hcju/proofwikips huggingface dataset:
Mean of Random Sample from Chi-Squared Distribution has Gamma Distribution
Let $n$ be a strictly positive integer.  Let $X_1, X_2, \ldots, X_k$ form a random sample of size $k$ from the chi-squared distribution with $n$ degrees of freedom.  Then:  :$\displaystyle \overline X = \frac 1 k \sum_{i \mathop = 1}^k X_i \sim \map \Gamma {\frac {n k} 2, \frac k 2}$ where $\map \Gamma {\dfrac {n k} 2, \dfrac k 2}$ is the gamma distribution with parameters $\dfrac {n k} 2$ and $\dfrac k 2$.
Equivalence of Definitions of Separated Sets
Let $T = \struct {S, \tau}$ be a topological space. Let $A, B \subseteq S$. {{TFAE|def = Separated Sets}}
Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2
Let $A, B \subseteq S$ satisfy: :$A^- \cap B = A \cap B^- = \O$ where $A^-$ denotes the closure of $A$ in $T$, and $\O$ denotes the empty set.
Equivalence of Definitions of Separated Sets/Definition 2 implies Definition 1
Let $A, B \subseteq S$. Let $U, V \in \tau$ satisfy: :$A \subset U$ and $U \cap B = \O$ :$B \subset V$ and $V \cap A = \O$
Excess Kurtosis of Student's t-Distribution
Let $k$ be a strictly positive integer.  Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then the excess kurtosis $\gamma_2$ of $X$ is given by:  :$\gamma_2 = \dfrac 6 {k - 4}$ for $k > 4$, and does not exist otherwise.
Connected Set in Subspace
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq B \subseteq S$. Let $T_B = \struct {B, \tau_B}$ be the topological space where $\tau_B$ is the subspace topology on $B$. Then :$A$ is connected in $T_B$  $A$ is connected in $T$.
Expectation of Erlang Distribution
Let $k$ be a strictly positive integer.  Let $\lambda$ be a strictly positive real number.  Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$. Then the expectation of $X$ is given by: :$\expect X = \dfrac k \lambda$
Variance of Erlang Distribution
Let $k$ be a strictly positive integer.  Let $\lambda$ be a strictly positive real number.  Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$. Then the variance of $X$ is given by:  :$\var X = \dfrac k {\lambda^2}$
Covariance as Expectation of Product minus Product of Expectations
Let $X$ and $Y$ be random variables.  Let the expectations of $X$ and $Y$ exist and be finite.  Then the covariance of $X$ and $Y$ is given by:  :$\cov {X, Y} = \expect {X Y} - \expect X \expect Y$
Covariance of Independent Random Variables is Zero
Let $X$ and $Y$ be independent random variables. Let the expectations of $X$ and $Y$ exist and be finite.  Then the covariance of $X$ and $Y$ is $0$.
Variance of Linear Combination of Random Variables
Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ be finite.  Let $a$ and $b$ be real numbers.  Then the variance of $a X + b Y$ is given by:  :$\var {a X + b Y} = a^2 \, \var X + b^2 \, \var Y + 2 a b \, \cov {X, Y}$ where $\cov {X, Y}$ is the covariance of $X$ and $Y$.
Square of Expectation of Product is Less Than or Equal to Product of Expectation of Squares
Let $X$ and $Y$ be random variables.  Let the expectation of $X Y$, $\expect {X Y}$, exist and be finite.  Then:  :$\paren {\expect {X Y} }^2 \le \expect {X^2} \expect {Y^2}$
Square of Covariance is Less Than or Equal to Product of Variances
Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ exist and be finite.  Then:  :$\paren {\cov {X, Y} }^2 \le \var X \, \var Y$ where $\cov {X, Y}$ denotes the covariance of $X$ and $Y$.
Absolute Value of Pearson Correlation Coefficient is Less Than or Equal to 1
Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ exist and be finite.  Then:  :$\size {\map \rho {X, Y} } \le 1$ where $\map \rho {X, Y}$ denotes the Pearson correlation coefficient of $X$ and $Y$.
Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 2
Let each point of $T$ have a local basis consisting entirely of connected sets in $T$.
Equivalence of Definitions of Locally Connected Space/Definition 2 implies Definition 1
Let $T$ be weakly locally connected at each point of $T$. That is, each point of $T$ has a neighborhood basis consisting of connected sets of $T$.
Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 3
Let each point $x$ of $T$ have a local basis $\mathcal D_x$ consisting entirely of connected sets in $T$.
Equivalence of Definitions of Locally Connected Space/Definition 3 implies Definition 1
Let $T$ have a basis $\mathcal B$ consisting of connected sets in $T$.
Equivalence of Definitions of Locally Connected Space/Definition 4 implies Definition 3
Let the components of the open sets of $T$ are also open in $T$.
Connected Component is Closed
Let $T = \struct{S, \tau}$ be a topological space. Then every connected component of $T$ is closed.
Constant Mapping is Non-Commutative
Let $S$ be a set whose cardinality is greater than one. Let $f: S \to S$ and $g: S \to S$ be constant mappings on $S$. Then: :$f \circ g \ne g \circ f$ where $\circ$ denotes composition of mappings.
Quotient Mapping is Injection iff Equality
Let $\mathcal R$ be an equivalence relation on $S$. Then the quotient mapping $q_{\mathcal R}: S \to S / \mathcal R$ is an injection  $\mathcal R$ is the equality relation.
Closure of Subset in Subspace
Let $T = \struct{S, \tau}$ be a topological space. Let $H$ be a subset of $S$. Let $T_H = \struct {H, \tau_H}$ be the topological subspace on $H$. Let $A$ be a subset of $H$. Then: :$\map {\cl_H} A = H \cap \map \cl A$ where  :$\map {\cl_H} A$ denotes the closure of $A$ in $T_H$ :$\map \cl A$ denotes the closure of $A$ in $T$
Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 2
Let each point of $T$ have a local basis consisting entirely of path-connected sets in $T$.
Equivalence of Definitions of Locally Path-Connected Space/Definition 2 implies Definition 1
Let each point of $T$ have a neighborhood basis consisting of path-connected sets in $T$.
Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 3
Let each point $x$ of $T$ have a local basis $\mathcal D_x$ consisting entirely of path-connected sets in $T$.
Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 1
Let $T$ have a basis $\mathcal B$ consisting of path-connected sets in $T$.
Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 4
Let $T$ have a basis consisting of path-connected sets in $T$.
Equivalence of Definitions of Locally Path-Connected Space/Definition 4 implies Definition 3
Let the path components of open sets of $T$ be also open in $T$.
Path-Connected Set in Subspace
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq B \subseteq S$. Let $T_B = \struct {B, \tau_B}$ be the topological space where $\tau_B$ is the subspace topology on $B$. Then :$A$ is path-connected in $T_B$  $A$ is path-connected in $T$.
Precisely One Function in terms of And, Or and Not
Let $\map P {A, B, C}$ denote the precisely one function on the statements $A$, $B$ and $C$. Then: :$\map P {A, B, C} \dashv \vdash \paren {A \land \neg B \land \neg C} \lor \paren {\neg A \land B \land \neg C} \lor \paren {\neg A \land \neg B \land C}$ where: :$\land$ denotes conjunction :$\lor$ denotes disjunction :$\neg$ denotes negation
Lévy's Continuity Theorem
Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of discrete random variables with characteristic functions $\map {\phi_n} t := E \sqbrk {e^{i t X_n} }$. Let the sequence $\sequence {\phi_n}$ converge to some real function $\phi$: :$\forall t \in \R: \map {\phi_n} t \to \map \phi t$. Then the following statements are equivalent: :$(1): \quad$ The $\sequence {X_n}$ converges in distribution to some random variable $X$: ::$X_n \stackrel {dist} {\to} X$ with characteristic function $\map {\phi_X} t := \map \phi t$ :$(2): \quad$ The sequence $\sequence {X_n}$ is tight, that is: ::$\displaystyle \lim_{M \mathop \to \infty} \sup_{n \mathop \ge 1} P \sqbrk {\size {X_n} \ge M} = 0$
Composition of Three Mappings which form Identity Mapping
Let $A$, $B$ and $C$ be non-empty sets. Let $f: A \to B$, $g: B \to C$ and $h: C \to A$ be mappings. Let the following hold: {{begin-eqn}} {{eqn | l = h \circ g \circ f       | r = I_A }} {{eqn | l = f \circ h \circ g       | r = I_B }} {{eqn | l = g \circ f \circ h       | r = I_C }} {{end-eqn}} where: :$g \circ f$ (and so on) denote composition of mappings :$I_A$ (and so on) denote the identity mappings. Then each of $f$, $g$ and $h$ are bijections, and: {{begin-eqn}} {{eqn | l = f^{-1}       | r = h \circ g }} {{eqn | l = g^{-1}       | r = f \circ h }} {{eqn | l = h^{-1}       | r = g \circ f }} {{end-eqn}} where $f^{-1}$ (and so on) denote the inverse mappings.
Composition of Product Mappings on Natural Numbers
Let $a \in \N$ be a natural number. Let $\mu_a: \N \to \N$ be the mapping defined as: :$\forall x \in \N: \map {\mu_a} x = x a$ Then: :$\mu_{a b} = \mu_b \circ \mu_a$
Moment Generating Function of Linear Transformation of Random Variable
Let $X$ be a random variable.  Let $\alpha$ and $\beta$ be real numbers. Let $Z = \alpha X + \beta$. Let $M_X$ be the moment generating function of $X$.  Then the moment generating function of $Z$, $M_Z$, is given by:  :$\map {M_Z} t = e^{\beta t} \map {M_X} {\alpha t}$
Standard Gaussian Random Variable as Transformation of Gaussian Random Variable
Let $\mu$ be a real number.  Let $\sigma$ be a positive real number.  Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the Gaussian distribution with parameters $\mu$ and $\sigma^2$.  Then:  :$\dfrac {X - \mu} \sigma \sim \Gaussian 0 1$ where $\Gaussian 0 1$ is the standard Gaussian distribution.
Composition of Addition Mappings on Natural Numbers
Let $a \in \N$ be a natural number. Let $\alpha_a: \N \to \N$ be the mapping defined as: :$\forall x \in \N: \map {\alpha_a} x = x + a$ Then: :$\alpha_{a + b} = \alpha_b \circ \alpha_a$
Reciprocal of Random Variable with F-Distribution has F-Distribution
Let $n, m$ be strictly positive integers.  Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom. Then:  :$\dfrac 1 X \sim F_{m, n}$
Square of Random Variable with t-Distribution has F-Distribution
Let $k$ be a strictly positive integer.  Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then:  :$X^2 \sim F_{1, k}$ where $F_{1, k}$ is the $F$-distribution with $\tuple {1, k}$ degrees of freedom.
Union of Connected Sets with Common Point is Connected
Let $T = \struct {S, \tau}$ be a topological space. Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a family of connected sets of $T$. Let $\exists x \in \displaystyle \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$. Then :$\displaystyle \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$ is a connected set of $T$.
Connected Subset of Union of Disjoint Open Sets
Let $T = \struct{S, \tau}$ be a topological space. Let $A$ be a connected set of $T$. Let $U, V$ be disjoint open sets. Let $A \subseteq U \cup V$. Then :either $A \subseteq U$ or $A \subseteq V$.
Rearrangement of Variables in Total Differential Equation
In a total differential equation, any one of the variables can be regarded as an independent variable, while the remainder can be treated as dependent variables.
Archimedean Principle/Variant
Let $x$ and $y$ be a natural numbers. Then there exists a natural number $n$ such that: :$n x \ge y$
Set Intersection Preserves Subsets/Families of Sets/Intersection is Empty Implies Intersection of Subsets is Empty
Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$. Let: :$\forall \beta \in I: A_\beta \subseteq B_\beta$ Then: :$\displaystyle \bigcap_{\alpha \mathop \in I} B_\alpha = \O \implies \bigcap_{\alpha \mathop \in I} A_\alpha = \O$
Union of Path-Connected Sets with Common Point is Path-Connected
Let $T = \struct {S, \tau}$ be a topological space. Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a family of path-connected sets of $T$. Let $\exists x \in \displaystyle \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$. Then :$\displaystyle \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$ is a path-connected set of $T$.
Upper Bound for Lucas Number
Let $L_n$ denote the $n$th Lucas number. Then: :$L_n < \paren {\dfrac 7 4}^n$
Condition for Increasing Binomial Coefficients
Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Let $\dbinom n k$ denote a binomial coefficient for $k \in \N$. Then: :$\dbinom n k < \dbinom n {k + 1} \iff 0 \le k < \dfrac {n - 1} 2$
Condition for Equality of Adjacent Binomial Coefficients
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\dbinom n k$ denote a binomial coefficient for $k \in \Z$. Then: :$\dbinom n k = \dbinom n {k + 1}$ : :$n$ is an odd integer :$k = \dfrac {n - 1} 2$
Equivalence of Definitions of Component/Equivalence Class equals Union of Connected Sets
Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$. Let $\CC_x = \set {A \subseteq S: x \in A \land A \text{ is connected in } T}$ Let $C = \bigcup \CC_x$ Let $\sim$ be the equivalence relation defined by: :$y \sim z$  $y$ and $z$ are connected in $T$. Let $C’$ be the equivalence class of $\sim$ containing $x$. Then $C = C'$.
Equivalence of Definitions of Component/Union of Connected Sets is Maximal Connected Set
Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$. Let $\mathcal C_x = \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$ Let $C = \bigcup \mathcal C_x$ Then $C$ is a maximal connected set of $T$.
Equivalence of Definitions of Component/Maximal Connected Set is Union of Connected Sets
Let $\tilde C$ be a maximal connected set of $T$ that contains $x$.
Binomial Coefficient n Choose j in terms of n-2 Choose r
Let $n \in \Z$ such that $n \ge 4$. Let $\dbinom n k$ denote a binomial coefficient for $k \in \Z$. Then: :$\dbinom n k = \dbinom {n - 2} {k - 2} + 2 \dbinom {n - 2} {k - 1} + \dbinom {n - 2} k$ for $2 \le k \le n - 2$.
Sum of Sequence of Binomial Coefficients by Powers of 2
{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^n 2^j \binom n j       | r = \dbinom n 0 + 2 \dbinom n 1 + 2^2 \dbinom n 2 + \dotsb + 2^n \dbinom n n       | c =  }} {{eqn | r = 3^n       | c =  }} {{end-eqn}}
Sum of Sequence of n Choose 2
Let $n \in \Z$ be an integer such that $n > 2$. {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 2}^n \dbinom j 2       | r = \dbinom 2 2 + \dbinom 3 2 + \dbinom 4 2 + \dotsb + \dbinom n 2       | c =  }} {{eqn | r = \dbinom {n + 1} 3       | c =  }} {{end-eqn}} where $\dbinom n j$ denotes a binomial coefficient.
If n is Triangular then so is 25n + 3
Let $n$ be a triangular number. Then $25 n + 3$ is also triangular.
If n is Triangular then so is 49n + 6
Let $n$ be a triangular number. Then $49 n + 6$ is also triangular.
Sum of Sequence of Triangular Numbers
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $T_n$ denote the $n$th triangular number. Then: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^n T_j       | r = T_1 + T_2 + T_3 + \dotsb + T_n       | c =  }} {{eqn | r = \dfrac {n \paren {n + 1} \paren {n + 2} } 6       | c =  }} {{end-eqn}}
Square of Odd Multiple of 3 is Difference between Triangular Numbers
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $T_n$ denote the $n$th triangular number. Let $m = 2 n + 1$ be an odd integer Then: :$\paren {3 m}^2 = T_{9 n + 4} - T_{3 n + 1}$
Square Sum of Three Consecutive Triangular Numbers
Let $T_n$ denote the $n$th triangular number for $n \in \Z_{>0}$ a (strictly) positive integer. Let $T_n + T_{n + 1} + T_{n + 2}$ be a square number. Then at least one value of $n$ fulfils this condition: :$n = 5$
Sufficient Condition for Square of Product to be Triangular
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $2 n^2 \pm 1 = m^2$ be a square number. Then $\paren {m n}^2$ is a triangular number.
Topology on Singleton is Indiscrete Topology
Let $S$ be a singleton. The only possible topology on $S$ is the indiscrete topology.
Existence of Divisor with Remainder between 2b and 3b
For every pair of integers $a, b$ where $b > 0$, there exist unique integers $q$ and $r$ where $2 b \le r < 3 b$ such that: :$a = q b + r$
Integer of form 6k + 5 is of form 3k + 2 but not Conversely
Let $n \in \Z$ be an integer of the form: :$n = 6 k + 5$ where $k \in \Z$. Then $n$ can also be expressed in the form: :$n = 3 k + 2$ for some other $k \in \Z$. However it is not necessarily the case that if $n$ can be expressed in the form: :$n = 3 k + 2$ then it can also be expressed in the form: :$n = 6 k + 5$
Odd Integer Modulo 4
Let $n$ be an odd integer. Then $n$ can be expressed either as: :$n = 4 k + 1$ or as: :$n = 4 k + 3$
N (n + 1) (2n + 1) over 6 is Integer
Let $n \in \Z$ be an integer. Then $\dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$ is also an integer.
Number which is Square and Cube Modulo 7
Let $n \in \Z$ be an integer. Let $n$ be both a square and a cube at the same time. Then either: :$n \equiv 0 \pmod 7$ or: :$n \equiv 1 \pmod 7$
Weak Law of Large Numbers
Let $P$ be a population. Let $P$ have mean $\mu$ and finite variance.  Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of random variables forming a random sample from $P$. Let:  :$\ds {\overline X}_n = \frac 1 n \sum_{i \mathop = 1}^n X_i$ Then: :${\overline X}_n \xrightarrow p \mu$ where $\xrightarrow p$ denotes convergence in probability.
Variance of Linear Combination of Random Variables/Corollary
Let $X$ and $Y$ be independent random variables.
Singleton is Connected in Topological Space
Let $T = \struct {S, \tau}$ be a topological space. Let $x \in S$. Then the singleton $\set{x}$ is connected.
Equivalence of Definitions of Component/Lemma 1
:$C$ is connected in $T$ and $C \in \mathcal C_x$.
Variance of Sample Mean
Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$. Let:  :$\displaystyle \overline X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ Then:  :$\var {\overline X} = \dfrac {\sigma^2} n$
Common Divisor Divides Integer Combination/General Result
Let $c$ be a common divisor of a set of integers $A := \set {a_1, a_2, \dotsc, a_n}$. That is: :$\forall x \in A: c \divides x$ Then $c$ divides any integer combination of elements of $A$: :$\forall x_1, x_2, \dotsc, x_n \in \Z: c \divides \paren {a_1 x_2 + a_2 x_2 + \dotsb + a_n x_n}$
One is Common Divisor of Integers
Let $a, b \in \Z$ be integers. Then $1$ is a common divisor of $a$ and $b$.
Equivalence of Definitions of Path Component
{{TFAE|def = Path Component|view = Path Component|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$.
GCD of Integer and its Negative
Let $a \in \Z$ be an integer. Then: :$\gcd \set {a, -a} = \size a$ where: :$\gcd$ denotes greatest common divisor :$\size a$ denotes the absolute value of $a$.
Equivalence of Definitions of Path Component/Lemma 1
:$C$ is path-connected in $T$ and  $C \in \CC_x$.
Square Divides Product of Multiples
Let $a, b, c, \in \Z$ be integers. Let: :$a \divides b, a \divides c$ where $\divides$ denotes divisibility. Then: :$a^2 \divides b c$
Abel's Limit Theorem
Let $\sequence {a_i}$ be a convergent series. The limit which is assigned by the Abel summation method exists and equals the sum of the series.
Equivalence of Definitions of Path Component/Equivalence Class equals Union of Path-Connected Sets
Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$. Let $\CC_x = \left\{ {A \subseteq S : x \in A \land A }\right.$ is path-connected in $\left. {T}\right\}$. Let $C = \bigcup \CC_x$ Let $\sim$ be the equivalence relation defined by: :$y \sim z$  $y$ and $z$ are path-connected in $T$. Let $C'$ be the equivalence class of $\sim$ containing $x$. Then $C = C'$.
Image of Path is Path-Connected Set
Let $T = \struct {S, \tau}$ be a topological space. Let $I \subset \R$ be the closed real interval $\closedint a b$. Let $\gamma: I \to S$ be a path. Then: :$\map \gamma I$ is a path-connected set of $T$.
Closure of Hadamard Product
Let $\struct {S, \cdot}$ be an algebraic structure. Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$. For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$. The operation $\circ$ is closed on $\map {\MM_S} {m, n}$  $\cdot$ is closed on $\struct {S, \cdot}$.
Associativity of Hadamard Product
Let $\struct {S, \cdot}$ be an algebraic structure. Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$. For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$. The operation $\circ$ is associative on $\map {\MM_S} {m, n}$  $\cdot$ is associative on $\struct {S, \cdot}$.
Commutativity of Hadamard Product
Let $\struct {S, \cdot}$ be an algebraic structure. Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$. For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$. The operation $\circ$ is commutative on $\map {\MM_S} {m, n}$  $\cdot$ is commutative on $\struct {S, \cdot}$.
Matrix Entrywise Addition over Ring is Closed
Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$. For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation $+$ is closed on $\map {\MM_R} {m, n}$.
Points are Path-Connected iff Contained in Path-Connected Set
Let $T = \struct {S, \tau}$ be a topological space. Let $x, y \in S$ Then: :$x, y$ are path-connected points in $T$  there exists a path-connected set of $T$ containing $x$ and $y$.
Compound Angle Formulas
=== Sine of Sum === {{:Sine of Sum}} === Cosine of Sum === {{:Cosine of Sum}} === Tangent of Sum === {{:Tangent of Sum}}
Equivalence of Definitions of Path Component/Union of Path-Connected Sets is Maximal Path-Connected Set
Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$. Let $\CC_x = \left\{ {A \subseteq S : x \in A \land A } \right.$ is path-connected in $\left. {T} \right\}$ Let $C = \bigcup \CC_x$ Then $C$ is a maximal path-connected set of $T$.
Solutions to Diophantine Equation 16x^2+32x+20 = y^2+y
The indeterminate Diophantine equation: :$16x^2 + 32x + 20 = y^2 + y$ has exactly $4$ solutions: :$\tuple {0, 4}, \tuple {-2, 4}, \tuple {0, -5}, \tuple {-2, -5}$
Equivalence of Definitions of Path Component/Maximal Path-Connected Set is Union of Path-Connected Sets
Let $\tilde C$ be a maximal path-connected set of $T$ that contains $x$.
Power Set is Closed under Set Complement
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A \in \powerset S: \relcomp S A \in \powerset S$
Normal Subgroup of Symmetric Group Order Greater than 4 is Alternating Group
Let $n \in \N$ be a natural number such that $n > 4$. Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ denote the alternating group on $n$ letters. $A_n$ is the only proper non-trivial normal subgroup of $S_n$.
Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets
Let the path components of $T$ be open sets.
Path Components are Open iff Union of Open Path-Connected Sets/Space is Union of Open Path-Connected Sets implies Path Components are Open
Let $T = \left({S, \tau}\right)$ be a topological space. Let $S$ be the union of open path-connected sets of $T$. Then: :The path components of $T$ are open sets.
Path Components are Open iff Union of Open Path-Connected Sets/Lemma 1
::$U \cap C \ne \O$  $U \ne \O$ and $U \subseteq C$
Integral Points of Elliptic Curve y^2 = x^3+3x
The elliptic curve: :$y^2 = x^3 + 3x$ has exactly $7$ lattice points: :$\tuple {0, 0}, \tuple {1, \pm 2}, \tuple {3, \pm 6}, \tuple {12, \pm 42}$