition:Mass",
"Definition:Speed of Light"
] | [
"Einstein's Law of Motion",
"Definition:Force",
"Definition:Body",
"Definition:Acceleration",
"Definition:Body",
"Definition:Magnitude",
"Definition:Velocity",
"Definition:Body",
"Definition:Speed of Light",
"Definition:Rest Mass",
"Definition:Body",
"Definition:Stationary",
"Definition:Coor... |
proofwiki-2934 | Dimension of Universal Gravitational Constant | The dimension of the universal gravitational constant $G$ is $M^{-1} L^3 T^{-2}$. | From Newton's Law of Universal Gravitation:
:$\mathbf F = \dfrac {G m_1 m_2 \mathbf r} {r^3}$
We have that:
: The dimension of force is $M L T^{-2}$
: The dimension of displacement is $L$
: The dimension of mass is $M$.
Let $x$ be the dimension of $G$.
Then we have:
:$M L T^{-2} = x \dfrac {M^2 L}{L^3}$
Hence, after al... | The [[Definition:Dimension of Measurement|dimension]] of the [[Definition:Universal Gravitational Constant|universal gravitational constant]] $G$ is $M^{-1} L^3 T^{-2}$. | From [[Newton's Law of Universal Gravitation]]:
:$\mathbf F = \dfrac {G m_1 m_2 \mathbf r} {r^3}$
We have that:
: The [[Definition:Dimension of Measurement|dimension]] of [[Definition:Force|force]] is $M L T^{-2}$
: The [[Definition:Dimension of Measurement|dimension]] of [[Definition:Displacement|displacement]] is $L... | Dimension of Universal Gravitational Constant | https://proofwiki.org/wiki/Dimension_of_Universal_Gravitational_Constant | https://proofwiki.org/wiki/Dimension_of_Universal_Gravitational_Constant | [
"Universal Gravitational Constant",
"Dimensional Analysis"
] | [
"Definition:Dimension (Measurement)",
"Definition:Universal Gravitational Constant"
] | [
"Newton's Law of Universal Gravitation",
"Definition:Dimension (Measurement)",
"Definition:Force",
"Definition:Dimension (Measurement)",
"Definition:Displacement",
"Definition:Dimension (Measurement)",
"Definition:Mass",
"Definition:Dimension (Measurement)",
"Category:Universal Gravitational Constan... |
proofwiki-2935 | Dimension of Spring Force Constant | The dimension of a spring force constant is $\mathsf {M T}^{-2}$. | From Hooke's Law, we have:
:$\mathbf F = -k \mathbf x$
where:
:$\mathbf F$ is a force, of dimension $\mathsf {M L T}^{-2}$
:$k$ is the spring force constant
:$\mathbf x$ is a displacement, of dimension $\mathsf L$.
Let the dimension of $k$ be $D$.
Then we have:
:$\mathsf {M L T}^{-2} = D \mathsf L$
from which the resul... | The [[Definition:Dimension of Measurement|dimension]] of a [[Definition:Spring Force Constant|spring force constant]] is $\mathsf {M T}^{-2}$. | From [[Hooke's Law]], we have:
:$\mathbf F = -k \mathbf x$
where:
:$\mathbf F$ is a [[Definition:Force|force]], of [[Definition:Dimension of Measurement|dimension]] $\mathsf {M L T}^{-2}$
:$k$ is the [[Definition:Spring Force Constant|spring force constant]]
:$\mathbf x$ is a [[Definition:Displacement|displacement]], ... | Dimension of Spring Force Constant | https://proofwiki.org/wiki/Dimension_of_Spring_Force_Constant | https://proofwiki.org/wiki/Dimension_of_Spring_Force_Constant | [
"Dimensional Analysis"
] | [
"Definition:Dimension (Measurement)",
"Definition:Spring/Force Constant"
] | [
"Hooke's Law",
"Definition:Force",
"Definition:Dimension (Measurement)",
"Definition:Spring/Force Constant",
"Definition:Displacement",
"Definition:Dimension (Measurement)",
"Definition:Dimension (Measurement)",
"Category:Dimensional Analysis"
] |
proofwiki-2936 | Vieta's Formula for Pi | :$\pi = 2 \times \dfrac 2 {\sqrt 2} \times \dfrac 2 {\sqrt {2 + \sqrt 2} } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt 2} } } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt 2 } } } } \times \cdots$ | {{begin-eqn}}
{{eqn | l = 1
| r = \sin \frac \pi 2
| c = Sine of Half-Integer Multiple of Pi
}}
{{eqn | r = 2 \sin \frac \pi 4 \cos \frac \pi 4
| c = Double Angle Formula for Sine
}}
{{eqn | r = 2 \paren {2 \sin \frac \pi 8 \cos \frac \pi 8} \cos \frac \pi 4
| c = Double Angle Formula for Sine
}... | :$\pi = 2 \times \dfrac 2 {\sqrt 2} \times \dfrac 2 {\sqrt {2 + \sqrt 2} } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt 2} } } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt 2 } } } } \times \cdots$ | {{begin-eqn}}
{{eqn | l = 1
| r = \sin \frac \pi 2
| c = [[Sine of Half-Integer Multiple of Pi]]
}}
{{eqn | r = 2 \sin \frac \pi 4 \cos \frac \pi 4
| c = [[Double Angle Formula for Sine]]
}}
{{eqn | r = 2 \paren {2 \sin \frac \pi 8 \cos \frac \pi 8} \cos \frac \pi 4
| c = [[Double Angle Formula ... | Vieta's Formula for Pi | https://proofwiki.org/wiki/Vieta's_Formula_for_Pi | https://proofwiki.org/wiki/Vieta's_Formula_for_Pi | [
"Vieta's Formula for Pi",
"Formulas for Pi",
"Examples of Infinite Products"
] | [] | [
"Sine of Half-Integer Multiple of Pi",
"Double Angle Formulas/Sine",
"Double Angle Formulas/Sine",
"Double Angle Formulas/Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Half Angle Formulas/Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"De... |
proofwiki-2937 | Leibniz's Formula for Pi | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | First we note that:
:$(1): \quad \dfrac 1 {1 + t^2} = 1 - t^2 + t^4 - t^6 + \cdots + t^{4 n} - \dfrac {t^{4 n + 2} } {1 + t^2}$
which is demonstrated here.
Now consider the real number $x \in \R: 0 \le x \le 1$.
We can integrate expression $(1)$ {{WRT|Integration}} $t$ from $0$ to $x$:
:$(2): \quad \ds \int_0^x \frac {... | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | First we note that:
:$(1): \quad \dfrac 1 {1 + t^2} = 1 - t^2 + t^4 - t^6 + \cdots + t^{4 n} - \dfrac {t^{4 n + 2} } {1 + t^2}$
which is demonstrated [[Leibniz's Formula for Pi/Lemma|here]].
Now consider the [[Definition:Real Number|real number]] $x \in \R: 0 \le x \le 1$.
We can [[Definition:Integration|integrate... | Leibniz's Formula for Pi/Elementary Proof | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi/Elementary_Proof | [
"Leibniz's Formula for Pi",
"Formulas for Pi"
] | [] | [
"Leibniz's Formula for Pi/Lemma",
"Definition:Real Number",
"Definition:Primitive (Calculus)/Integration",
"Square of Real Number is Non-Negative",
"Relative Sizes of Definite Integrals",
"Definition:Basic Null Sequence",
"Squeeze Theorem",
"Derivative of Arctangent Function",
"Fundamental Theorem o... |
proofwiki-2938 | Leibniz's Formula for Pi | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | :500px
The area $OAT$ is a quarter-circle whose area is $\dfrac \pi 4$ by Area of Circle.
Now consider the area $C$ of the segment $OPQT$, bounded by the arc $OT$ and the chord $OT$.
Consider the area $OPQ$, bounded by the line segments $OP$ and $OQ$ and the arc $PQ$.
As $P$ and $Q$ approach each other, the arc $PQ$ te... | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | :[[File:LeibnizFormula.png|500px]]
The area $OAT$ is a quarter-[[Definition:Circle|circle]] whose area is $\dfrac \pi 4$ by [[Area of Circle]].
Now consider the area $C$ of the [[Definition:Segment of Circle|segment]] $OPQT$, bounded by the [[Definition:Arc of Circle|arc]] $OT$ and the [[Definition:Chord of Circle|ch... | Leibniz's Formula for Pi/Leibniz's Proof | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi/Leibniz's_Proof | [
"Leibniz's Formula for Pi",
"Formulas for Pi"
] | [] | [
"File:LeibnizFormula.png",
"Definition:Circle",
"Area of Circle",
"Definition:Segment of Circle",
"Definition:Circle/Arc",
"Definition:Circle/Chord",
"Definition:Line/Segment",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Definition:Line/Segment",
"Definition:Triangle (Geometry)",
"Defin... |
proofwiki-2939 | Leibniz's Formula for Pi | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | {{begin-eqn}}
{{eqn | l = 2 b \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {a + b k}
| r = \map \psi {\dfrac a {2 b} + 1} - \map \psi {\dfrac a {2 b} + \dfrac 1 2}
| c = {{Corollary|Reciprocal times Derivative of Gamma Function|2}}
}}
{{eqn | ll= \leadsto
| l = -4 \sum_{k \mathop = 1}^\in... | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | {{begin-eqn}}
{{eqn | l = 2 b \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {a + b k}
| r = \map \psi {\dfrac a {2 b} + 1} - \map \psi {\dfrac a {2 b} + \dfrac 1 2}
| c = {{Corollary|Reciprocal times Derivative of Gamma Function|2}}
}}
{{eqn | ll= \leadsto
| l = -4 \sum_{k \mathop = 1}^\in... | Leibniz's Formula for Pi/Proof by Digamma Function | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi/Proof_by_Digamma_Function | [
"Leibniz's Formula for Pi",
"Formulas for Pi"
] | [] | [
"Digamma Function of Three Fourths",
"Digamma Function/Examples/Digamma Function of Five Fourths"
] |
proofwiki-2940 | Leibniz's Formula for Pi | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | {{Recall|Dirichlet Beta Function}}
{{:Definition:Dirichlet Beta Function}}
From Dirichlet Beta Function at Odd Positive Integers, we obtain:
:$\map \beta {2 n + 1} = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}$
Therefore, setting $n = 0$ above:
{{begin-eqn}}
{{eqn | l = \map \beta 1
| ... | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | {{Recall|Dirichlet Beta Function}}
{{:Definition:Dirichlet Beta Function}}
From [[Dirichlet Beta Function at Odd Positive Integers]], we obtain:
:$\map \beta {2 n + 1} = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}$
Therefore, setting $n = 0$ above:
{{begin-eqn}}
{{eqn | l = \map \beta 1
... | Leibniz's Formula for Pi/Proof by Dirichlet Beta Function | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi/Proof_by_Dirichlet_Beta_Function | [
"Leibniz's Formula for Pi",
"Formulas for Pi"
] | [] | [
"Dirichlet Beta Function at Odd Positive Integers",
"Dirichlet Beta Function at Odd Positive Integers",
"Factorial/Examples/0"
] |
proofwiki-2941 | Leibniz's Formula for Pi | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | From {{Corollary|Mittag-Leffler Expansion for Cotangent Function|1}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi {2 n} \map \cot {\frac {\pi m} {2 n} }
| r = \frac 1 m + \sum_{k \mathop = 1}^\infty \paren {\frac 1 {2 k n + m} - \frac 1 {2 k n - m} }
| c =
}}
{{eqn | r = \frac 1 m + \paren {\frac 1 {2 n +... | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | From {{Corollary|Mittag-Leffler Expansion for Cotangent Function|1}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi {2 n} \map \cot {\frac {\pi m} {2 n} }
| r = \frac 1 m + \sum_{k \mathop = 1}^\infty \paren {\frac 1 {2 k n + m} - \frac 1 {2 k n - m} }
| c =
}}
{{eqn | r = \frac 1 m + \paren {\frac 1 {2 n ... | Leibniz's Formula for Pi/Proof by Mittag-Leffler Expansion for Cotangent Function | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi/Proof_by_Mittag-Leffler_Expansion_for_Cotangent_Function | [
"Leibniz's Formula for Pi",
"Formulas for Pi"
] | [] | [] |
proofwiki-2942 | Leibniz's Formula for Pi | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | From {{Corollary|Mittag-Leffler Expansion for Tangent Function}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi {2 n} \map \tan {\frac {\pi m} {2 n} }
| r = \sum_{k \mathop = 0}^\infty \paren {\frac 1 {\paren {2 k + 1} n - m} - \frac 1 {\paren {2 k + 1} n + m} }
| c =
}}
{{eqn | r = \paren {\frac 1 {n - m} ... | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | From {{Corollary|Mittag-Leffler Expansion for Tangent Function}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi {2 n} \map \tan {\frac {\pi m} {2 n} }
| r = \sum_{k \mathop = 0}^\infty \paren {\frac 1 {\paren {2 k + 1} n - m} - \frac 1 {\paren {2 k + 1} n + m} }
| c =
}}
{{eqn | r = \paren {\frac 1 {n - m}... | Leibniz's Formula for Pi/Proof by Mittag-Leffler Expansion for Tangent Function | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi/Proof_by_Mittag-Leffler_Expansion_for_Tangent_Function | [
"Leibniz's Formula for Pi",
"Formulas for Pi"
] | [] | [] |
proofwiki-2943 | Leibniz's Formula for Pi | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | From Power Series Expansion for Real Arctangent Function, we obtain:
:$\arctan x = x - \dfrac {x^3} 3 + \dfrac {x^5} 5 - \dfrac {x^7} 7 + \dfrac {x^9} 9 - \cdots$
Substituting $x = 1$ gives the required result.
{{qed}} | :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$ | From [[Power Series Expansion for Real Arctangent Function]], we obtain:
:$\arctan x = x - \dfrac {x^3} 3 + \dfrac {x^5} 5 - \dfrac {x^7} 7 + \dfrac {x^9} 9 - \cdots$
Substituting $x = 1$ gives the required result.
{{qed}} | Leibniz's Formula for Pi/Proof by Taylor Expansion | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi/Proof_by_Taylor_Expansion | [
"Leibniz's Formula for Pi",
"Formulas for Pi"
] | [] | [
"Power Series Expansion for Real Arctangent Function"
] |
proofwiki-2944 | Derivative of Arc Length | Let $C$ be a curve in the cartesian plane described by the equation $y = \map f x$.
Let $s$ be the length along the arc of the curve from some reference point $P$.
Then the derivative of $s$ with respect to $x$ is given by:
:$\dfrac {\d s} {\d x} = \sqrt {1 + \paren {\dfrac {\d y} {\d x} }^2}$ | Consider a length $\d s$ of $C$, short enough for it to be approximated to a straight line segment:
:250px
By Pythagoras's Theorem, we have:
:$\d s^2 = \d x^2 + \d y^2$
Dividing by $\d x^2$ we have:
{{begin-eqn}}
{{eqn | l = \paren {\frac {\d s} {\d x} }^2
| r = \paren {\frac {\d x} {\d x} }^2 + \paren {\frac {\d... | Let $C$ be a [[Definition:Curve|curve]] in the [[Definition:Cartesian Plane|cartesian plane]] described by the equation $y = \map f x$.
Let $s$ be the [[Definition:Linear Measure|length]] along the [[Definition:Arc of Curve|arc]] of the curve from some reference point $P$.
Then the [[Definition:Derivative|derivative]... | Consider a length $\d s$ of $C$, short enough for it to be approximated to a [[Definition:Line Segment|straight line segment]]:
:[[File:DSbyDX.png|250px]]
By [[Pythagoras's Theorem]], we have:
:$\d s^2 = \d x^2 + \d y^2$
Dividing by $\d x^2$ we have:
{{begin-eqn}}
{{eqn | l = \paren {\frac {\d s} {\d x} }^2
... | Derivative of Arc Length/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Arc_Length | https://proofwiki.org/wiki/Derivative_of_Arc_Length/Proof_1 | [
"Analytic Geometry",
"Differential Calculus",
"Integral Calculus",
"Derivative of Arc Length"
] | [
"Definition:Line/Curve",
"Definition:Cartesian Plane",
"Definition:Linear Measure",
"Definition:Curve/Arc",
"Definition:Derivative"
] | [
"Definition:Line/Segment",
"File:DSbyDX.png",
"Pythagoras's Theorem",
"Definition:Square Root/Complex Number/Principal Square Root"
] |
proofwiki-2945 | Derivative of Arc Length | Let $C$ be a curve in the cartesian plane described by the equation $y = \map f x$.
Let $s$ be the length along the arc of the curve from some reference point $P$.
Then the derivative of $s$ with respect to $x$ is given by:
:$\dfrac {\d s} {\d x} = \sqrt {1 + \paren {\dfrac {\d y} {\d x} }^2}$ | From Continuously Differentiable Curve has Finite Arc Length, $s$ exists and is given by:
{{begin-eqn}}
{{eqn | l = s
| r = \int_P^x \sqrt {1 + \paren {\frac {\d y} {\d u} }^2} \rd u
}}
{{eqn | ll= \leadsto
| l = \frac {\d s} {\d x}
| r = \frac {\d} {\d x} \int_P^x \sqrt {1 + \paren {\frac {\d y} {\d ... | Let $C$ be a [[Definition:Curve|curve]] in the [[Definition:Cartesian Plane|cartesian plane]] described by the equation $y = \map f x$.
Let $s$ be the [[Definition:Linear Measure|length]] along the [[Definition:Arc of Curve|arc]] of the curve from some reference point $P$.
Then the [[Definition:Derivative|derivative]... | From [[Continuously Differentiable Curve has Finite Arc Length]], $s$ exists and is given by:
{{begin-eqn}}
{{eqn | l = s
| r = \int_P^x \sqrt {1 + \paren {\frac {\d y} {\d u} }^2} \rd u
}}
{{eqn | ll= \leadsto
| l = \frac {\d s} {\d x}
| r = \frac {\d} {\d x} \int_P^x \sqrt {1 + \paren {\frac {\d y}... | Derivative of Arc Length/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Arc_Length | https://proofwiki.org/wiki/Derivative_of_Arc_Length/Proof_2 | [
"Analytic Geometry",
"Differential Calculus",
"Integral Calculus",
"Derivative of Arc Length"
] | [
"Definition:Line/Curve",
"Definition:Cartesian Plane",
"Definition:Linear Measure",
"Definition:Curve/Arc",
"Definition:Derivative"
] | [
"Continuously Differentiable Curve has Finite Arc Length",
"Definition:Differentiation",
"Fundamental Theorem of Calculus/First Part"
] |
proofwiki-2946 | Duality Principle for Sets | Any identity in set theory which uses any or all of the operations:
:Set intersection $\cap$
:Set union $\cup$
:Empty set $\O$
:Universal set $\mathbb U$
and none other, remains valid if:
:$\cap$ and $\cup$ are exchanged throughout
:$\O$ and $\mathbb U$ are exchanged throughout. | Follows from:
* Algebra of Sets is Huntington Algebra
* Principle of Duality of Huntington Algebras
{{qed}} | Any identity in [[Definition:Set Theory|set theory]] which uses any or all of the operations:
:[[Definition:Set Intersection|Set intersection]] $\cap$
:[[Definition:Set Union|Set union]] $\cup$
:[[Definition:Empty Set|Empty set]] $\O$
:[[Definition:Universal Set|Universal set]] $\mathbb U$
and none other, remains valid... | Follows from:
* [[Algebra of Sets is Huntington Algebra]]
* [[Principle of Duality of Huntington Algebras]]
{{qed}} | Duality Principle for Sets | https://proofwiki.org/wiki/Duality_Principle_for_Sets | https://proofwiki.org/wiki/Duality_Principle_for_Sets | [
"Named Theorems",
"Set Theory"
] | [
"Definition:Set Theory",
"Definition:Set Intersection",
"Definition:Set Union",
"Definition:Empty Set",
"Definition:Universal Set"
] | [
"Algebra of Sets is Huntington Algebra",
"Principle of Duality of Huntington Algebras"
] |
proofwiki-2947 | Rational Addition is Commutative | The operation of addition on the set of rational numbers $\Q$ is commutative:
:$\forall x, y \in \Q: x + y = y + x$ | Follows directly from the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
So $\struct {\Q, +, \times}$ is a field, and therefore a fortiori $+$ is commutative on $\Q$.
{{qed}} | The operation of [[Definition:Rational Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \Q: x + y = y + x$ | Follows directly from the [[Definition:Rational Number|definition of rational numbers]] as the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers]].
So $\struct {\Q, +, \times}$ is a [[Definition:Field (A... | Rational Addition is Commutative | https://proofwiki.org/wiki/Rational_Addition_is_Commutative | https://proofwiki.org/wiki/Rational_Addition_is_Commutative | [
"Rational Addition",
"Examples of Commutative Operations",
"Commutative Law of Addition"
] | [
"Definition:Addition/Rational Numbers",
"Definition:Set",
"Definition:Rational Number",
"Definition:Commutative/Operation"
] | [
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Definition:Field (Abstract Algebra)",
"Definition:A Fortiori",
"Definition:Commutative/Operation"
] |
proofwiki-2948 | Rational Addition is Associative | The operation of addition on the set of rational numbers $\Q$ is associative:
:$\forall x, y, z \in \Q: x + \paren {y + z} = \paren {x + y} + z$ | Follows directly from the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
So $\struct {\Q, +, \times}$ is a field, and therefore a fortiori $+$ is associative on $\Q$.
{{qed}} | The operation of [[Definition:Rational Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \Q: x + \paren {y + z} = \paren {x + y} + z$ | Follows directly from the [[Definition:Rational Number|definition of rational numbers]] as the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers]].
So $\struct {\Q, +, \times}$ is a [[Definition:Field (A... | Rational Addition is Associative | https://proofwiki.org/wiki/Rational_Addition_is_Associative | https://proofwiki.org/wiki/Rational_Addition_is_Associative | [
"Rational Addition",
"Examples of Associative Operations",
"Associative Law of Addition"
] | [
"Definition:Addition/Rational Numbers",
"Definition:Set",
"Definition:Rational Number",
"Definition:Associative Operation"
] | [
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Definition:Field (Abstract Algebra)",
"Definition:A Fortiori",
"Definition:Associative Operation"
] |
proofwiki-2949 | Subtraction on Numbers is Anticommutative | The operation of subtraction on the numbers is anticommutative.
That is:
:$a - b = b - a \iff a = b$ | === Natural Numbers ===
{{:Subtraction on Numbers is Anticommutative/Natural Numbers}} | The operation of [[Definition:Subtraction|subtraction]] on the [[Definition:Number|numbers]] is [[Definition:Anticommutative|anticommutative]].
That is:
:$a - b = b - a \iff a = b$ | === [[Subtraction on Numbers is Anticommutative/Natural Numbers|Natural Numbers]] ===
{{:Subtraction on Numbers is Anticommutative/Natural Numbers}} | Subtraction on Numbers is Anticommutative | https://proofwiki.org/wiki/Subtraction_on_Numbers_is_Anticommutative | https://proofwiki.org/wiki/Subtraction_on_Numbers_is_Anticommutative | [
"Numbers",
"Subtraction",
"Examples of Anticommutativity",
"Subtraction on Numbers is Anticommutative"
] | [
"Definition:Subtraction",
"Definition:Number",
"Definition:Anticommutative"
] | [
"Subtraction on Numbers is Anticommutative/Natural Numbers"
] |
proofwiki-2950 | Integer Addition is Commutative | The operation of addition on the set of integers $\Z$ is commutative:
:$\forall x, y \in \Z: x + y = y + x$ | From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.
From Integers under Addition form Abelian Group, the integers under addition form an abelian group, from which commutativity follows {{afortiori}}.
{{qed}} | The operation of [[Definition:Integer Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \Z: x + y = y + x$ | From the [[Definition:Integer/Formal Definition|formal definition of integers]], $\eqclass {a, b} {}$ is an [[Definition:Equivalence Class|equivalence class]] of [[Definition:Ordered Pair|ordered pairs]] of [[Definition:Natural Numbers|natural numbers]].
From [[Integers under Addition form Abelian Group]], the [[Defin... | Integer Addition is Commutative/Proof 1 | https://proofwiki.org/wiki/Integer_Addition_is_Commutative | https://proofwiki.org/wiki/Integer_Addition_is_Commutative/Proof_1 | [
"Integer Addition is Commutative",
"Integer Addition",
"Commutative Law of Addition",
"Examples of Commutative Operations"
] | [
"Definition:Addition/Integers",
"Definition:Set",
"Definition:Integer",
"Definition:Commutative/Operation"
] | [
"Definition:Integer/Formal Definition",
"Definition:Equivalence Class",
"Definition:Ordered Pair",
"Definition:Natural Numbers",
"Integers under Addition form Abelian Group",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Abelian Group",
"Definition:Commutative/Operation"
] |
proofwiki-2951 | Integer Addition is Commutative | The operation of addition on the set of integers $\Z$ is commutative:
:$\forall x, y \in \Z: x + y = y + x$ | Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for some $x, y \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x + y
| r = \eqclass {a, b} {} + \eqclass {c, d} {}
| c = {{Defof|Integer|subdef = Formal Definition}}
}}
{{eqn | r = \eqclass {a + c, b + d} {}
| c = {{Defof|Integer Addition}}
}}
{{eqn |... | The operation of [[Definition:Integer Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \Z: x + y = y + x$ | Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for some $x, y \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x + y
| r = \eqclass {a, b} {} + \eqclass {c, d} {}
| c = {{Defof|Integer|subdef = Formal Definition}}
}}
{{eqn | r = \eqclass {a + c, b + d} {}
| c = {{Defof|Integer Addition}}
}}
{{eqn ... | Integer Addition is Commutative/Proof 2 | https://proofwiki.org/wiki/Integer_Addition_is_Commutative | https://proofwiki.org/wiki/Integer_Addition_is_Commutative/Proof_2 | [
"Integer Addition is Commutative",
"Integer Addition",
"Commutative Law of Addition",
"Examples of Commutative Operations"
] | [
"Definition:Addition/Integers",
"Definition:Set",
"Definition:Integer",
"Definition:Commutative/Operation"
] | [
"Natural Number Addition is Commutative"
] |
proofwiki-2952 | Integer Addition is Associative | The operation of addition on the set of integers $\Z$ is associative:
:$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$ | From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.
From Integers under Addition form Abelian Group, the integers under addition form a group, from which associativity follows from {{Group-axiom|1}}.
{{qed}} | The operation of [[Definition:Integer Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$ | From the [[Definition:Integer/Formal Definition|formal definition of integers]], $\eqclass {a, b} {}$ is an [[Definition:Equivalence Class|equivalence class]] of [[Definition:Ordered Pair|ordered pairs]] of [[Definition:Natural Numbers|natural numbers]].
From [[Integers under Addition form Abelian Group]], the [[Defin... | Integer Addition is Associative/Proof 1 | https://proofwiki.org/wiki/Integer_Addition_is_Associative | https://proofwiki.org/wiki/Integer_Addition_is_Associative/Proof_1 | [
"Integer Addition",
"Examples of Associative Operations",
"Integer Addition is Associative",
"Associative Law of Addition"
] | [
"Definition:Addition/Integers",
"Definition:Set",
"Definition:Integer",
"Definition:Associative Operation"
] | [
"Definition:Integer/Formal Definition",
"Definition:Equivalence Class",
"Definition:Ordered Pair",
"Definition:Natural Numbers",
"Integers under Addition form Abelian Group",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Group",
"Definition:Associative Operation"
] |
proofwiki-2953 | Integer Addition is Associative | The operation of addition on the set of integers $\Z$ is associative:
:$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$ | Let $a, b, c, d, e, f \in \N$ such that:
:$x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$.
Then:
{{begin-eqn}}
{{eqn | l = x + \paren {y + z}
| r = \eqclass {a, b} {} + \paren {\eqclass {c, d} {} + \eqclass {e, f} {} }
| c = {{Defof|Integer|subdef = Formal Definition}}
}}
{{e... | The operation of [[Definition:Integer Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$ | Let $a, b, c, d, e, f \in \N$ such that:
:$x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$.
Then:
{{begin-eqn}}
{{eqn | l = x + \paren {y + z}
| r = \eqclass {a, b} {} + \paren {\eqclass {c, d} {} + \eqclass {e, f} {} }
| c = {{Defof|Integer|subdef = Formal Definition}}
}}
... | Integer Addition is Associative/Proof 2 | https://proofwiki.org/wiki/Integer_Addition_is_Associative | https://proofwiki.org/wiki/Integer_Addition_is_Associative/Proof_2 | [
"Integer Addition",
"Examples of Associative Operations",
"Integer Addition is Associative",
"Associative Law of Addition"
] | [
"Definition:Addition/Integers",
"Definition:Set",
"Definition:Integer",
"Definition:Associative Operation"
] | [
"Natural Number Addition is Associative"
] |
proofwiki-2954 | Natural Number Addition is Commutative | The operation of addition on the set of natural numbers $\N$ is commutative:
:$\forall m, n \in \N: m + n = n + m$ | Consider the natural numbers defined as a naturally ordered semigroup.
By definition, the operation in a naturally ordered semigroup is commutative.
Hence the result.
{{qed}} | The operation of [[Definition:Natural Number Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Commutative Operation|commutative]]:
:$\forall m, n \in \N: m + n = n + m$ | Consider the [[Definition:Natural Numbers|natural numbers]] defined as a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
By definition, the [[Definition:Binary Operation|operation]] in a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]] is [[Definition:Commutative Operatio... | Natural Number Addition is Commutative/Proof 1 | https://proofwiki.org/wiki/Natural_Number_Addition_is_Commutative | https://proofwiki.org/wiki/Natural_Number_Addition_is_Commutative/Proof_1 | [
"Natural Number Addition is Commutative",
"Natural Number Addition",
"Commutative Law of Addition",
"Examples of Commutative Operations"
] | [
"Definition:Addition/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Commutative/Operation"
] | [
"Definition:Natural Numbers",
"Definition:Naturally Ordered Semigroup",
"Definition:Operation/Binary Operation",
"Definition:Naturally Ordered Semigroup",
"Definition:Commutative/Operation"
] |
proofwiki-2955 | Natural Number Addition is Commutative | The operation of addition on the set of natural numbers $\N$ is commutative:
:$\forall m, n \in \N: m + n = n + m$ | Proof by induction.
Consider the natural numbers $\N$ defined as the elements of the minimally inductive set $\omega$.
From the definition of addition in $\omega$, we have that:
{{begin-eqn}}
{{eqn | q = \forall m, n \in \N
| l = m + 0
| r = m
}}
{{eqn | l = m + n^+
| r = \paren {m + n}^+
}}
{{end-eqn... | The operation of [[Definition:Natural Number Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Commutative Operation|commutative]]:
:$\forall m, n \in \N: m + n = n + m$ | Proof by [[Principle of Mathematical Induction|induction]].
Consider the [[Definition:Natural Numbers|natural numbers]] $\N$ defined as the [[Definition:Element|elements]] of the [[Definition:Minimally Inductive Set|minimally inductive set]] $\omega$.
From the definition of [[Definition:Addition in Minimally Inducti... | Natural Number Addition is Commutative/Proof 2 | https://proofwiki.org/wiki/Natural_Number_Addition_is_Commutative | https://proofwiki.org/wiki/Natural_Number_Addition_is_Commutative/Proof_2 | [
"Natural Number Addition is Commutative",
"Natural Number Addition",
"Commutative Law of Addition",
"Examples of Commutative Operations"
] | [
"Definition:Addition/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Commutative/Operation"
] | [
"Principle of Mathematical Induction",
"Definition:Natural Numbers",
"Definition:Element",
"Definition:Minimally Inductive Set",
"Definition:Addition in Minimally Inductive Set",
"Definition:Proposition",
"Natural Number Addition Commutes with Zero",
"Definition:True",
"Definition:Basis for the Indu... |
proofwiki-2956 | Natural Number Addition is Commutative | The operation of addition on the set of natural numbers $\N$ is commutative:
:$\forall m, n \in \N: m + n = n + m$ | Using the following axioms:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
Let $x \in \N_{> 0}$ be arbitrary.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$x + n = n + x$
=== Basis for the Induction ===
From Natural Number Commutes with 1 under Addition, we have that:
:$\forall x \in \N_{> 0}: x ... | The operation of [[Definition:Natural Number Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Commutative Operation|commutative]]:
:$\forall m, n \in \N: m + n = n + m$ | Using the [[Axiom:Axiomatization of 1-Based Natural Numbers|following axioms]]:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
Let $x \in \N_{> 0}$ be arbitrary.
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$x + n = n + x$
=== Basis for the Induction ===
From [[Nat... | Natural Number Addition is Commutative/Proof 3 | https://proofwiki.org/wiki/Natural_Number_Addition_is_Commutative | https://proofwiki.org/wiki/Natural_Number_Addition_is_Commutative/Proof_3 | [
"Natural Number Addition is Commutative",
"Natural Number Addition",
"Commutative Law of Addition",
"Examples of Commutative Operations"
] | [
"Definition:Addition/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Commutative/Operation"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Proposition",
"Natural Number Commutes with 1 under Addition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Natural Number Addition is Associative",
"Natural Number Addition is Comm... |
proofwiki-2957 | Natural Number Addition is Associative | The operation of addition on the set of natural numbers $\N$ is associative:
:$\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$ | Consider the natural numbers defined as a naturally ordered semigroup.
By definition, the operation in a semigroup is associative.
Hence the result.
{{qed}} | The operation of [[Definition:Natural Number Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$ | Consider the [[Definition:Natural Numbers|natural numbers]] defined as a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
By definition, the [[Definition:Binary Operation|operation]] in a [[Definition:Semigroup|semigroup]] is [[Definition:Associative Operation|associative]].
Hence the result.... | Natural Number Addition is Associative/Proof 1 | https://proofwiki.org/wiki/Natural_Number_Addition_is_Associative | https://proofwiki.org/wiki/Natural_Number_Addition_is_Associative/Proof_1 | [
"Natural Number Addition",
"Examples of Associative Operations",
"Natural Number Addition is Associative",
"Associative Law of Addition"
] | [
"Definition:Addition/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Associative Operation"
] | [
"Definition:Natural Numbers",
"Definition:Naturally Ordered Semigroup",
"Definition:Operation/Binary Operation",
"Definition:Semigroup",
"Definition:Associative Operation"
] |
proofwiki-2958 | Natural Number Addition is Associative | The operation of addition on the set of natural numbers $\N$ is associative:
:$\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$ | Consider the von Neumann construction of natural numbers $\N$, as elements of the minimally inductive set $\omega$.
We are to show that:
:$\paren {x + y} + n = x + \paren {y + n}$
for all $x, y, n \in \N$.
From the definition of addition, we have that:
{{begin-eqn}}
{{eqn | q = \forall m, n \in \N
| l = m + 0
... | The operation of [[Definition:Natural Number Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$ | Consider the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction of natural numbers]] $\N$, as [[Definition:Element|elements]] of the [[Definition:Minimally Inductive Set|minimally inductive set]] $\omega$.
We are to show that:
:$\paren {x + y} + n = x + \paren {y + n}$
for all $x, y, n \... | Natural Number Addition is Associative/Proof 2 | https://proofwiki.org/wiki/Natural_Number_Addition_is_Associative | https://proofwiki.org/wiki/Natural_Number_Addition_is_Associative/Proof_2 | [
"Natural Number Addition",
"Examples of Associative Operations",
"Natural Number Addition is Associative",
"Associative Law of Addition"
] | [
"Definition:Addition/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Associative Operation"
] | [
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Element",
"Definition:Minimally Inductive Set",
"Definition:Addition in Minimally Inductive Set",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Natura... |
proofwiki-2959 | Natural Number Addition is Associative | The operation of addition on the set of natural numbers $\N$ is associative:
:$\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$ | Using the following axioms:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
Let $x, y \in \N_{> 0}$ be arbitrary.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\paren {x + y} + n = x + \paren {y + n}$
=== Basis for the Induction ===
From Axiom $\text C$, we have by definition that:
:$\forall x, y ... | The operation of [[Definition:Natural Number Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$ | Using the [[Axiom:Axiomatization of 1-Based Natural Numbers|following axioms]]:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
Let $x, y \in \N_{> 0}$ be arbitrary.
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\paren {x + y} + n = x + \paren {y + n}$
=== Basis for ... | Natural Number Addition is Associative/Proof 3 | https://proofwiki.org/wiki/Natural_Number_Addition_is_Associative | https://proofwiki.org/wiki/Natural_Number_Addition_is_Associative/Proof_3 | [
"Natural Number Addition",
"Examples of Associative Operations",
"Natural Number Addition is Associative",
"Associative Law of Addition"
] | [
"Definition:Addition/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Associative Operation"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Proposition",
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Natural Number Addition is Associative/Proof 3",
"Natural Number Additi... |
proofwiki-2960 | Rational Multiplication is Commutative | The operation of multiplication on the set of rational numbers $\Q$ is commutative:
:$\forall x, y \in \Q: x \times y = y \times x$ | Follows directly from the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
So $\struct {\Q, +, \times}$ is a field, and therefore a fortiori $\times$ is commutative on $\Q$.
{{qed}} | The operation of [[Definition:Rational Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \Q: x \times y = y \times x$ | Follows directly from the [[Definition:Rational Number|definition of rational numbers]] as the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers]].
So $\struct {\Q, +, \times}$ is a [[Definition:Field (A... | Rational Multiplication is Commutative | https://proofwiki.org/wiki/Rational_Multiplication_is_Commutative | https://proofwiki.org/wiki/Rational_Multiplication_is_Commutative | [
"Rational Multiplication",
"Examples of Commutative Operations",
"Commutative Law of Multiplication"
] | [
"Definition:Multiplication/Rational Numbers",
"Definition:Set",
"Definition:Rational Number",
"Definition:Commutative/Operation"
] | [
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Definition:Field (Abstract Algebra)",
"Definition:A Fortiori",
"Definition:Commutative/Operation"
] |
proofwiki-2961 | Rational Multiplication is Associative | The operation of multiplication on the set of rational numbers $\Q$ is associative:
:$\forall x, y, z \in \Q: x \times \paren {y \times z} = \paren {x \times y} \times z$ | Follows directly from the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
So $\struct {\Q, +, \times}$ is a field, and therefore a fortiori $\times$ is associative on $\Q$.
{{qed}} | The operation of [[Definition:Rational Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \Q: x \times \paren {y \times z} = \paren {x \times y} \times z$ | Follows directly from the [[Definition:Rational Number|definition of rational numbers]] as the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers]].
So $\struct {\Q, +, \times}$ is a [[Definition:Field (A... | Rational Multiplication is Associative | https://proofwiki.org/wiki/Rational_Multiplication_is_Associative | https://proofwiki.org/wiki/Rational_Multiplication_is_Associative | [
"Rational Multiplication",
"Examples of Associative Operations",
"Associative Law of Multiplication"
] | [
"Definition:Multiplication/Rational Numbers",
"Definition:Set",
"Definition:Rational Number",
"Definition:Associative Operation"
] | [
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Definition:Field (Abstract Algebra)",
"Definition:A Fortiori",
"Definition:Associative Operation"
] |
proofwiki-2962 | Natural Number Multiplication is Commutative | The operation of multiplication on the set of natural numbers $\N$ is commutative:
:$\forall x, y \in \N: x \times y = y \times x$ | Let $A, B$ be two (natural) numbers, and let $A$ by multiplying $B$ make $C$, and $B$ by multiplying $A$ make $D$.
We need to show that $C = D$.
:350px
We have that $A \times B = C$.
So $B$ measures $C$ according to the units of $A$.
But the unit $E$ also measures $A$ according to the units in it.
So $E$ measures $A$ t... | The operation of [[Definition:Natural Number Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \N: x \times y = y \times x$ | Let $A, B$ be two [[Definition:Natural Number|(natural) numbers]], and let $A$ by [[Definition:Natural Number Multiplication|multiplying]] $B$ make $C$, and $B$ by [[Definition:Natural Number Multiplication|multiplying]] $A$ make $D$.
We need to show that $C = D$.
:[[File:Euclid-VII-16.png|350px]]
We have that $A \t... | Natural Number Multiplication is Commutative/Euclid's Proof | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Commutative | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Commutative/Euclid's_Proof | [
"Commutative Law of Multiplication",
"Natural Number Multiplication",
"Examples of Commutative Operations",
"Natural Number Multiplication is Commutative"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Commutative/Operation"
] | [
"Definition:Natural Numbers",
"Definition:Multiplication/Natural Numbers",
"Definition:Multiplication/Natural Numbers",
"File:Euclid-VII-16.png",
"Definition:Divisor (Algebra)/Integer",
"Definition:Unit (One)",
"Definition:One",
"Definition:Divisor (Algebra)/Integer",
"Definition:Unit (One)",
"Def... |
proofwiki-2963 | Natural Number Multiplication is Commutative | The operation of multiplication on the set of natural numbers $\N$ is commutative:
:$\forall x, y \in \N: x \times y = y \times x$ | Natural number multiplication is recursively defined as:
:$\forall m, n \in \N: \begin{cases} m \times 0 & = 0 \\ m \times \paren {n + 1} & = m \times n + m \end{cases}$
From the Principle of Recursive Definition, there is only one mapping $f$ satisfying this definition; that is, such that:
:$\forall n \in \N: \begin{c... | The operation of [[Definition:Natural Number Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \N: x \times y = y \times x$ | [[Definition:Natural Number Multiplication|Natural number multiplication]] is [[Definition:Recursively Defined Mapping|recursively defined]] as:
:$\forall m, n \in \N: \begin{cases} m \times 0 & = 0 \\ m \times \paren {n + 1} & = m \times n + m \end{cases}$
From the [[Principle of Recursive Definition]], there is onl... | Natural Number Multiplication is Commutative/Proof 1 | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Commutative | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Commutative/Proof_1 | [
"Commutative Law of Multiplication",
"Natural Number Multiplication",
"Examples of Commutative Operations",
"Natural Number Multiplication is Commutative"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Commutative/Operation"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Recursively Defined Mapping",
"Principle of Recursive Definition",
"Definition:Mapping",
"Zero is Zero Element for Natural Number Multiplication",
"Natural Number Multiplication Distributes over Addition",
"Principle of Recursive Definition"
] |
proofwiki-2964 | Natural Number Multiplication is Commutative | The operation of multiplication on the set of natural numbers $\N$ is commutative:
:$\forall x, y \in \N: x \times y = y \times x$ | Proof by induction:
From the definition of natural number multiplication, we have that:
{{begin-eqn}}
{{eqn | q = \forall m, n \in \N
| l = m \times 0
| r = 0
}}
{{eqn | l = m \times n^+
| r = \paren {m \times n} + m
}}
{{end-eqn}}
For all $n \in \N$, let $\map P n$ be the proposition:
:$\forall m \in... | The operation of [[Definition:Natural Number Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \N: x \times y = y \times x$ | Proof by [[Principle of Mathematical Induction|induction]]:
From the definition of [[Definition:Natural Number Multiplication|natural number multiplication]], we have that:
{{begin-eqn}}
{{eqn | q = \forall m, n \in \N
| l = m \times 0
| r = 0
}}
{{eqn | l = m \times n^+
| r = \paren {m \times n} +... | Natural Number Multiplication is Commutative/Proof 2 | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Commutative | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Commutative/Proof_2 | [
"Commutative Law of Multiplication",
"Natural Number Multiplication",
"Examples of Commutative Operations",
"Natural Number Multiplication is Commutative"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Commutative/Operation"
] | [
"Principle of Mathematical Induction",
"Definition:Multiplication/Natural Numbers",
"Definition:Proposition",
"Zero is Zero Element for Natural Number Multiplication",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Principle of Mathematical Induction",
"Natural Number Additi... |
proofwiki-2965 | Natural Number Multiplication is Commutative | The operation of multiplication on the set of natural numbers $\N$ is commutative:
:$\forall x, y \in \N: x \times y = y \times x$ | Using the following axioms:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\forall a \in \N_{> 0}: a \times n = n \times a$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = a \times 1
| r = a
| c = Axiom $\... | The operation of [[Definition:Natural Number Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Commutative Operation|commutative]]:
:$\forall x, y \in \N: x \times y = y \times x$ | Using the [[Axiom:Axiomatization of 1-Based Natural Numbers|following axioms]]:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\forall a \in \N_{> 0}: a \times n = n \times a$
=== Basis for the Induction ===
$\map P 1$ i... | Natural Number Multiplication is Commutative/Proof 3 | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Commutative | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Commutative/Proof_3 | [
"Commutative Law of Multiplication",
"Natural Number Multiplication",
"Examples of Commutative Operations",
"Natural Number Multiplication is Commutative"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Commutative/Operation"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Proposition",
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Left Distributive La... |
proofwiki-2966 | Natural Number Multiplication is Associative | The operation of multiplication on the set of natural numbers $\N$ is associative:
:$\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$ | From Index Laws for Semigroup: Product of Indices we have:
:$+^{z \times y} x = \map {+^z} {+^y x}$
By definition of multiplication, this amounts to:
:$x \times \paren {z \times y} = \paren {x \times y} \times z$
From Natural Number Multiplication is Commutative, we have:
:$x \times \paren {z \times y} = x \times \pare... | The operation of [[Definition:Natural Number Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$ | From [[Index Laws for Semigroup/Product of Indices|Index Laws for Semigroup: Product of Indices]] we have:
:$+^{z \times y} x = \map {+^z} {+^y x}$
By definition of [[Definition:Natural Number Multiplication|multiplication]], this amounts to:
:$x \times \paren {z \times y} = \paren {x \times y} \times z$
From [[Nat... | Natural Number Multiplication is Associative/Proof 1 | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Associative | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Associative/Proof_1 | [
"Natural Number Multiplication",
"Examples of Associative Operations",
"Natural Number Multiplication is Associative",
"Associative Law of Multiplication"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Associative Operation"
] | [
"Index Laws/Product of Indices/Semigroup",
"Definition:Multiplication/Natural Numbers",
"Natural Number Multiplication is Commutative"
] |
proofwiki-2967 | Natural Number Multiplication is Associative | The operation of multiplication on the set of natural numbers $\N$ is associative:
:$\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$ | We are to show that:
:$\paren {x \times y} \times n = x \times \paren {y \times n}$
for all $x, y, n \in \N$.
From the definition of natural number multiplication, we have that:
{{begin-eqn}}
{{eqn | q = \forall m, n \in \N
| l = m \times 0
| r = 0
| c =
}}
{{eqn | l = m \times \paren {n + 1}
|... | The operation of [[Definition:Natural Number Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$ | We are to show that:
:$\paren {x \times y} \times n = x \times \paren {y \times n}$
for all $x, y, n \in \N$.
From the definition of [[Definition:Natural Number Multiplication|natural number multiplication]], we have that:
{{begin-eqn}}
{{eqn | q = \forall m, n \in \N
| l = m \times 0
| r = 0
| c =... | Natural Number Multiplication is Associative/Proof 2 | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Associative | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Associative/Proof_2 | [
"Natural Number Multiplication",
"Examples of Associative Operations",
"Natural Number Multiplication is Associative",
"Associative Law of Multiplication"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Associative Operation"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Natural Number Multiplication is Associative/Proof 2",
"Natural Number Addition is Commutative",
"Natural Number ... |
proofwiki-2968 | Natural Number Multiplication is Associative | The operation of multiplication on the set of natural numbers $\N$ is associative:
:$\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$ | Using the following axioms:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
Let $x, y \in \N_{> 0}$ be arbitrary.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\paren {x \times y} \times n = x \times \paren {y \times n}$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn |... | The operation of [[Definition:Natural Number Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ is [[Definition:Associative Operation|associative]]:
:$\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$ | Using the [[Axiom:Axiomatization of 1-Based Natural Numbers|following axioms]]:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
Let $x, y \in \N_{> 0}$ be arbitrary.
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\paren {x \times y} \times n = x \times \paren {y \times... | Natural Number Multiplication is Associative/Proof 3 | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Associative | https://proofwiki.org/wiki/Natural_Number_Multiplication_is_Associative/Proof_3 | [
"Natural Number Multiplication",
"Examples of Associative Operations",
"Natural Number Multiplication is Associative",
"Associative Law of Multiplication"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Associative Operation"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Proposition",
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Axiom:Axiomatization... |
proofwiki-2969 | Natural Number Multiplication Distributes over Addition | The operation of multiplication is distributive over addition on the set of natural numbers $\N$:
:$\forall x, y, z \in \N:$
::$\paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$
::$z \times \paren {x + y} = \paren {z \times x} + \paren {z \times y}$ | {{begin-eqn}}
{{eqn | l = \paren {x + y} \times z
| r = +^z \paren {x + y}
| c = {{Defof|Natural Number Multiplication}}
}}
{{eqn | r = \paren {+^z x} + \paren {+^z y}
| c = Power of Product of Commuting Elements in Semigroup equals Product of Powers
}}
{{eqn | r = x \times z + y \times z
}}
{{end-eqn... | The operation of [[Definition:Natural Number Multiplication|multiplication]] is [[Definition:Distributive Operation|distributive]] over [[Definition:Natural Number Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$:
:$\forall x, y, z \in \N:$
::$\paren {x + y} \time... | {{begin-eqn}}
{{eqn | l = \paren {x + y} \times z
| r = +^z \paren {x + y}
| c = {{Defof|Natural Number Multiplication}}
}}
{{eqn | r = \paren {+^z x} + \paren {+^z y}
| c = [[Power of Product of Commuting Elements in Semigroup equals Product of Powers]]
}}
{{eqn | r = x \times z + y \times z
}}
{{end... | Natural Number Multiplication Distributes over Addition/Proof 1 | https://proofwiki.org/wiki/Natural_Number_Multiplication_Distributes_over_Addition | https://proofwiki.org/wiki/Natural_Number_Multiplication_Distributes_over_Addition/Proof_1 | [
"Natural Number Addition",
"Natural Number Multiplication",
"Natural Number Multiplication Distributes over Addition",
"Examples of Distributive Operations"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Distributive Operation",
"Definition:Addition/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers"
] | [
"Power of Product of Commuting Elements in Semigroup equals Product of Powers",
"Index Laws/Sum of Indices/Semigroup"
] |
proofwiki-2970 | Natural Number Multiplication Distributes over Addition | The operation of multiplication is distributive over addition on the set of natural numbers $\N$:
:$\forall x, y, z \in \N:$
::$\paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$
::$z \times \paren {x + y} = \paren {z \times x} + \paren {z \times y}$ | We are to show that:
:$\forall x, y, z \in \N: \paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$
From the definition of natural number multiplication, we have by definition that:
{{begin-eqn}}
{{eqn | q = \forall m, n \in \N
| l = m \times 0
| r = 0
}}
{{eqn | l = m \times n^+
| r =... | The operation of [[Definition:Natural Number Multiplication|multiplication]] is [[Definition:Distributive Operation|distributive]] over [[Definition:Natural Number Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$:
:$\forall x, y, z \in \N:$
::$\paren {x + y} \time... | We are to show that:
:$\forall x, y, z \in \N: \paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$
From the definition of [[Definition:Natural Number Multiplication|natural number multiplication]], we have by definition that:
{{begin-eqn}}
{{eqn | q = \forall m, n \in \N
| l = m \times 0
... | Natural Number Multiplication Distributes over Addition/Proof 2 | https://proofwiki.org/wiki/Natural_Number_Multiplication_Distributes_over_Addition | https://proofwiki.org/wiki/Natural_Number_Multiplication_Distributes_over_Addition/Proof_2 | [
"Natural Number Addition",
"Natural Number Multiplication",
"Natural Number Multiplication Distributes over Addition",
"Examples of Distributive Operations"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Distributive Operation",
"Definition:Addition/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Natural Number Multiplication Distributes over Addition/Proof 2",
"Natural Number Addition is Commutative/Proof 2",... |
proofwiki-2971 | Natural Number Multiplication Distributes over Addition | The operation of multiplication is distributive over addition on the set of natural numbers $\N$:
:$\forall x, y, z \in \N:$
::$\paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$
::$z \times \paren {x + y} = \paren {z \times x} + \paren {z \times y}$ | Using the following axioms:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
=== Left Distributive Law for Natural Numbers ===
First we show that:
:$n \times \paren {x + y} = \paren {n \times x} + \paren {n \times y}$
{{:Left Distributive Law for Natural Numbers}}{{qed|lemma}}
=== Right Distributive Law for Natural... | The operation of [[Definition:Natural Number Multiplication|multiplication]] is [[Definition:Distributive Operation|distributive]] over [[Definition:Natural Number Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$:
:$\forall x, y, z \in \N:$
::$\paren {x + y} \time... | Using the [[Axiom:Axiomatization of 1-Based Natural Numbers|following axioms]]:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
=== [[Left Distributive Law for Natural Numbers]] ===
First we show that:
:$n \times \paren {x + y} = \paren {n \times x} + \paren {n \times y}$
{{:Left Distributive Law for Natural N... | Natural Number Multiplication Distributes over Addition/Proof 3 | https://proofwiki.org/wiki/Natural_Number_Multiplication_Distributes_over_Addition | https://proofwiki.org/wiki/Natural_Number_Multiplication_Distributes_over_Addition/Proof_3 | [
"Natural Number Addition",
"Natural Number Multiplication",
"Natural Number Multiplication Distributes over Addition",
"Examples of Distributive Operations"
] | [
"Definition:Multiplication/Natural Numbers",
"Definition:Distributive Operation",
"Definition:Addition/Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Left Distributive Law for Natural Numbers",
"Right Distributive Law for Natural Numbers"
] |
proofwiki-2972 | Distributive Laws/Arithmetic | On all the number systems:
:natural numbers $\N$
:integers $\Z$
:rational numbers $\Q$
:real numbers $\R$
:complex numbers $\C$
the operation of multiplication is distributive over addition:
:$m \paren {n + p} = m n + m p$
:$\paren {m + n} p = m p + n p$ | This is demonstrated in these pages:
:Natural Number Multiplication Distributes over Addition
:Integer Multiplication Distributes over Addition
:Rational Multiplication Distributes over Addition
:Real Multiplication Distributes over Addition
:Complex Multiplication Distributes over Addition
{{qed}} | On all the number systems:
:[[Definition:Natural Numbers|natural numbers]] $\N$
:[[Definition:Integer|integers]] $\Z$
:[[Definition:Rational Number|rational numbers]] $\Q$
:[[Definition:Real Number|real numbers]] $\R$
:[[Definition:Complex Number|complex numbers]] $\C$
the operation of [[Definition:Multiplication|multi... | This is demonstrated in these pages:
:[[Natural Number Multiplication Distributes over Addition]]
:[[Integer Multiplication Distributes over Addition]]
:[[Rational Multiplication Distributes over Addition]]
:[[Real Multiplication Distributes over Addition]]
:[[Complex Multiplication Distributes over Addition]]
{{qed}} | Distributive Laws/Arithmetic | https://proofwiki.org/wiki/Distributive_Laws/Arithmetic | https://proofwiki.org/wiki/Distributive_Laws/Arithmetic | [
"Distributive Laws of Arithmetic",
"Distributive Laws",
"Multiplication",
"Addition",
"Examples of Distributive Operations",
"Arithmetic",
"Algebra"
] | [
"Definition:Natural Numbers",
"Definition:Integer",
"Definition:Rational Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Multiplication",
"Definition:Distributive Operation",
"Definition:Addition"
] | [
"Natural Number Multiplication Distributes over Addition",
"Integer Multiplication Distributes over Addition",
"Rational Multiplication Distributes over Addition",
"Real Multiplication Distributes over Addition",
"Complex Multiplication Distributes over Addition"
] |
proofwiki-2973 | Rational Multiplication Distributes over Addition | The operation of multiplication on the set of rational numbers $\Q$ is distributive over addition:
:$\forall x, y, z \in \Q: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$
:$\forall x, y, z \in \Q: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$ | Follows directly from the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.
So $\struct {\Q, +, \times}$ is a field, and therefore {{afortiori}} $\times$ is distributive over $+$ on $\Q$.
{{qed}}
Category:Rational Addition
Category:Rational Multiplica... | The operation of [[Definition:Rational Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ is [[Definition:Distributive Operation|distributive]] over [[Definition:Rational Addition|addition]]:
:$\forall x, y, z \in \Q: x \times \paren {y + z} = \paren {... | Follows directly from the [[Definition:Rational Number|definition of rational numbers]] as the [[Definition:Field of Quotients|field of quotients]] of the [[Definition:Integral Domain|integral domain]] $\struct {\Z, +, \times}$ of [[Definition:Integer|integers]].
So $\struct {\Q, +, \times}$ is a [[Definition:Field (A... | Rational Multiplication Distributes over Addition | https://proofwiki.org/wiki/Rational_Multiplication_Distributes_over_Addition | https://proofwiki.org/wiki/Rational_Multiplication_Distributes_over_Addition | [
"Rational Addition",
"Rational Multiplication",
"Examples of Distributive Operations"
] | [
"Definition:Multiplication/Rational Numbers",
"Definition:Set",
"Definition:Rational Number",
"Definition:Distributive Operation",
"Definition:Addition/Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Field of Quotients",
"Definition:Integral Domain",
"Definition:Integer",
"Definition:Field (Abstract Algebra)",
"Definition:Distributive Operation",
"Category:Rational Addition",
"Category:Rational Multiplication",
"Category:Examples of Distributive Operatio... |
proofwiki-2974 | Associative Law of Multiplication | :$\forall x, y, z \in \mathbb F: x \times \paren {y \times z} = \paren {x \times y} \times z$
That is, the operation of multiplication on the standard number sets is associative. | Let a first magnitude $A$ be the same multiple of a second $B$ that a third $C$ is of a fourth $D$.
Let equimultiples $EF, GH$ be taken of $A, C$.
We need to show that $EF$ is the same multiple of $B$ that $GH$ is of $D$.
We have that $EF$ is the same multiple of $A$ that $GH$ is of $C$.
Therefore as many magnitudes as... | :$\forall x, y, z \in \mathbb F: x \times \paren {y \times z} = \paren {x \times y} \times z$
That is, the operation of [[Definition:Multiplication|multiplication]] on the [[Definition:Number|standard number sets]] is [[Definition:Associative Operation|associative]]. | Let a first [[Definition:Strictly Positive Real Number|magnitude]] $A$ be the same [[Definition:Multiple|multiple]] of a second $B$ that a third $C$ is of a fourth $D$.
Let [[Definition:Equimultiples|equimultiples]] $EF, GH$ be taken of $A, C$.
We need to show that $EF$ is the same multiple of $B$ that $GH$ is of $D$... | Associative Law of Multiplication/Euclid's Proof | https://proofwiki.org/wiki/Associative_Law_of_Multiplication | https://proofwiki.org/wiki/Associative_Law_of_Multiplication/Euclid's_Proof | [
"Associative Law of Multiplication",
"Numbers",
"Examples of Associative Operations",
"Multiplication"
] | [
"Definition:Multiplication",
"Definition:Number",
"Definition:Associative Operation"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Multiple",
"Definition:Equimultiples",
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Positive/R... |
proofwiki-2975 | Subtraction on Numbers is Not Associative | The operation of subtraction on the numbers is not associative.
That is, in general:
:$a - \paren {b - c} \ne \paren {a - b} - c$ | By definition of subtraction:
{{begin-eqn}}
{{eqn | l = a - \paren {b - c}
| r = a + \paren {-\paren {b + \paren {-c} } }
| c =
}}
{{eqn | r = a + \paren {-b} + c
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {a - b} - c
| r = \paren {a + \paren {-b} } + \paren {-c}
| c =
}}
{{e... | The operation of [[Definition:Subtraction|subtraction]] on the [[Definition:Number|numbers]] is not [[Definition:Associative Operation|associative]].
That is, in general:
:$a - \paren {b - c} \ne \paren {a - b} - c$ | By definition of [[Definition:Subtraction|subtraction]]:
{{begin-eqn}}
{{eqn | l = a - \paren {b - c}
| r = a + \paren {-\paren {b + \paren {-c} } }
| c =
}}
{{eqn | r = a + \paren {-b} + c
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {a - b} - c
| r = \paren {a + \paren {-b} } + \... | Subtraction on Numbers is Not Associative | https://proofwiki.org/wiki/Subtraction_on_Numbers_is_Not_Associative | https://proofwiki.org/wiki/Subtraction_on_Numbers_is_Not_Associative | [
"Subtraction on Numbers is Not Associative",
"Numbers",
"Subtraction",
"Examples of Associative Operations"
] | [
"Definition:Subtraction",
"Definition:Number",
"Definition:Associative Operation"
] | [
"Definition:Subtraction"
] |
proofwiki-2976 | Identity Element of Multiplication on Numbers | On all the number systems:
* natural numbers $\N$
* integers $\Z$
* rational numbers $\Q$
* real numbers $\R$
* complex numbers $\C$
the identity element of multiplication is one ($1$). | This is demonstrated in these pages:
* Identity Element of Natural Number Multiplication is One
* Integer Multiplication Identity is One
* Rational Multiplication Identity is One
* Real Multiplication Identity is One
* Complex Multiplication Identity is One
{{qed}} | On all the number systems:
* [[Definition:Natural Numbers|natural numbers]] $\N$
* [[Definition:Integer|integers]] $\Z$
* [[Definition:Rational Number|rational numbers]] $\Q$
* [[Definition:Real Number|real numbers]] $\R$
* [[Definition:Complex Number|complex numbers]] $\C$
the [[Definition:Identity Element|identity el... | This is demonstrated in these pages:
* [[Identity Element of Natural Number Multiplication is One]]
* [[Integer Multiplication Identity is One]]
* [[Rational Multiplication Identity is One]]
* [[Real Multiplication Identity is One]]
* [[Complex Multiplication Identity is One]]
{{qed}} | Identity Element of Multiplication on Numbers | https://proofwiki.org/wiki/Identity_Element_of_Multiplication_on_Numbers | https://proofwiki.org/wiki/Identity_Element_of_Multiplication_on_Numbers | [
"Examples of Identity Elements",
"Multiplication",
"Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Integer",
"Definition:Rational Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Multiplication",
"Definition:One"
] | [
"Identity Element of Natural Number Multiplication is One",
"Integer Multiplication Identity is One",
"Rational Multiplication Identity is One",
"Real Multiplication Identity is One",
"Complex Multiplication Identity is One"
] |
proofwiki-2977 | Identity Element of Addition on Numbers | On all the number systems:
* natural numbers $\N$
* integers $\Z$
* rational numbers $\Q$
* real numbers $\R$
* complex numbers $\C$
the identity element of addition is zero ($0$). | This is demonstrated in these pages:
* Identity Element of Natural Number Addition is Zero
* Integer Addition Identity is Zero
* Rational Addition Identity is Zero
* Real Addition Identity is Zero
* Complex Addition Identity is Zero
{{qed}} | On all the number systems:
* [[Definition:Natural Numbers|natural numbers]] $\N$
* [[Definition:Integer|integers]] $\Z$
* [[Definition:Rational Number|rational numbers]] $\Q$
* [[Definition:Real Number|real numbers]] $\R$
* [[Definition:Complex Number|complex numbers]] $\C$
the [[Definition:Identity Element|identity el... | This is demonstrated in these pages:
* [[Identity Element of Natural Number Addition is Zero]]
* [[Integer Addition Identity is Zero]]
* [[Rational Addition Identity is Zero]]
* [[Real Addition Identity is Zero]]
* [[Complex Addition Identity is Zero]]
{{qed}} | Identity Element of Addition on Numbers | https://proofwiki.org/wiki/Identity_Element_of_Addition_on_Numbers | https://proofwiki.org/wiki/Identity_Element_of_Addition_on_Numbers | [
"Examples of Identity Elements",
"Addition",
"Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Integer",
"Definition:Rational Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Addition",
"Definition:Zero (Number)"
] | [
"Identity Element of Natural Number Addition is Zero",
"Integer Addition Identity is Zero",
"Rational Addition Identity is Zero",
"Real Addition Identity is Zero",
"Complex Addition Identity is Zero"
] |
proofwiki-2978 | Zero Element is Unique | Let $\struct {S, \circ}$ be an algebraic structure that has a zero element $z \in S$.
Then $z$ is unique. | Suppose $z_1$ and $z_2$ are both zeroes of $\struct {S, \circ}$.
Then by the definition of zero element:
:$z_2 \circ z_1 = z_1$ by dint of $z_1$ being a zero
:$z_2 \circ z_1 = z_2$ by dint of $z_2$ being a zero.
So $z_1 = z_2 \circ z_1 = z_2$.
So $z_1 = z_2$ and there is only one zero after all.
{{qed}} | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]] that has a [[Definition:Zero Element|zero element]] $z \in S$.
Then $z$ is [[Definition:Unique|unique]]. | Suppose $z_1$ and $z_2$ are both zeroes of $\struct {S, \circ}$.
Then by the definition of [[Definition:Zero Element|zero element]]:
:$z_2 \circ z_1 = z_1$ by dint of $z_1$ being a [[Definition:Zero Element|zero]]
:$z_2 \circ z_1 = z_2$ by dint of $z_2$ being a [[Definition:Zero Element|zero]].
So $z_1 = z_2 \circ z_... | Zero Element is Unique | https://proofwiki.org/wiki/Zero_Element_is_Unique | https://proofwiki.org/wiki/Zero_Element_is_Unique | [
"Zero Elements"
] | [
"Definition:Algebraic Structure",
"Definition:Zero Element",
"Definition:Unique"
] | [
"Definition:Zero Element",
"Definition:Zero Element",
"Definition:Zero Element",
"Definition:Zero Element"
] |
proofwiki-2979 | Group with Zero Element is Trivial | Let $\struct {G, \circ}$ be a group.
Let $\struct {G, \circ}$ have a zero element.
Then $\struct {G, \circ}$ is the trivial group. | Let $e \in G$ be the identity element of $G$.
Let $z \in G$ be a zero element.
Let $x \in G$ be any arbitrary element of $\struct {G, \circ}$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = x \circ e
| c = {{Group-axiom|2}}
}}
{{eqn | r = x \circ \paren {z \circ z^{-1} }
| c = {{Group-axiom|3}}
}}
{{eqn | r ... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $\struct {G, \circ}$ have a [[Definition:Zero Element|zero element]].
Then $\struct {G, \circ}$ is the [[Definition:Trivial Group|trivial group]]. | Let $e \in G$ be the [[Definition:Identity Element|identity element]] of $G$.
Let $z \in G$ be a [[Definition:Zero Element|zero element]].
Let $x \in G$ be any arbitrary [[Definition:Element|element]] of $\struct {G, \circ}$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = x \circ e
| c = {{Group-axiom|2}}
}}
{... | Group with Zero Element is Trivial | https://proofwiki.org/wiki/Group_with_Zero_Element_is_Trivial | https://proofwiki.org/wiki/Group_with_Zero_Element_is_Trivial | [
"Group Theory",
"Trivial Group",
"Zero Elements"
] | [
"Definition:Group",
"Definition:Zero Element",
"Definition:Trivial Group"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Zero Element",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Trivial Group"
] |
proofwiki-2980 | Count of Binary Operations on Set | Let $S$ be a set whose cardinality is $n$.
The number $N$ of different binary operations that can be applied to $S$ is given by:
:$N = n^{\paren {n^2} }$ | A binary operation on $S$ is by definition a mapping from the cartesian product $S \times S$ to the set $S$.
Thus we are looking to evaluate:
:$N = \card {\set {f: S \times S \to S} }$
The domain of $f$ has $n^2$ elements, from Cardinality of Cartesian Product of Finite Sets of Finite Sets.
The result follows from Car... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is $n$.
The number $N$ of different [[Definition:Binary Operation|binary operations]] that can be applied to $S$ is given by:
:$N = n^{\paren {n^2} }$ | A [[Definition:Binary Operation|binary operation]] on $S$ is by definition a [[Definition:Mapping|mapping]] from the [[Definition:Cartesian Product|cartesian product]] $S \times S$ to the set $S$.
Thus we are looking to evaluate:
:$N = \card {\set {f: S \times S \to S} }$
The [[Definition:Domain of Mapping|domain]] ... | Count of Binary Operations on Set | https://proofwiki.org/wiki/Count_of_Binary_Operations_on_Set | https://proofwiki.org/wiki/Count_of_Binary_Operations_on_Set | [
"Combinatorics",
"Abstract Algebra",
"Count of Binary Operations on Set"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Operation/Binary Operation"
] | [
"Definition:Operation/Binary Operation",
"Definition:Mapping",
"Definition:Cartesian Product",
"Definition:Domain (Set Theory)/Mapping",
"Cardinality of Cartesian Product of Finite Sets of Finite Sets",
"Cardinality of Set of All Mappings"
] |
proofwiki-2981 | Count of Commutative Binary Operations on Set | Let $S$ be a set whose cardinality is $n$.
The number $N$ of possible different commutative binary operations that can be applied to $S$ is given by:
:$N = n^{\frac {n \paren {n + 1} } 2}$ | Let $\struct {S, \circ}$ be a magma.
From Cardinality of Cartesian Product of Finite Sets, there are $n^2$ elements in $S \times S$.
The binary operations $\circ$ is commutative {{iff}}:
:$\forall x, y \in S: x \circ y = y \circ x$
Thus for every pair of elements $\tuple {x, y} \in S \times S$, it is required that $\tu... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is $n$.
The number $N$ of possible different [[Definition:Commutative Operation|commutative]] [[Definition:Binary Operation|binary operations]] that can be applied to $S$ is given by:
:$N = n^{\frac {n \paren {n + 1} } 2}$ | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
From [[Cardinality of Cartesian Product of Finite Sets]], there are $n^2$ elements in $S \times S$.
The [[Definition:Binary Operation|binary operations]] $\circ$ is [[Definition:Commutative Operation|commutative]] {{iff}}:
:$\forall x, y \in S: x \circ y = y \... | Count of Commutative Binary Operations on Set | https://proofwiki.org/wiki/Count_of_Commutative_Binary_Operations_on_Set | https://proofwiki.org/wiki/Count_of_Commutative_Binary_Operations_on_Set | [
"Combinatorics",
"Abstract Algebra",
"Commutativity",
"Count of Commutative Binary Operations on Set"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Commutative/Operation",
"Definition:Operation/Binary Operation"
] | [
"Definition:Magma",
"Cardinality of Cartesian Product of Finite Sets",
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation",
"Definition:Element",
"Definition:Doubleton",
"Cardinality of Set of Subsets",
"Definition:Doubleton",
"Cardinality of Set of All Mappings"
] |
proofwiki-2982 | Count of Binary Operations with Fixed Identity | Let $S$ be a set whose cardinality is $n$.
Let $x \in S$.
The number $N$ of possible different binary operations such that $x$ is an identity element that can be applied to $S$ is given by:
:$N = n^{\paren {\paren {n - 1}^2} }$ | Let $S$ be a set such that $\card S = n$.
Let $x \in S$ be an identity element.
From Count of Binary Operations on Set, there are $n^{\paren {n^2} }$ binary operations in total.
We also know that $a \in S \implies a \circ x = a = x \circ a$, so all operations on $x$ are already specified.
It remains to count all possib... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is $n$.
Let $x \in S$.
The number $N$ of possible different [[Definition:Binary Operation|binary operations]] such that $x$ is an [[Definition:Identity Element|identity element]] that can be applied to $S$ is given by:
:$N = n^{\paren {... | Let $S$ be a [[Definition:Set|set]] such that $\card S = n$.
Let $x \in S$ be an [[Definition:Identity Element|identity element]].
From [[Count of Binary Operations on Set]], there are $n^{\paren {n^2} }$ [[Definition:Binary Operation|binary operations]] in total.
We also know that $a \in S \implies a \circ x = a = ... | Count of Binary Operations with Fixed Identity | https://proofwiki.org/wiki/Count_of_Binary_Operations_with_Fixed_Identity | https://proofwiki.org/wiki/Count_of_Binary_Operations_with_Fixed_Identity | [
"Combinatorics",
"Abstract Algebra",
"Count of Binary Operations with Fixed Identity"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Set",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Count of Binary Operations on Set",
"Definition:Operation/Binary Operation",
"Count of Binary Operations on Set",
"Definition:Algebraic Structure/One Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-2983 | Count of Commutative Binary Operations with Fixed Identity | Let $S$ be a set whose cardinality is $n$.
Let $x \in S$.
The number $N$ of possible different commutative binary operations such that $x$ is an identity element that can be applied to $S$ is given by:
:$N = n^{\frac {n \paren {n - 1} } 2}$ | This follows by the arguments of Count of Binary Operations with Fixed Identity and Count of Commutative Binary Operations on Set.
From Count of Binary Operations on Set, there are $n^{\paren {n^2} }$ binary operations in total.
We also know that:
:$a \in S \implies a \circ x = a = x \circ a$
so all binary operations o... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is $n$.
Let $x \in S$.
The number $N$ of possible different [[Definition:Commutative Operation|commutative]] [[Definition:Binary Operation|binary operations]] such that $x$ is an [[Definition:Identity Element|identity element]] that can ... | This follows by the arguments of [[Count of Binary Operations with Fixed Identity]] and [[Count of Commutative Binary Operations on Set]].
From [[Count of Binary Operations on Set]], there are $n^{\paren {n^2} }$ [[Definition:Binary Operation|binary operations]] in total.
We also know that:
:$a \in S \implies a \cir... | Count of Commutative Binary Operations with Fixed Identity | https://proofwiki.org/wiki/Count_of_Commutative_Binary_Operations_with_Fixed_Identity | https://proofwiki.org/wiki/Count_of_Commutative_Binary_Operations_with_Fixed_Identity | [
"Combinatorics",
"Abstract Algebra",
"Commutativity",
"Count of Commutative Binary Operations with Fixed Identity"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Commutative/Operation",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Count of Binary Operations with Fixed Identity",
"Count of Commutative Binary Operations on Set",
"Count of Binary Operations on Set",
"Definition:Operation/Binary Operation",
"Definition:Operation/Binary Operation",
"Definition:Mapping",
"Definition:Doubleton",
"Cardinality of Set of Subsets",
"De... |
proofwiki-2984 | Count of Binary Operations with Identity | Let $S$ be a set whose cardinality is $n$.
The number $N$ of possible different binary operations which have an identity element that can be applied to $S$ is given by:
:$N = n^{\paren {n - 1}^2 + 1}$ | From Count of Binary Operations with Fixed Identity, there are $n^{\paren {n - 1}^2}$ such binary operations for each individual element of $S$.
As Identity is Unique, if $x$ is the identity, no other element can also be an identity.
As there are $n$ different ways of choosing such an identity, there are $n \times n^{\... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is $n$.
The number $N$ of possible different [[Definition:Binary Operation|binary operations]] which have an [[Definition:Identity Element|identity element]] that can be applied to $S$ is given by:
:$N = n^{\paren {n - 1}^2 + 1}$ | From [[Count of Binary Operations with Fixed Identity]], there are $n^{\paren {n - 1}^2}$ such [[Definition:Binary Operation|binary operations]] for each individual [[Definition:Element|element]] of $S$.
As [[Identity is Unique]], if $x$ is the [[Definition:Identity Element|identity]], no other [[Definition:Element|el... | Count of Binary Operations with Identity | https://proofwiki.org/wiki/Count_of_Binary_Operations_with_Identity | https://proofwiki.org/wiki/Count_of_Binary_Operations_with_Identity | [
"Combinatorics",
"Abstract Algebra",
"Count of Binary Operations with Identity"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Count of Binary Operations with Fixed Identity",
"Definition:Operation/Binary Operation",
"Definition:Element",
"Identity is Unique",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (A... |
proofwiki-2985 | Count of Commutative Binary Operations with Identity | Let $S$ be a set whose cardinality is $n$.
The number $N$ of possible different commutative binary operations that can be applied to $S$ which have an identity element is given by:
:$N = n^{\frac {n \paren {n - 1} } 2 + 1}$ | From Count of Commutative Binary Operations with Fixed Identity, there are $n^{\frac {n \paren {n - 1} } 2}$ such binary operations for each individual element of $S$.
As Identity is Unique, if $x$ is the identity, no other element can also be an identity.
As there are $n$ different ways of choosing such an identity, t... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is $n$.
The number $N$ of possible different [[Definition:Commutative Operation|commutative]] [[Definition:Binary Operation|binary operations]] that can be applied to $S$ which have an [[Definition:Identity Element|identity element]] is g... | From [[Count of Commutative Binary Operations with Fixed Identity]], there are $n^{\frac {n \paren {n - 1} } 2}$ such [[Definition:Binary Operation|binary operations]] for each individual [[Definition:Element|element]] of $S$.
As [[Identity is Unique]], if $x$ is the [[Definition:Identity Element|identity]], no other ... | Count of Commutative Binary Operations with Identity | https://proofwiki.org/wiki/Count_of_Commutative_Binary_Operations_with_Identity | https://proofwiki.org/wiki/Count_of_Commutative_Binary_Operations_with_Identity | [
"Combinatorics",
"Abstract Algebra",
"Commutativity",
"Count of Commutative Binary Operations with Identity"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Commutative/Operation",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Count of Commutative Binary Operations with Fixed Identity",
"Definition:Operation/Binary Operation",
"Definition:Element",
"Identity is Unique",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition... |
proofwiki-2986 | Count of Binary Operations Without Identity | Let $S$ be a set whose cardinality is $n$.
The number $N$ of possible different binary operations which do not have an identity element that can be applied to $S$ is given by:
:$N = n^{\paren {\paren {n - 1}^2 + 1} } \paren {n^{2 \paren {n - 1} } - 1}$ | From Count of Binary Operations on Set, the total number of operations is $n^{\paren {n^2} }$.
From Count of Binary Operations with Identity, the total number of operations with an identity is $n^{\paren {n - 1}^2 + 1}$.
So the total number of operations without an identity is:
:$n^{\paren {n^2} } - n^{\paren {n - 1}^2... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is $n$.
The number $N$ of possible different [[Definition:Binary Operation|binary operations]] which do not have an [[Definition:Identity Element|identity element]] that can be applied to $S$ is given by:
:$N = n^{\paren {\paren {n - 1}^... | From [[Count of Binary Operations on Set]], the total number of operations is $n^{\paren {n^2} }$.
From [[Count of Binary Operations with Identity]], the total number of operations with an identity is $n^{\paren {n - 1}^2 + 1}$.
So the total number of operations without an identity is:
:$n^{\paren {n^2} } - n^{\pare... | Count of Binary Operations Without Identity | https://proofwiki.org/wiki/Count_of_Binary_Operations_Without_Identity | https://proofwiki.org/wiki/Count_of_Binary_Operations_Without_Identity | [
"Combinatorics",
"Abstract Algebra",
"Count of Binary Operations Without Identity"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Count of Binary Operations on Set",
"Count of Binary Operations with Identity"
] |
proofwiki-2987 | Power of Element in Subgroup | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $\struct {H, \circ}$ be a subgroup of $\struct {G, \circ}$.
Let $x \in H$.
Then:
:$\forall n \in \Z: x^n \in H$ | Proof by induction:
For all $n \in \N^*$, let $\map P n$ be the compound proposition:
:$x^n \in H \text{ and } x^{-n} \in H$.
$\map P 0$ is true, as this just says $x^0 \in H$.
By Powers of Group Elements, $x^0 = e$.
This follows by Identity of Subgroup. | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\struct {H, \circ}$ be a [[Definition:Subgroup|subgroup]] of $\struct {G, \circ}$.
Let $x \in H$.
Then:
:$\forall n \in \Z: x^n \in H$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N^*$, let $\map P n$ be the compound [[Definition:Proposition|proposition]]:
:$x^n \in H \text{ and } x^{-n} \in H$.
$\map P 0$ is true, as this just says $x^0 \in H$.
By [[Powers of Group Elements]], $x^0 = e$.
This follows by [[Identity ... | Power of Element in Subgroup | https://proofwiki.org/wiki/Power_of_Element_in_Subgroup | https://proofwiki.org/wiki/Power_of_Element_in_Subgroup | [
"Subgroups"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Powers of Group Elements",
"Identity of Subgroup",
"Powers of Group Elements",
"Principle of Mathematical Induction"
] |
proofwiki-2988 | Element to Power of Remainder | :$\forall n \in \Z: n = q k + r: 0 \le r < k \iff a^n = a^r$ | Let $n \in \Z$.
We have:
{{begin-eqn}}
{{eqn | l = n
| r = q k + r
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = n - r
| r = q k
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = k
| o = \divides
| r = \paren {n - r}
| c =
}}
{{end-eqn}}
The result follows from Equal Power... | :$\forall n \in \Z: n = q k + r: 0 \le r < k \iff a^n = a^r$ | Let $n \in \Z$.
We have:
{{begin-eqn}}
{{eqn | l = n
| r = q k + r
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = n - r
| r = q k
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = k
| o = \divides
| r = \paren {n - r}
| c =
}}
{{end-eqn}}
The result follows from [[Equal ... | Element to Power of Remainder | https://proofwiki.org/wiki/Element_to_Power_of_Remainder | https://proofwiki.org/wiki/Element_to_Power_of_Remainder | [
"Group Theory"
] | [] | [
"Equal Powers of Finite Order Element"
] |
proofwiki-2989 | Element to Power of Multiple of Order is Identity | :$\forall n \in \Z: k \divides n \iff a^n = e$ | Let $k \in \N$ be the smallest such that $a^k = e$ as per the hypothesis. | :$\forall n \in \Z: k \divides n \iff a^n = e$ | Let $k \in \N$ be the smallest such that $a^k = e$ as per the hypothesis. | Element to Power of Multiple of Order is Identity | https://proofwiki.org/wiki/Element_to_Power_of_Multiple_of_Order_is_Identity | https://proofwiki.org/wiki/Element_to_Power_of_Multiple_of_Order_is_Identity | [
"Identity Elements",
"Order of Group Elements"
] | [] | [] |
proofwiki-2990 | List of Elements in Finite Cyclic Group | :$\set {a^0, a^1, a^2, \ldots, a^{k - 1} }$ is a complete repetition-free list of the elements of $\gen a$ | By Element to Power of Remainder, every power of $a$ is equal to one appearing in the list $a^0, a^1, a^2, \ldots, a^{k - 1}$.
This list has to be repetition free, otherwise it would contain $a^m = a^n$ with $0 \le m < n < k$ which contradicts the hypothesis.
{{qed}} | :$\set {a^0, a^1, a^2, \ldots, a^{k - 1} }$ is a complete repetition-free list of the elements of $\gen a$ | By [[Element to Power of Remainder]], every power of $a$ is equal to one appearing in the list $a^0, a^1, a^2, \ldots, a^{k - 1}$.
This list has to be repetition free, otherwise it would contain $a^m = a^n$ with $0 \le m < n < k$ which contradicts the hypothesis.
{{qed}} | List of Elements in Finite Cyclic Group | https://proofwiki.org/wiki/List_of_Elements_in_Finite_Cyclic_Group | https://proofwiki.org/wiki/List_of_Elements_in_Finite_Cyclic_Group | [
"Finite Cyclic Groups"
] | [] | [
"Element to Power of Remainder"
] |
proofwiki-2991 | Equivalence of Definitions of Order of Group Element | {{TFAE|def = Order of Group Element}}
Let $G$ be a group whose identity is $e$.
Let $x \in G$. | Let $k$ be the order of $x$ in $G$ according to Definition 1.
Let $l$ be the order of $x$ in $G$ according to Definition 3. | {{TFAE|def = Order of Group Element}}
Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $x \in G$. | Let $k$ be the order of $x$ in $G$ according to [[Definition:Order of Group Element/Definition 1|Definition 1]].
Let $l$ be the order of $x$ in $G$ according to [[Definition:Order of Group Element/Definition 3|Definition 3]]. | Equivalence of Definitions of Order of Group Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Order_of_Group_Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Order_of_Group_Element | [
"Subgroups",
"Order of Group Elements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Order of Group Element/Definition 1",
"Definition:Order of Group Element/Definition 3",
"Definition:Order of Group Element/Definition 3"
] |
proofwiki-2992 | Rising Sum of Binomial Coefficients | :$\ds \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1} = \binom {n + m + 1} m$ | Proof by induction:
Let $n \in \Z$.
For all $m \in \N$, let $\map P m$ be the proposition:
:$\ds \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1}$
$\map P 0$ is true, as this just says:
:$\dbinom n n = \dbinom {n + 1} {n + 1}$
But $\dbinom n n = \dbinom {n + 1} {n + 1} = 1$ from the {{Defof|Binomial... | :$\ds \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1} = \binom {n + m + 1} m$ | Proof by [[Principle of Mathematical Induction|induction]]:
Let $n \in \Z$.
For all $m \in \N$, let $\map P m$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1}$
$\map P 0$ is true, as this just says:
:$\dbinom n n = \dbinom {n + 1} {n + 1}$
B... | Rising Sum of Binomial Coefficients/Proof by Induction | https://proofwiki.org/wiki/Rising_Sum_of_Binomial_Coefficients | https://proofwiki.org/wiki/Rising_Sum_of_Binomial_Coefficients/Proof_by_Induction | [
"Rising Sum of Binomial Coefficients",
"Binomial Coefficients"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Rising Sum of Binomial Coefficients",
"Pascal's Rule",
"Principle of Mathematical Induction",
"Symmetry Rule for Binomial Coefficien... |
proofwiki-2993 | Sum of Even Index Binomial Coefficients | :$\ds \sum_{i \mathop \ge 0} \binom n {2 i} = 2^{n - 1}$ | From Sum of Binomial Coefficients over Lower Index we have:
:$\ds \sum_{i \mathop \in \Z} \binom n i = 2^n$
That is:
:$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 + \cdots + \dbinom n n = 2^n$
as $\dbinom n i = 0$ for $i < 0$ and $i > n$.
This can be written more conveniently as:
:$\dbinom n 0 + \dbinom n 1 +... | :$\ds \sum_{i \mathop \ge 0} \binom n {2 i} = 2^{n - 1}$ | From [[Sum of Binomial Coefficients over Lower Index]] we have:
:$\ds \sum_{i \mathop \in \Z} \binom n i = 2^n$
That is:
:$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 + \cdots + \dbinom n n = 2^n$
as $\dbinom n i = 0$ for $i < 0$ and $i > n$.
This can be written more conveniently as:
:$\dbinom n 0 + \dbino... | Sum of Even Index Binomial Coefficients/Proof 1 | https://proofwiki.org/wiki/Sum_of_Even_Index_Binomial_Coefficients | https://proofwiki.org/wiki/Sum_of_Even_Index_Binomial_Coefficients/Proof_1 | [
"Sum of Even Index Binomial Coefficients",
"Binomial Coefficients"
] | [] | [
"Sum of Binomial Coefficients over Lower Index",
"Alternating Sum and Difference of Binomial Coefficients for Given n"
] |
proofwiki-2994 | Sum of Even Index Binomial Coefficients | :$\ds \sum_{i \mathop \ge 0} \binom n {2 i} = 2^{n - 1}$ | Let ${\N_n}^*$ be the initial segment of natural numbers, one-based.
Let:
:$E_n = \set {X : \paren {X \subset {\N_n}^*} \land \paren {2 \divides \size X} }$
:$O_n = \set {X : \paren {X \subset {\N_n}^*} \land \paren {2 \nmid \size X} }$
That is:
:$E_n$ is the set of all subsets of ${\N_n}^*$ with an even number of elem... | :$\ds \sum_{i \mathop \ge 0} \binom n {2 i} = 2^{n - 1}$ | Let ${\N_n}^*$ be the [[Definition:Initial Segment of Natural Numbers|initial segment of natural numbers]], one-based.
Let:
:$E_n = \set {X : \paren {X \subset {\N_n}^*} \land \paren {2 \divides \size X} }$
:$O_n = \set {X : \paren {X \subset {\N_n}^*} \land \paren {2 \nmid \size X} }$
That is:
:$E_n$ is the [[Defin... | Sum of Even Index Binomial Coefficients/Proof 2 | https://proofwiki.org/wiki/Sum_of_Even_Index_Binomial_Coefficients | https://proofwiki.org/wiki/Sum_of_Even_Index_Binomial_Coefficients/Proof_2 | [
"Sum of Even Index Binomial Coefficients",
"Binomial Coefficients"
] | [] | [
"Definition:Initial Segment of Natural Numbers",
"Definition:Set",
"Definition:Subset",
"Definition:Even Integer",
"Definition:Element",
"Definition:Set",
"Definition:Subset",
"Definition:Odd Integer",
"Definition:Element",
"Cardinality of Set of Subsets",
"Principle of Mathematical Induction",
... |
proofwiki-2995 | Sum of Odd Index Binomial Coefficients | :$\ds \sum_{i \mathop \ge 0} \binom n {2 i + 1} = 2^{n - 1}$ | From Sum of Binomial Coefficients over Lower Index we have:
:$\ds \sum_{i \mathop \in \Z} \binom n i = 2^n$
That is:
:$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 + \cdots + \dbinom n n = 2^n$
as $\dbinom n i = 0$ for $i < 0$ and $i > n$.
This can be written more conveniently as:
:$(1): \quad \dbinom n 0 + \d... | :$\ds \sum_{i \mathop \ge 0} \binom n {2 i + 1} = 2^{n - 1}$ | From [[Sum of Binomial Coefficients over Lower Index]] we have:
:$\ds \sum_{i \mathop \in \Z} \binom n i = 2^n$
That is:
:$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 + \cdots + \dbinom n n = 2^n$
as $\dbinom n i = 0$ for $i < 0$ and $i > n$.
This can be written more conveniently as:
:$(1): \quad \dbinom n... | Sum of Odd Index Binomial Coefficients | https://proofwiki.org/wiki/Sum_of_Odd_Index_Binomial_Coefficients | https://proofwiki.org/wiki/Sum_of_Odd_Index_Binomial_Coefficients | [
"Binomial Coefficients"
] | [] | [
"Sum of Binomial Coefficients over Lower Index",
"Alternating Sum and Difference of Binomial Coefficients for Given n",
"Definition:Even Integer"
] |
proofwiki-2996 | Increasing Sum of Binomial Coefficients | :$\ds \sum_{j \mathop = 0}^n j \binom n j = n 2^{n - 1}$ | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^n j \binom n j
| r = \sum_{j \mathop = 1}^n j \binom n j
| c = as $0 \dbinom n 0 = 0$
}}
{{eqn | r = \sum_{j \mathop = 1}^n n \binom {n - 1} {j - 1}
| c = Factors of Binomial Coefficient
}}
{{eqn | r = n \sum_{j \mathop = 0}^{n - 1} \binom {n - 1} j
... | :$\ds \sum_{j \mathop = 0}^n j \binom n j = n 2^{n - 1}$ | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^n j \binom n j
| r = \sum_{j \mathop = 1}^n j \binom n j
| c = as $0 \dbinom n 0 = 0$
}}
{{eqn | r = \sum_{j \mathop = 1}^n n \binom {n - 1} {j - 1}
| c = [[Factors of Binomial Coefficient]]
}}
{{eqn | r = n \sum_{j \mathop = 0}^{n - 1} \binom {n - 1} j
... | Increasing Sum of Binomial Coefficients/Proof 1 | https://proofwiki.org/wiki/Increasing_Sum_of_Binomial_Coefficients | https://proofwiki.org/wiki/Increasing_Sum_of_Binomial_Coefficients/Proof_1 | [
"Increasing Sum of Binomial Coefficients",
"Binomial Coefficients"
] | [] | [
"Factors of Binomial Coefficient",
"Translation of Index Variable of Summation",
"Sum of Binomial Coefficients over Lower Index"
] |
proofwiki-2997 | Increasing Sum of Binomial Coefficients | :$\ds \sum_{j \mathop = 0}^n j \binom n j = n 2^{n - 1}$ | From the Binomial Theorem:
:$(1): \quad \paren {1 + x}^n = \ds \sum_{j \mathop = 0}^n \dbinom n j x^n$
Differentiating $(1)$ {{WRT|Differentiation}} $x$:
{{begin-eqn}}
{{eqn | l = n \paren {1 + x}^{n - 1}
| r = \sum_{j \mathop = 1}^n j \dbinom n j x^{j - 1}
| c = Power Rule for Derivatives
}}
{{eqn | ll= \l... | :$\ds \sum_{j \mathop = 0}^n j \binom n j = n 2^{n - 1}$ | From the [[Binomial Theorem]]:
:$(1): \quad \paren {1 + x}^n = \ds \sum_{j \mathop = 0}^n \dbinom n j x^n$
[[Definition:Differentiation|Differentiating]] $(1)$ {{WRT|Differentiation}} $x$:
{{begin-eqn}}
{{eqn | l = n \paren {1 + x}^{n - 1}
| r = \sum_{j \mathop = 1}^n j \dbinom n j x^{j - 1}
| c = [[Power... | Increasing Sum of Binomial Coefficients/Proof 2 | https://proofwiki.org/wiki/Increasing_Sum_of_Binomial_Coefficients | https://proofwiki.org/wiki/Increasing_Sum_of_Binomial_Coefficients/Proof_2 | [
"Increasing Sum of Binomial Coefficients",
"Binomial Coefficients"
] | [] | [
"Binomial Theorem",
"Definition:Differentiation",
"Power Rule for Derivatives"
] |
proofwiki-2998 | Increasing Sum of Binomial Coefficients | :$\ds \sum_{j \mathop = 0}^n j \binom n j = n 2^{n - 1}$ | {{begin-eqn}}
{{eqn | n = 1
| l = \sum_{j \mathop = 0}^n j \binom n j
| r = 0 \binom n 0 + 1 \binom n 1 + 2 \binom n 2 + 3 \binom n 3 + \cdots + \paren {n - 2} \binom n {n - 2} + \paren {n - 1} \binom n {n - 1} + n \binom n n
| c =
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sum_{j \mathop = 0... | :$\ds \sum_{j \mathop = 0}^n j \binom n j = n 2^{n - 1}$ | {{begin-eqn}}
{{eqn | n = 1
| l = \sum_{j \mathop = 0}^n j \binom n j
| r = 0 \binom n 0 + 1 \binom n 1 + 2 \binom n 2 + 3 \binom n 3 + \cdots + \paren {n - 2} \binom n {n - 2} + \paren {n - 1} \binom n {n - 1} + n \binom n n
| c =
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sum_{j \mathop = 0... | Increasing Sum of Binomial Coefficients/Proof 3 | https://proofwiki.org/wiki/Increasing_Sum_of_Binomial_Coefficients | https://proofwiki.org/wiki/Increasing_Sum_of_Binomial_Coefficients/Proof_3 | [
"Increasing Sum of Binomial Coefficients",
"Binomial Coefficients"
] | [] | [
"Symmetry Rule for Binomial Coefficients",
"Sum of Binomial Coefficients over Lower Index"
] |
proofwiki-2999 | Increasing Alternating Sum of Binomial Coefficients | :$\ds \sum_{j \mathop = 0}^n \paren {-1}^{n + 1} j \binom n j = 0$ | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^n \paren {-1}^{n + 1} j \binom n j
| r = \sum_{j \mathop = 1}^n \paren {-1}^{n + 1} j \binom n j
| c = as $0 \dbinom n 0 = 0$
}}
{{eqn | r = \sum_{j \mathop = 1}^n \paren {-1}^{n + 1} n \binom {n - 1} {j - 1}
| c = Factors of Binomial Coefficient
}}
{{eqn... | :$\ds \sum_{j \mathop = 0}^n \paren {-1}^{n + 1} j \binom n j = 0$ | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^n \paren {-1}^{n + 1} j \binom n j
| r = \sum_{j \mathop = 1}^n \paren {-1}^{n + 1} j \binom n j
| c = as $0 \dbinom n 0 = 0$
}}
{{eqn | r = \sum_{j \mathop = 1}^n \paren {-1}^{n + 1} n \binom {n - 1} {j - 1}
| c = [[Factors of Binomial Coefficient]]
}}
{... | Increasing Alternating Sum of Binomial Coefficients | https://proofwiki.org/wiki/Increasing_Alternating_Sum_of_Binomial_Coefficients | https://proofwiki.org/wiki/Increasing_Alternating_Sum_of_Binomial_Coefficients | [
"Binomial Coefficients"
] | [] | [
"Factors of Binomial Coefficient",
"Translation of Index Variable of Summation",
"Alternating Sum and Difference of Binomial Coefficients for Given n"
] |
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