hcju/proofwikips huggingface dataset:
Probability Generating Function of Binomial Distribution
Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$. Then the p.g.f. of $X$ is: :$\Pi_X \left({s}\right) = \left({q + ps}\right)^n$ where $q = 1 - p$.
Probability Generating Function of Shifted Geometric Distribution
Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the p.g.f. of $X$ is: :$\map {\Pi_X} s = \dfrac {p s} {1 - q s}$ where $q = 1 - p$.
Probability Generating Function of Geometric Distribution
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the p.g.f. of $X$ is: :$\map {\Pi_X} s = \dfrac q {1 - p s}$ where $q = 1 - p$.
Probability Generating Function of Poisson Distribution
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Then the p.g.f. of $X$ is: :$\Pi_X \left({s}\right) = e^{-\lambda \left({1-s}\right)}$
Expectation of Discrete Random Variable from PGF
Let $X$ be a discrete random variable whose probability generating function is $\map {\Pi_X} s$. Then the expectation of $X$ is the value of the first derivative of $\map {\Pi_X} s$ {{WRT|Differentiation}} $s$ at $s = 1$. That is: :$\expect X = \map {\Pi'_X} 1$
Variance of Discrete Random Variable from PGF
Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$. Then the variance of $X$ can be easily obtained from the value of the second derivative of $\Pi_X \left({s}\right)$ WRT $s$ at $x=1$: :$\operatorname{var} \left({X}\right) = \Pi''_X \left({1}\right) + \mu - \mu^2$ where $\mu = E \left({X}\right)$ is the expectation of $X$.
Properties of Probability Generating Function
Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$. Then $\Pi_X \left({s}\right)$ has the following properties:
Nth Derivative of Mth Power
Let $m \in \Z$ be an integer such that $m \ge 0$. The $n$th derivative of $x^m$ {{WRT|Differentiation}} $x$ is: :$\dfrac {\d^n} {\d x^n} x^m = \begin{cases} m^{\underline n} x^{m - n} & : n \le m \\ 0 & : n > m \end{cases}$ where $m^{\underline n}$ denotes the falling factorial.
Derivative of Complex Power Series
Let $\xi \in \C$ be a complex number. Let $\sequence {a_n}$ be a sequence in $\C$. Let $\displaystyle f \paren z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a power series in a complex variable $z \in \C$ about $\xi$. Let $R$ be the radius of convergence of the series defining $f \paren z$. Let $\cmod {z - \xi} < R$. Then $f$ is complex-differentiable and its derivative is: :$\displaystyle f' \paren z = \sum_{n \mathop = 1}^\infty n a_n \paren {z - \xi}^{n - 1}$
Probability Generating Function of Discrete Uniform Distribution
Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$. Then the p.g.f. of $X$ is: :$\displaystyle \map {\Pi_X} s = \frac {s \paren {1 - s^n} } {n \paren {1 - s} }$
Derivatives of PGF of Bernoulli Distribution
Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$. Then the derivatives of the PGF of $X$ w.r.t. $s$ are: :$\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = \begin{cases} p & : k = 1 \\ 0 & : k > 1 \end{cases}$
Derivatives of PGF of Binomial Distribution
Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$. Then the derivatives of the PGF of $X$ w.r.t. $s$ are: :$\dfrac {\mathrm d^k} {\mathrm ds^k} \Pi_X \left({s}\right) = \begin{cases} n^{\underline k} p^k \left({q + ps}\right)^{n-k} & : k \le n \\ 0 & : k > n \end{cases}$ where: : $n^{\underline k}$ is the falling factorial : $q = 1 - p$
Higher Derivatives of Exponential Function
Let $\exp x$ be the exponential function. Then: :$\map {\dfrac {\d^n} {\d x^n} } {\exp x} = \exp x$
Derivatives of PGF of Poisson Distribution
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {d^k} {\d s^k} \, \map {\Pi_X} s = \lambda^k e^{- \lambda \paren {1 - s} }$
Derivatives of Function of a x + b
Let $f$ be a real function which is differentiable on $\R$. Let $a, b \in \R$ be constants. Then: : $\dfrac {\d^n} {\d x^n} \left({f \left({a x + b}\right)}\right) = a^n \dfrac {\d^n} {\d z^n} \left({f \left({z}\right)}\right)$ where $z = a x + b$.
Nth Derivative of Reciprocal of Mth Power
Let $m \in \Z$ be an integer such that $m > 0$. The $n$th derivative of $\dfrac 1 {x^m}$ {{WRT|Differentiation}} $x$ is: :$\dfrac {\d^n} {\d x^n} \dfrac 1 {x^m} = \dfrac {\paren {-1}^n m^{\overline n}} {x^{m + n}}$ where $m^{\overline n}$ denotes the rising factorial.
Derivatives of PGF of Geometric Distribution
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {\d^n} {\d s^n} \map {\Pi_X} s = \dfrac {q p^n n!} {\paren {1 - p s}^{n + 1} }$ where $q = 1 - p$.
Derivatives of PGF of Shifted Geometric Distribution
Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {\mathrm d^n} {\mathrm d s^n} \Pi_X \left({s}\right) = \dfrac {p q^{n - 1} \left({n - 1}\right)!} {\left({1 - q s}\right)^{n + 1} }$ where $q = 1 - p$.
Total Expectation Theorem
Let $\mathcal E = \struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $x$ be a discrete random variable on $\mathcal E$. Let $\set {B_1 \mid B_2 \mid \cdots}$ be a partition of $\omega$ such that $\map \Pr {B_i} > 0$ for each $i$. Then: :$\displaystyle \expect X = \sum_i \expect {X \mid B_i} \, \map \Pr {B_i}$ whenever this sum converges absolutely. In the above: :$\expect X$ denotes the expectation of $X$ :$\expect {X \mid B_i}$ denotes the conditional expectation of $X$ given $B_i$.
Function of Two Discrete Random Variables
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$. Let $g: \R^2 \to \R$ be a real-valued function. Then $Z = g \left({X, Y}\right)$, defined as: :$\forall \omega \in \Omega: Z \left({\omega}\right) = g \left({X \left({\omega}\right), Y \left({\omega}\right)}\right)$ is also a discrete random variable.
Expectation of Function of Joint Probability Mass Distribution
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$. Let $E \left({X}\right)$ be the expectation of $X$. Let $g: \R^2 \to \R$ be a real-valued function Let $p_{X, Y}$ be the joint probability mass function of $X$ and $Y$. :$\displaystyle E \left({g \left({X, Y}\right)}\right) = \sum_{x \mathop \in \Omega_X} \sum_{y \mathop \in \Omega_Y} g \left({x, y}\right) p_{X, Y} \left({x, y}\right)$ whenever the sum is absolutely convergent.
Linearity of Expectation Function
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $\expect X$ denote the expectation of $X$. Then: :$\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \, \expect X + \beta \, \expect Y$
Condition for Independence from Product of Expectations
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $\expect X$ denote the expectation of $X$. Then $X$ and $Y$ are independent : :$\expect {\map g x \map h y} = \expect {\map g x} \expect {\map h y}$ for all functions $g, h: \R \to \R$ for which the latter two expectations exist.
PGF of Sum of Independent Discrete Random Variables
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be independent discrete random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $Z$ be a discrete random variable such that $Z = X + Y$. Then: :$\map {\Pi_Z} s = \map {\Pi_X} s \, \map {\Pi_Y} s$ where $\map {\Pi_Z} s$ is the probability generating function of $Z$.
Brahmagupta Theorem
If a cyclic quadrilateral has diagonals which are perpendicular, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. Specifically: Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular, crossing at $M$. Let $EF$ be a line passing through $M$ and crossing opposite sides $BC$ and $AD$ of $ABCD$. Then $EF$ is perpendicular to $BC$  $F$ is the midpoint of $AD$.
Clavius's Law
If, from the negation of a proposition $p,$ we can derive $p$, we may conclude $p$:
Viète's Formulas
Let $P$ be a polynomial of degree $n$ with real or complex coefficients: {{begin-eqn}} {{eqn | l = \map P x        | r = \sum_{i \mathop = 0}^n a_i x^i }} {{eqn | r = a_n x^n + a_{n - 1} x^{n - 1} + \dotsb + a_1 x + a_0 }} {{end-eqn}} where $a_n \ne 0$. Let $z_1, \ldots, z_k$ be real or complex roots of $P$, not assumed distinct. Let $P$ be expressible in the form: :$\displaystyle \map P x = a_n \prod_{k \mathop = 1}^n \paren {x - z_k}$ Then: {{begin-eqn}} {{eqn | l = \paren {-1}^k \dfrac {a_{n - k} } {a_n}       | r = e_k \paren {\set {z_1, \ldots, z_n} }       | c = that is, the elementary symmetric function on $\set {z_1, \ldots, z_n}$ }} {{eqn | r = \sum_{1 \mathop \le i_1 \mathop < \dotsb \mathop < i_k \mathop \le n} z_{i_1} \dotsm z_{i_k}        | c = for $k = 1, 2, \ldots, n$. }} {{end-eqn}}
Inscribed Angle Theorem
An inscribed angle is equal to half the angle that is subtended by that arc. :300px Thus, in the figure above: :$\angle ABC = \frac 1 2 \angle ADC$  :''In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.'' {{EuclidPropRef|III|20}}
Existence and Uniqueness Theorem for 1st Order IVPs
Let $x' = \map f {t, x}$, $\map x {t_0} = x_0$ be an explicit ODE of dimension $n$. Let there exist an open ball $V = \sqbrk {t_0 - \ell_0, t_0 + \ell_0} \times \map {\overline B} {x_0, \epsilon}$ of $\tuple {t_0, x_0}$ in phase space $\R \times \R^n$ such that $f$ is Lipschitz continuous on $V$. {{explain|Notation needs to be explained: $\sqbrk {t_0 - \ell_0, t_0 + \ell_0}$ looks as though it should be an interval (and so needs to be written in Wirth interval notation $\closedint {t_0 - \ell_0} {t_0 + \ell_0}$ so as to abide by house style rules), and $\tuple {t_0, x_0}$ is probably an ordered pair. It's not clear enough. The immediate confusion arises because as $\closedint {t_0 - \ell_0} {t_0 + \ell_0}$ is a closed interval it is counter-intuitive for it to be one of the factors of an open ball expressed as a Cartesian product.}} Then there exists $\ell < \ell_0$ such that there exists a unique solution $\map x t$ defined for $t \in \closedint {t_0 - \ell_0} {t_0 + \ell_0}$. {{explain|what is an "IVP"?}}
Maximum Modulus Principle
A non-constant analytic function $f$ in an open connected set $D$ does not have any interior maximum points. That is, for each $z \in D$ and $\delta > 0$, there exists some $\omega \in B_\delta \left({z}\right) \cap D$, such that: :$\left\vert{f \left({\omega}\right)}\right\vert > \left\vert{f \left({z}\right)}\right\vert$.
Euler Polyhedron Formula
For any convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces:  :$V - E + F = 2$
Line Joining Centers of Two Circles Touching Internally
Let two circles touch internally. Then the straight line joining their centers passes through the point where they touch. {{:Euclid:Proposition/III/11}}
Properties of Matrix Exponential
In the following: : $\mathrm A$ and $\mathrm B$ are constant square matrices : $P$ is a nonsingular matrix : $t, s \in \R$ The matrix exponential $e^{\mathrm A t}$ has the following properties:
Lyapunov's Stability Theorem
Let $V$ be a Lyapunov function on an open set containing an equilibrium point $x_0$. {{Disambiguate|Definition:Open Set}} Then $x_0$ is stable. If $V$ is strict, then $x_0$ is asymptotically stable. {{explain|strict}}
Bendixson-Dulac Theorem
Suppose there exists a continuously differentiable function $\alpha \left({x, y}\right)$ on a simply connected domain. {{Explain|What ''is'' the domain? Reals, complex, or what?}} Suppose that: :$\nabla \cdot \left({\alpha F}\right)$  is either always positive or always negative. Then the two-dimensional autonomous system: :$ \left({x, y}\right)' = F \left({x, y}\right)$ does not have a periodic solution.
General Vector Solution of Fundamental Matrix
Let $\map \Phi t$ be a fundamental matrix of the system $x' = \map A t x$. Then: :$\map \Phi t c$ is a general solution of $x' = \map A t x$.
General Fundamental Matrix
Let $\map \Phi t$ be a fundamental matrix of the system $x' = \map A t x$. Then: :$\map \Phi t C$ is a general fundamental matrix of $x' = \map A t x$, where $C$ is ''any'' nonsingular matrix.
Condition for Composite Mapping on Left
Let $A, B, C$ be sets. Suppose that $C$ is non-empty. Let $f: A \to B$ and $g: A \to C$ be mappings. Let $\RR: B \to C$ be a relation such that $g = \RR \circ f$ is the composite of $f$ and $\RR$. Then $\RR$ may be a mapping : :$\forall x, y \in A: \map f x = \map f y \implies \map g x = \map g y$ That is: :$\forall x, y \in A: \map f x = \map f y \implies \map g x = \map g y$ : :$\exists h: B \to C$ such that $h$ is a mapping and $h \circ f = g$
Condition for Composite Mapping on Right
Let $A, B, C$ be sets. Let $f: B \to A$ and $g: C \to A$ be mappings. Let $\mathcal R: C \to B$ be a relation such that $g = f \circ \mathcal R$ is the composite of $\mathcal R$ and $f$. Then $\mathcal R$ may be a mapping : :$\Img g \subseteq \Img f$ That is: :$\Img g \subseteq \Img f$ : :$\exists h: C \to B$ such that $h$ is a mapping and $f \circ h = g$
Set Union Preserves Subsets
Let $A, B, S, T$ be sets. Then: :$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$
Set Complement inverts Subsets
:$S \subseteq T \iff \map \complement T \subseteq \map \complement S$
Set Difference with Superset is Empty Set
:$S \subseteq T \iff S \setminus T = \O$
Intersection with Subset is Subset
:$S \subseteq T \iff S \cap T = S$
Union with Superset is Superset
:$S \subseteq T \iff S \cup T = T$
Intersection with Complement is Empty iff Subset
:$S \subseteq T \iff S \cap \map \complement T = \O$
Complement Union with Superset is Universe
:$S \subseteq T \iff \map \complement S \cup T = \mathbb U$
Set Difference is not Associative
Let $R, S, T$ be sets. The expression: :$\paren {R \setminus S} \setminus T = R \setminus \paren {S \setminus T}$ holds exactly when $R \cap T = \O$. Here $R \setminus S$ denotes set difference. Thus, set difference is not associative.
Cardinality of Set Union
=== Union of 2 Sets === {{:Cardinality of Set Union/2 Sets}} === Union of 3 Sets === {{:Cardinality of Set Union/3 Sets}}
Group Isomorphism Preserves Identity
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group isomorphism. Let: :$e_G$ be the identity of $\struct {G, \circ}$ :$e_H$ be the identity of $\struct {H, *}$. Then: :$\map \phi {e_G} = e_H$
Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group
Let $\Q_{> 0}$ be the set of strictly positive rational numbers, i.e. $\Q_{> 0} = \set {x \in \Q: x > 0}$. The structure $\struct {\Q_{> 0}, \times}$ is a countably infinite abelian group.
Multinomial Theorem
Let $x_1, x_2, \ldots, x_k \in F$, where $F$ is a field. Then: :$\displaystyle \paren {x_1 + x_2 + \cdots + x_m}^n = \sum_{k_1 \mathop + k_2 \mathop + \mathop \cdots \mathop + k_m \mathop = n} \binom n {k_1, k_2, \ldots, k_m} {x_1}^{k_1} {x_2}^{k_2} \cdots {x_m}^{k_m}$ where: :$m \in \Z_{> 0}$ is a positive integer :$n \in \Z_{\ge 0}$ is a non-negative integer :$\dbinom n {k_1, k_2, \ldots, k_m} = \dfrac {n!} {k_1! \, k_2! \, \cdots k_m!}$ denotes a multinomial coefficient. The sum is taken for all non-negative integers $k_1, k_2, \ldots, k_m$ such that $k_1 + k_2 + \cdots + k_m = n$, and with the understanding that wherever $0^0$ may appear  it shall be considered to have a value of $1$. The '''multinomial theorem''' is a generalization of the Binomial Theorem.
Mapping is Constant iff Increasing and Decreasing
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping. Then $\phi$ is a constant mapping  $\phi$ is both increasing and decreasing.
Equivalence of Definitions of Euler's Number
{{TFAE|def = Euler's Number}} === Limit of Series === {{:Definition:Euler's Number/Limit of Series}} === Limit of Sequence === {{:Definition:Euler's Number/Limit of Sequence}} === Base of Logarithm === {{:Definition:Euler's Number/Base of Logarithm}} === Exponential Function === {{:Definition:Euler's Number/Exponential Function}}
Powers Drown Logarithms
Let $r \in \R_{>0}$ be a (strictly) positive real number. Then: :$\displaystyle \lim_{x \mathop \to \infty} x^{-r} \ln x = 0$
Equivalence of Well-Ordering Principle and Induction
The Well-Ordering Principle, the Principle of Finite Induction and the Principle of Complete Finite Induction are logically equivalent. That is: :Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: :: $0 \in S$ :: $n \in S \implies n + 1 \in S$ :then $S = \N$. : :Principle of Complete Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: :: $0 \in S$ :: $\set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$ :then $S = \N$. : :Well-Ordering Principle: Every non-empty subset of $\N$ has a minimal element.
Lower Bound for Subset
Let $\left({S, \preceq}\right)$ be an ordered set. Let $L$ be a lower bound for $S$. Let $\left({T, \preceq}\right)$ be a subset of $\left({S, \preceq}\right)$. Then $L$ is a lower bound for $T$.
Upper Bound for Subset
Let $\left({S, \preceq}\right)$ be an ordered set. Let $U$ be an upper bound for $S$. Let $\left({T, \preceq}\right)$ be a subset of $\left({S, \preceq}\right)$. Then $U$ is an upper bound for $T$.
Common Divisor Divides GCD
Let $a, b \in \Z$ such that not both of $a$ and $b$ are zero. Let $c$ be any common divisor of $a$ and $b$. That is, let $c \in \Z: c \divides a, c \divides b$. Then: :$c \divides \gcd \set {a, b}$ where $\gcd \set {a, b}$ is the greatest common divisor of $a$ and $b$.
Square Modulo 5
Let $x \in \Z$ be an integer. Then one of the following holds: {{begin-eqn}} {{eqn | l = x^2       | o = \equiv       | r = 0 \pmod 5       | c =  }} {{eqn | l = x^2       | o = \equiv       | r = 1 \pmod 5       | c =  }} {{eqn | l = x^2       | o = \equiv       | r = 4 \pmod 5       | c =  }} {{end-eqn}}
Square Modulo 3
Let $x \in \Z$ be an integer. Then one of the following holds: {{begin-eqn}} {{eqn | l = x^2       | o = \equiv       | r = 0 \pmod 3       | c =  }} {{eqn | l = x^2       | o = \equiv       | r = 1 \pmod 3       | c =  }} {{end-eqn}}
Additive Group of Integers is Countably Infinite Abelian Group
The set of integers under addition $\struct {\Z, +}$ forms a countably infinite abelian group.
Division Laws for Groups
Let $G$ be a group. Let $a, b, x \in G$. Then: : $(1): \quad a x = b \iff x = a^{-1} b$ : $(2): \quad x a = b \iff x = b a^{-1}$
Center is Intersection of Centralizers
The center of a group is the intersection of all the centralizers of the elements of that group: :$\displaystyle \map Z G = \bigcap_{g \mathop \in G} \map {C_G} g$
Equivalent Statements for Congruence Modulo Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$.
Congruence Modulo Subgroup is Equivalence Relation
Let $G$ be a group, and let $H$ be a subgroup of $G$. Let $x, y \in G$.
Group Homomorphism Preserves Identity
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism. Let: :$e_G$ be the identity of $G$ :$e_H$ be the identity of $H$. Then: :$\map \phi {e_G} = e_H$
Group Homomorphism Preserves Inverses
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: \struct {G, \circ} \to\struct {H, *}$ be a group homomorphism. Let: :$e_G$ be the identity of $G$ :$e_H$ be the identity of $H$ Then: :$\forall x \in G: \map \phi {x^{-1} } = \paren {\map \phi x}^{-1}$
Subset Product with Normal Subgroup is Subgroup
Let $G$ be a group whose identity is $e$. Let: :$(1): \quad H$ be a subgroup of $G$ :$(2): \quad N$ be a normal subgroup of $G$. Let $H N$ denote subset product. Then $H N$ and $N H$ are both subgroups of $G$.
Coset by Identity
Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$.
Element of Group is in its own Coset
Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $x \in G$.
Element in Coset iff Product with Inverse in Subgroup
==== Element in Left Coset iff Product with Inverse in Subgroup ==== {{:Element in Left Coset iff Product with Inverse in Subgroup}} ==== Element in Right Coset iff Product with Inverse in Subgroup ==== {{:Element in Right Coset iff Product with Inverse in Subgroup}}
Cosets are Equal iff Product with Inverse in Subgroup
==== Left Cosets are Equal iff Product with Inverse in Subgroup ==== {{:Left Cosets are Equal iff Product with Inverse in Subgroup}} ==== Right Cosets are Equal iff Product with Inverse in Subgroup ==== {{:Right Cosets are Equal iff Product with Inverse in Subgroup}}
Coset Equals Subgroup iff Element in Subgroup
==== Left Coset Equals Subgroup iff Element in Subgroup ==== {{:Left Coset Equals Subgroup iff Element in Subgroup}} ==== Right Coset Equals Subgroup iff Element in Subgroup ==== {{:Right Coset Equals Subgroup iff Element in Subgroup}}
Elements in Same Coset iff Product with Inverse in Subgroup
==== Elements in Same Left Coset iff Product with Inverse in Subgroup ==== {{:Elements in Same Left Coset iff Product with Inverse in Subgroup}} ==== Elements in Same Right Coset iff Product with Inverse in Subgroup ==== {{:Elements in Same Right Coset iff Product with Inverse in Subgroup}}
Regular Representation on Subgroup is Bijection to Coset
* The mapping $\lambda_x: H \to x H$, where $\lambda_x$ is the left regular representation of $H$ with respect to $x$, is a bijection from $H$ to $x H$. * The mapping $\rho_x: H \to H x$, where $\rho_x$ is the right regular representation of $H$ with respect to $x$, is a bijection from $H$ to $H x$.
Ring Homomorphism Preserves Zero
Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. Let: :$0_{R_1}$ be the zero of $R_1$ :$0_{R_2}$ be the zero of $R_2$. Then: :$\map \phi {0_{R_1} } = 0_{R_2}$
Image of Set Difference under Mapping
Let $f: S \to T$ be a mapping. The image of the set difference of two subsets of $S$ is a subset of the set difference of the images. That is: Let $S_1$ and $S_2$ be subsets of $S$. Then: :$f \sqbrk {S_1} \setminus f \sqbrk {S_2} \subseteq f \sqbrk {S_1 \setminus S_2}$ where $\setminus$ denotes set difference.
Convergence by Multiple of Error Term
Let $\sequence {s_n}$ be a real sequence. Suppose that $\exists \epsilon \in \R, \epsilon > 0$ such that: :$\exists N \in \N: \forall n \ge N: \size {s_n - l} < K \epsilon$ for any $K \in \R, K > 0$, independent of both $\epsilon$ and $N$. Then $\sequence {s_n}$ converges to $l$.
Cauchy Sequence Converges on Real Number Line
Let $\sequence {a_n}$ be a Cauchy sequence in $\R$. Then $\sequence {a_n}$ is convergent. In other words, $\struct {\R, \size {\,\cdot\,}}$ is a Banach space.
Cauchy's Convergence Criterion
Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is convergent  $\sequence {x_n}$ is a Cauchy sequence.
Wedderburn's Theorem
Every finite division ring $D$ is a field.
Preimages All Exist iff Surjection
Let $f: S \to T$ be a mapping. Let $f^{-1}$ be the inverse of $f$. Let $f^{-1} \paren t$ be the preimage of $t \in T$. Then $f^{-1} \paren t$ is empty for no $t \in T$  $f$ is a surjection.
Non-Successor Element of Peano Structure is Unique
Let $\struct {P, s, 0}$ be a Peano structure. Then: :$P \setminus s \sqbrk P$ is a singleton where: :$\setminus$ denotes set difference :$s \sqbrk P$ denotes the image of the mapping $s$. It follows that the non-successor element $0$ is the only element of $P$ with this property.