nyuuzyou/wikis huggingface dataset:
Cotangent of Complement equals Tangent/Corollary 2

== Corollary to Cotangent of Complement equals Tangent == 
Let $x \ne \paren {4 n + 1} \dfrac \pi 4$
Then: 
:$\ds \map \cot {\dfrac \pi 4 - x } = \map \tan {\dfrac \pi 4 + x}$
where:
:$\cot$ and $\tan$ are cotangent and tangent respectively.
== Proof ==
The above is valid only where $\cos x \ne 0$, as otherwise $\dfrac {\sin x} {\cos x}$ is undefined.
Since $\theta \ne \paren {2 n + 1} \dfrac \pi 2$. and $\theta = \dfrac \pi 4 + x$
Then $x \ne \paren {4 n + 1} \dfrac \pi 4$
Category:Cotangent of Complement equals Tangent
Cotangent of Complex Number

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
=== Formulation 1 ===
=== Formulation 2 ===
=== Formulation 3 ===
where:
:$\cot$ denotes the cotangent function (real and complex)
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function
:$\coth$ denotes the hyperbolic cotangent function.
== Also see ==
* Sine of Complex Number
* Cosine of Complex Number
* Tangent of Complex Number
* Secant of Complex Number
* Cosecant of Complex Number
Category:Cotangent Function
Category:Complex Numbers
Category:Cotangent of Complex Number
Cotangent of Complex Number/Formulation 1

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\cot \paren {a + b i} = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}$
where:
:$\cot$ denotes the complex cotangent function
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function.
== Proof ==
== Also see ==
* Sine of Complex Number
* Cosine of Complex Number
* Tangent of Complex Number
* Cosecant of Complex Number
* Secant of Complex Number
Category:Cotangent of Complex Number
Cotangent of Complex Number/Formulation 2

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \cot {a + b i} = \dfrac {-1 - i \cot a \coth b} {\cot a - i \coth b}$
where:
:$\cot$ denotes the cotangent function (real and complex)
:$\coth$ denotes the hyperbolic cotangent function.
== Proof ==
== Also see ==
* Sine of Complex Number
* Cosine of Complex Number
* Tangent of Complex Number
* Cosecant of Complex Number
* Secant of Complex Number
Category:Cotangent of Complex Number
Cotangent of Complex Number/Formulation 3

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \cot {a + b i} = \dfrac {\cot a \coth^2 b - \cot a} {\cot^2 a + \coth^2 b} + \dfrac {-\cot^2 a \coth b - \coth b} {\cot^2 a + \coth^2 b} i$
where:
:$\cot$ denotes the cotangent function (real and complex)
:$\coth$ denotes the hyperbolic cotangent function.
== Proof ==
== Also see ==
* Sine of Complex Number
* Cosine of Complex Number
* Tangent of Complex Number
* Cosecant of Complex Number
* Secant of Complex Number
Category:Cotangent of Complex Number
Cotangent of Conjugate Angle

== Theorem ==
:$\map \cot {2 \pi - \theta} = -\cot \theta$
where $\cot$ denotes cotangent.
That is, the cotangent of an angle is the negative of its conjugate.
== Proof ==
== Also see ==
* Sine of Conjugate Angle
* Cosine of Conjugate Angle
* Tangent of Conjugate Angle
* Secant of Conjugate Angle
* Cosecant of Conjugate Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cotangent Function
Category:Conjugate Angles
Cotangent of Difference

== Corollary to Cotangent of Sum ==
:$\map \cot {a - b} = \dfrac {\cot a \cot b + 1} {\cot b - \cot a}$
where $\cot $ is cotangent.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.37$
Category:Cotangent of Difference
Category:Cotangent Function
Category:Trigonometric Subtraction Formulas
Cotangent of Full Angle

== Theorem ==
:$\cot 360^\circ = \cot 2 \pi$ is undefined
where $\cot$ denotes cotangent.
== Proof ==
From Cotangent is Cosine divided by Sine:
: $\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
From Cosine of Full Angle:
: $\cos 2 \pi = 1$
From Sine of Full Angle:
: $\sin 2 \pi = 0$
Thus $\cot \theta$ is undefined at this value.
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cotangent Function
Cotangent of Right Angle

== Theorem ==
:$\cot 90 \degrees = \cot \dfrac \pi 2 = 0$
where $\cot$ denotes cotangent.
== Proof ==
== Also see ==
* Sine of Right Angle
* Cosine of Right Angle
* Tangent of Right Angle
* Secant of Right Angle
* Cosecant of Right Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cotangent Function
Cotangent of Straight Angle

== Theorem ==
:$\cot 180^\circ = \cot \pi$ is undefined
where $\cot$ denotes cotangent.
== Proof ==
From Cotangent is Cosine divided by Sine:
: $\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
When $\sin \theta = 0$, $\dfrac {\cos \theta} {\sin \theta}$ can be defined only if $\cos \theta = 0$.
But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$.
When $\theta = \pi$, $\sin \theta = 0$.
Thus $\cot \theta$ is undefined at this value.
== Also see ==
* Sine of Straight Angle
* Cosine of Straight Angle
* Tangent of Straight Angle
* Secant of Straight Angle
* Cosecant of Straight Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cotangent Function
Cotangent of Sum

== Theorem ==
:$\map \cot {a + b} = \dfrac {\cot a \cot b - 1} {\cot b + \cot a}$
where $\cot $ is cotangent.
=== Corollary ===
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.37$
Category:Cotangent of Sum
Category:Cotangent Function
Category:Trigonometric Addition Formulas
Cotangent of Sum of Three Angles

== Theorem ==
:$\map \cot {A + B + C} = \dfrac {\cot A + \cot B + \cot C - \cot A \cot B \cot C} {1 - \cot B \cot C - \cot C \cot A - \cot A \cot B}$
== Proof ==
== Also see ==
* Tangent of Sum of Three Angles
Category:Cotangent of Sum
Category:Cotangent Function
Category:Trigonometric Addition Formulas
Cotangent of Supplementary Angle

== Theorem ==
:$\map \cot {\pi - \theta} = -\cot \theta$
where $\cot$ denotes tangent.
That is, the cotangent of an angle is the negative of its supplement.
== Proof ==
== Also see ==
* Sine of Supplementary Angle
* Cosine of Supplementary Angle
* Tangent of Supplementary Angle
* Secant of Supplementary Angle
* Cosecant of Supplementary Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cotangent Function
Category:Supplementary Angles
Cotangent of Three Right Angles

== Theorem ==
:$\cot 270 \degrees = \cot \dfrac {3 \pi} 2 = 0$
where $\cot$ denotes cotangent.
== Proof ==
== Also see ==
* Sine of Three Right Angles
* Cosine of Three Right Angles
* Tangent of Three Right Angles
* Secant of Three Right Angles
* Cosecant of Three Right Angles
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cotangent Function
Cotangent of Three Right Angles less Angle

== Theorem ==
:$\cot \left({\dfrac {3 \pi} 2 - \theta}\right) = \tan \theta$
where $\cot$ and $\tan$ are cotangent and tangent respectively.
== Proof ==
== Also see ==
* Sine of Three Right Angles less Angle
* Cosine of Three Right Angles less Angle
* Tangent of Three Right Angles less Angle
* Secant of Three Right Angles less Angle
* Cosecant of Three Right Angles less Angle
== Sources ==
* : $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
Category:Cotangent Function
Cotangent of Zero

== Theorem ==
:$\cot 0$ is undefined
where $\cot$ denotes cotangent.
== Proof ==
From Cotangent is Cosine divided by Sine:
:$\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
When $\sin \theta = 0$, $\dfrac {\cos \theta} {\sin \theta}$ can be defined only if $\cos \theta = 0$.
But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$.
When $\theta = 0$, $\sin \theta = 0$.
Thus $\cot \theta$ is undefined at this value.
== Also see ==
* Sine of Zero is Zero
* Cosine of Zero is One
* Tangent of Zero
* Secant of Zero
* Cosecant of Zero
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Cotangent Function
Cotangent of i

== Theorem ==
:$\cot i = \paren {\dfrac {1 + e^2} {1 - e^2} } i$
where $\cot$ denotes the complex cotangent function and $i$ is the imaginary unit.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Sine of $i$
* Cosine of $i$
* Tangent of $i$
* Secant of $i$
* Cosecant of $i$
Category:Cotangent of i
Category:Cotangent Function
Category:Imaginary Unit
Coulomb's Law of Electrostatics

== Physical Law ==
:
Let $a$ and $b$ be stationary particles in a vacuum, each carrying an electric charge of $q_a$ and $q_b$ respectively.
Then $a$ and $b$ exert a force upon each other whose magnitude and direction are given by Coulomb's law (of electrostatics):
:$\mathbf F_{a b} \propto \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$
where:
:$\mathbf F_{a b}$ is the force exerted on $b$ by the electric charge on $a$
:$\mathbf r_{a b}$ is the displacement vector from $a$ to $b$
:$r$ is the distance between $a$ and $b$.
:the constant of proportion is defined as being positive.
By exchanging $a$ and $b$ in the above, it is seen that $b$ exerts the same force on $a$ as $a$ does on $b$, but in the opposite direction.
== SI Units ==
== Also presented as ==
== Also known as ==
Coulomb's Law of Electrostatics is also known as just Coulomb's Law.
== Historical Note ==
== Sources ==
* : Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $9$. The Vector Field $\map \grad {k / r}$
* : $2$. Physical Constants and Conversion Factors
* : Chapter $1$: Force and energy in electrostatics
* : Chapter $1$: Force and energy in electrostatics: $1.1$ Electric Charge
Category:Electrostatics
Category:Coulomb's Law of Electrostatics
Coulomb's Law of Electrostatics/Also presented as

== Coulomb's Law of Electrostatics: Also presented as ==
:
Coulomb's Law of Electrostatics can also be presented as:
:$(1): \quad \mathbf F_{a b} \propto \dfrac {q_a q_b \hat {\mathbf r}_{a b} } {r^2}$
::::where $\hat {\mathbf r}_{a b}$ is the unit vector in the direction from $a$ to $b$.
:$(2): \quad \mathbf F_{a b} \propto \dfrac {q_a q_b} {\size {\mathbf r_a - \mathbf r_b}^3} \paren {\mathbf r_a - \mathbf r_b}$
::::where $\mathbf r_a$ and $\mathbf r_b$ are the position vectors of $q_a$ and $q_b$ respectively.
Category:Coulomb's Law of Electrostatics
Coulomb's Law of Electrostatics/SI Units

== Physical Law ==
Coulomb's Law of Electrostatics gives the force between two stationary charged particles $a$ and $b$ as:
:$\mathbf F_{a b} \propto \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$
where:
:$q_a$ and $q_b$ are the charges on $a$ and $b$ respectively
:$\mathbf F_{a b}$ is the force exerted on $b$ by the electric charge on $a$
:$\mathbf r_{a b}$ is the displacement vector from $a$ to $b$
:$r$ is the distance between $a$ and $b$.
:the constant of proportion is defined as being positive.
In SI units, the law becomes:
:$\mathbf F_{a b} = \dfrac 1 {4 \pi \varepsilon_0} \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$
where:
:$q_a$ and $q_b$ are measured in coulombs
:$r$ is measured in metres
:$\mathbf F_{a b}$ is measured in newtons
:$\varepsilon_0$ denotes the vacuum permittivity:
::$\varepsilon_0 \approx 8 \cdotp 85418 \, 78128 (13) \times 10^{-12} \, \mathrm C^2 \, \mathrm N^{-1} \, \mathrm m^{-2}$
== Also presented as ==
== Sources ==
* : Chapter $1$: Force and energy in electrostatics: $1.1$ Electric Charge
Category:Coulomb's Law of Electrostatics
Coulomb's Law of Electrostatics/SI Units/Also presented as

== Coulomb's Law of Electrostatics: Also presented as ==
:
Coulomb's Law of Electrostatics expressed in SI units can also be presented as:
:$(1): \quad \mathbf F_{a b} = \dfrac 1 {4 \pi \varepsilon_0} \dfrac {q_a q_b \hat {\mathbf r}_{a b} } {r^2}$
::::where $\hat {\mathbf r}_{a b}$ is the unit vector in the direction from $a$ to $b$.
:$(2): \quad \mathbf F_{a b} = \dfrac 1 {4 \pi \varepsilon_0} \dfrac {q_a q_b} {\size {\mathbf r_a - \mathbf r_b}^3} \paren {\mathbf r_a - \mathbf r_b}$
::::where $\mathbf r_a$ and $\mathbf r_b$ are the position vectors of $q_a$ and $q_b$ respectively.
== Sources ==
* : Appendix $\text A$: Units
Category:Coulomb's Law of Electrostatics
Count of All Permutations on n Objects

== Theorem ==
Let $S$ be a set of $n$ objects.
Let $N$ be the number of permutations of $k$ objects from $S$, where $1 \le k \le N$.
Then:
:$\ds N = n! \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {k!}$
=== Sequence ===
== Proof ==
The number of permutations on $k$ objects, from $n$ is denoted ${}^n P_k$.
From Number of Permutations:
:${}^n P_k = \dfrac {n!} {\paren {n - k}!}$
Hence:
== Examples ==
Category:Count of All Permutations on n Objects
Category:Permutations (Ordered Selections)
Count of All Permutations on n Objects/Examples/Even Integers from 1, 2, 3, 4

== Example of Count of All Permutations on $n$ Objects ==
Let $N$ be the number of even integers which can be made using one or more of the digits $1$, $2$, $3$ and $4$ no more than once each.
Then:
:$N = 32$
== Proof ==
From Count of All Permutations on $n$ Objects, the total number of integers which can be made using the digits $1$, $2$, $3$ and $4$ is given by:
By symmetry, exactly half of these integers end in $2$ or $4$.
The others end in $1$ or $3$.
Hence:
:$N = \dfrac {64} 2 = 32$
Category:Count of All Permutations on n Objects/Examples
Count of All Permutations on n Objects/Sequence

== Count of All Permutations on $n$ Objects: Sequence ==
The sequence $\sequence {s_n}$ defined as:
:$\ds s_n = n! \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {k!}$
begins:
:$1, 4, 15, 64, 325, 1956, 13699, 109600, 986409, 9864100, \ldots$
Category:Count of All Permutations on n Objects
Count of Binary Operations Without Identity

== Theorem ==
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:
:$N = n^{\paren {\paren {n - 1}^2 + 1} } \paren {n^{2 \paren {n - 1} } - 1}$
=== Sequence of Values of $N$ ===
== Proof ==
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:
:$n^{\paren {n^2} } - n^{\paren {n - 1}^2 + 1} = n^{\paren {\paren {n - 1}^2 + 1} } \paren {n^{2 \paren {n - 1} } - 1}$
Hence the result.
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.2$
Category:Combinatorics
Category:Abstract Algebra
Category:Count of Binary Operations Without Identity
Count of Binary Operations Without Identity/Sequence

== Theorem ==
Let $S$ be a set whose cardinality is $n$.
Let $N$ denote the number of different binary operations which do not have an identity element that can be applied to $S$:
:$N = n^{\paren {\paren {n - 1}^2 + 1} } \paren {n^{2 \paren {n - 1} } - 1}$
The sequence of $N$ for each $n$ begins:
$\begin{array} {c||r|r|r}
n & n^{\paren {n^2} } & n^{\paren {n - 1}^2 + 1} & n^{\paren {\paren {n - 1}^2 + 1} } \paren {n^{2 \paren {n - 1} } - 1} \\
\hline
1 & 1 & 1 & 0 \\
2 & 16 & 4 & 12 \\
3 & 19 \ 683 & 243 & 19 \ 440 \\
4 & 4 \ 294 \ 967 \ 296 & 1 \ 048 \ 576 & 4 \ 293 \ 918 \ 720 \\
\end{array}$
Category:Count of Binary Operations Without Identity
Count of Binary Operations on Set

== Theorem ==
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:
:$N = n^{\paren {n^2} }$
=== Sequence of Values of $N$ ===
== Proof ==
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 Cardinality of Set of All Mappings.
== Examples ==
== Also see ==
* Count of Truth Functions
* Binary Truth Functions
== Sources ==
* : Chapter $4$. Groups: Exercise $1$
* : Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.3$
Category:Combinatorics
Category:Abstract Algebra
Category:Count of Binary Operations on Set
Count of Binary Operations on Set/Examples/Order 2

== Example of Use of Count of Binary Operations on Set ==
The Cayley tables for the complete set of magmas of order $2$ are listed below.
The underlying set in all cases is $\set {a, b}$.
$\quad
\begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & a & a \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & a & b \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & b & a \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & b & b \\ \end{array}$
$\quad
\begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & a & a \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & a & b \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & b & a \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & b & b \\ \end{array}$
$\quad
\begin{array}{r|rr} & a & b \\ \hline a & b & a \\ b & a & a \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & b & a \\ b & a & b \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & b & a \\ b & b & a \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & b & a \\ b & b & b \\ \end{array}$
$\quad
\begin{array}{r|rr} & a & b \\ \hline a & b & b \\ b & a & a \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & b & b \\ b & a & b \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & b & b \\ b & b & a \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & b & b \\ b & b & b \\ \end{array}$
=== Equivalence Classes ===
== Sources ==
* : Chapter $4$. Groups: Exercise $1$
* : Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.2$
Category:Count of Binary Operations on Set/Examples
Count of Binary Operations on Set/Sequence

== Theorem ==
Let $S$ be a set whose cardinality is $n$.
Let $N$ denote the number of possible different binary operations that can be applied to $S$:
:$N = n^{\paren {n^2} }$
The sequence of $N$ for each $n$ begins:
$\begin{array} {c|rr}
n & n^2 & N = n^{\paren {n^2} } \\
\hline
1 & 1 & 1 \\
2 & 4 & 16 \\
3 & 9 & 19 \ 683 \\
4 & 16 & 4 \ 294 \ 967 \ 296 \\
\end{array}$
There are still only $4$ elements in a set, and already there are over $4$ thousand million different possible algebraic structures.
Category:Count of Binary Operations on Set
Count of Binary Operations with Fixed Identity

== Theorem ==
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:
:$N = n^{\paren {\paren {n - 1}^2} }$
=== Sequence of Values of $N$ ===
== Proof ==
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 possible combinations of the remaining $n - 1$ elements.
This is effectively counting the mappings $\paren {S \setminus \set x} \times \paren {S \setminus \set x} \to S$.
From Count of Binary Operations on Set, this is $n^{\paren {\paren {n - 1}^2} }$ structures with $x$ as the identity element.
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.2$
Category:Combinatorics
Category:Abstract Algebra
Category:Count of Binary Operations with Fixed Identity
Count of Binary Operations with Fixed Identity/Sequence

== Theorem ==
Let $S$ be a set whose cardinality is $n$.
Let $x \in S$.
Let $N$ denote the number of different binary operations such that $x$ is an identity element that can be applied to $S$:
:$N = n^{\paren {\paren {n - 1}^2} }$
The sequence of $N$ for each $n$ begins:
$\begin {array} {c|cr}
n & \paren {n - 1}^2 & n^{\paren {\paren {n - 1}^2} } \\
\hline
1 & 0 & 1 \\
2 & 1 & 2 \\
3 & 4 & 81 \\
4 & 9 & 262 \, 144 \\
\end{array}$
Category:Count of Binary Operations with Fixed Identity
Count of Binary Operations with Identity

== Theorem ==
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:
:$N = n^{\paren {n - 1}^2 + 1}$
=== Sequence of Values of $N$ ===
== Proof ==
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 $n \times n^{\paren {n - 1}^2}$ different magmas with an identity.
These are guaranteed not to overlap by the uniqueness of the identity.
Hence the result.
== Examples ==
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.2$
Category:Combinatorics
Category:Abstract Algebra
Category:Count of Binary Operations with Identity
Count of Binary Operations with Identity/Examples/Order 2

== Example of Use of Count of Binary Operations with Identity ==
The Cayley tables for the complete set of magmas of order $2$ which have an identity element are listed below.
The underlying set in all cases is $\set {a, b}$.
$\quad\begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & a & b \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & b & a \\ \end{array}$
$\quad\begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & b & b \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & b & a \\ b & a & b \\ \end{array}$
== Sources ==
* : Chapter $4$. Groups: Exercise $1 \ \text{(iii)}$
Category:Count of Binary Operations with Identity/Examples
Count of Binary Operations with Identity/Sequence

== Theorem ==
Let $S$ be a set whose cardinality is $n$.
Let $N$ denote the number of different binary operations which have an identity element that can be applied to $S$:
:$N = n^{\paren {n - 1}^2 + 1}$
The sequence of $N$ for each $n$ begins:
$\begin{array} {c|cr}
n & \paren {n - 1}^2 + 1 & n^{\paren {n - 1}^2 + 1}\\
\hline
1 & 1 & 1 \\
2 & 2 & 4 \\
3 & 5 & 243 \\
4 & 10 & 1 \ 048 \ 576 \\
\end{array}$
Category:Count of Binary Operations with Identity
Count of Binary Operations with Zero/Examples/Order 2

== Example of Use of Count of Binary Operations with Zero ==
The Cayley tables for the complete set of magmas of order $2$ which have a zero element are listed below.
The underlying set in all cases is $\set {a, b}$.
$\quad\begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & a & a \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & a & a \\ b & a & b \\ \end{array}$
$\quad\begin{array}{r|rr} & a & b \\ \hline a & a & b \\ b & b & b \\ \end{array}
\qquad
\begin{array}{r|rr} & a & b \\ \hline a & b & b \\ b & b & b \\ \end{array}$
== Sources ==
* : Chapter $4$. Groups: Exercise $1 \ \text{(iv)}$
Category:Count of Binary Operations with Zero