hcju/proofwikips huggingface dataset:
Characterization of Metacategory via Equations
Let $\mathbf C_0$ and $\mathbf C_1$ be collections of objects. Let $\operatorname{cod}$ and $\operatorname{dom}$ assign to every element of $\mathbf C_1$ an element of $\mathbf C_0$. Let $\operatorname{id}$ assign to every element of $\mathbf C_0$ an element of $\mathbf C_1$. Denote with $\mathbf C_2$ the collection of pairs $\left({f, g}\right)$ of elements of $\mathbf C_1$ satisfying: :$\operatorname{dom} g = \operatorname{cod} f$ Let $\circ$ assign to every such pair an element of $\mathbf C_1$. Then $\mathbf C_0, \mathbf C_1, \operatorname{cod}, \operatorname{dom}, \operatorname{id}$ and $\circ$ together determine a metacategory $\mathbf C$ iff the following seven axioms are satisfied: {{begin-eqn}} {{eqn|l = \operatorname{dom} \operatorname{id}_A = A      |o = \qquad      |r = \operatorname{cod} \operatorname{id}_A = A }} {{eqn|l = f \circ \operatorname{id}_{\operatorname{dom} f} = f      |o =       |r = \operatorname{id}_{\operatorname{cod} f} \circ f = f }} {{eqn|l = \operatorname{dom} \left({g \circ f}\right) = \operatorname{dom} f      |o =       |r = \operatorname{cod} \left({g \circ f}\right) = \operatorname{cod} g }} {{eqn|l = h \circ \left({g \circ f}\right)      |r = \left({h \circ g}\right) \circ f }} {{end-eqn}} with $A$ and $f,g,h$ arbitrary elements of $\mathbf C_0$ and $\mathbf C_1$, respectively. Further, in the last two lines it is presumed that all compositions are defined. Hence it follows that: * $\mathbf C_0$ and $\mathbf C_1$ represent the collections of objects and morphisms of $\mathbf C$ * $\operatorname{dom}$ and $\operatorname{cod}$ represent the domain and codomain of a morphism of $\mathbf C$ * $\operatorname{id}$ represents the identity morphisms of $\mathbf C$ * $\mathbf C_2$ represents the collection of composable morphisms of $\mathbf C$ * $\circ$ represents the composition of morphisms in $\mathbf C$
Kernel of Group Homomorphism Corresponds with Normal Subgroup of Domain
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism. Let $\map \ker \phi$ be the kernel of $\phi$. Then there exists $N \lhd G$, a normal subgroup of $G$ such that: :$N = \map \ker \phi$ Conversely, let $N \lhd G$ be normal subgroup of $G$. Then there exists $\phi: \struct {G, \circ} \to \struct {H, *}$, a group homomorphism, whose kernel $\map \ker \phi$ is such that: :$\map \ker \phi = N$
Inner Automorphism Maps Normal Subgroup to Itself
Let $G$ be a group. Let $x \in G$. Let $\kappa_x$ be the inner automorphism of $x$ in $G$. Let $N$ be a normal subgroup of $G$. Then: :$\kappa_x \sqbrk N = N$
Group Epimorphism Preserves Normal Subgroups
Let $\left({G, \circ}\right)$ and $\left({H, *}\right)$ be groups. Let $\phi: G \to H$ be a group epimorphism. Let $N \lhd G$, where $\lhd$ denotes that $N$ is a normal subgroup of $G$. Then $\phi \left({N}\right) \lhd H$. That is, the image under $\phi$ of a normal subgroup is itself normal.
Group Epimorphism Induces Bijection between Subgroups
Let $G_1$ and $G_2$ be groups whose identities are $e_{G_1}$ and $e_{G_2}$ respectively. Let $\phi: G_1 \to G_2$ be a group epimorphism. Let $K := \ker \left({\phi}\right)$ be the kernel of $\phi$. Let $\mathbb H_1 = \left\{{H \subseteq G_1: H \le G_1, K \subseteq H}\right\}$ be the set of subgroups of $G_1$ which contain $K$. Let $\mathbb H_2 = \left\{{H \subseteq G_2: H \le G_2}\right\}$ be the set of subgroups of $G_2$. Then there exists a bijection $Q: \mathbb H_1 \leftrightarrow \mathbb H_2$ such that: :$\forall N \lhd G_1: Q \left({N}\right) \lhd G_2$ :$\forall N \lhd G_2: Q^{-1} \left({N}\right) \lhd G_1$ where $N \lhd G_1$ denotes that $N$ is a normal subgroup of $G_1$. That is, normal subgroups map bijectively to normal subgroups under $Q$.
Homomorphic Image of Quotient Group under Epimorphism
Let $G_1$ and $G_2$ be groups whose identities are $e_{G_1}$ and $e_{G_2}$ respectively. Let $\phi: G_1 \to G_2$ be a group epimorphism. Let $K := \ker \left({\phi}\right)$ be the kernel of $\phi$. Let $N$ be a normal subgroup of $G_1$ such that $K \subseteq N$. Then: :$\dfrac {G_1} N \cong \dfrac {G_2}{\phi \left({N}\right)}$ where $\dfrac {G_1} N$ denotes the quotient group of $G_1$ by $N$.
Cauchy's Group Theorem
Let $G$ be a finite group whose identity is $e$. Let $p$ be a prime number which divides order of $G$. Then $G$ has a subgroup of order $p$.
Connected Subspace of Linearly Ordered Space
Let $\struct {S, \preceq, \tau}$ be a linearly ordered space. Let $Y \subseteq S$. Then $Y$ is connected in $\struct {S, \tau}$  both of the following hold: :$(1): \quad Y$ is convex in $S$ :$(2): \quad \struct {Y, \preceq \restriction_Y}$ is a linear continuum, where $\restriction$ denotes restriction.
Abnormal Subgroup is Self-Normalizing Subgroup
Let $G$ be a group. Let $H$ be an abnormal subgroup of $G$. Then $H$ is a self-normalizing subgroup of $G$.
Duality Principle (Category Theory)
In the study of metacategories and categories, the following two '''duality principles''' are very useful.
Duality Principle (Category Theory)/Conceptual Duality
Let $\Sigma$ be a statement about metacategories, be it in natural language or otherwise. Suppose that $\Sigma$ holds for all metacategories. Then so does its dual statement $\Sigma^*$.
Product of Complex Number with Conjugate
Let $z = a + i b \in \C$ be a complex number. Let $\overline z$ denote the complex conjugate of $z$. Then: :$z \overline z = a^2 + b^2 = \cmod z^2$ and thus is wholly real.
Compact Subspace of Linearly Ordered Space/Lemma
Let $\left({X, \preceq, \tau}\right)$ be a linearly ordered space. Let $Y \subseteq X$ be a non-empty subset of $X$. Let $Y$ be a compact subspace of $\left({X, \tau}\right)$. Then $\left({Y, \preceq \restriction_Y}\right)$ is a complete lattice, where $\restriction$ denotes restriction.
Inclusion Mapping on Subgroup is Homomorphism
Let $\struct {G, \circ}$ be a group. Let $\struct {H, \circ_{\restriction H} }$ be a subgroup of $G$. Let $i: H \to G$ be the inclusion mapping from $H$ to $G$. Then $i$ is a group homomorphism.
Mapping to Identity is Unique Constant Homomorphism
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups whose identities are $e_G$ and $e_H$ respectively. Then there exists a unique constant mapping from $G$ to $H$ which is a homomorphism: :$\phi_{e_H}: G \to H: \forall g \in G: \map {\phi_{e_H} } g = e_H$
Subgroup is Superset of Conjugate iff Normal
Let $\struct {G, \circ}$ be a group. Let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ (by definition 1) : : $\forall g \in G: g \circ N \circ g^{-1} \subseteq N$ : $\forall g \in G: g^{-1} \circ N \circ g \subseteq N$
Union Distributes over Union/General Result
Let $\left\langle{\mathbb S_i}\right\rangle_{i \in I}$ be an $I$-indexed family of sets of sets. Then: :$\displaystyle \bigcup_{i \mathop \in I} \bigcup \mathbb S_i = \bigcup \bigcup_{i \mathop \in I} \mathbb S_i$
Subgroup is Subset of Conjugate iff Normal
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ (by definition 1) : :$\forall g \in G: N \subseteq g \circ N \circ g^{-1}$ :$\forall g \in G: N \subseteq g^{-1} \circ N \circ g$
Existence and Uniqueness of Generated Topology
Let $X$ be a set. Let $\SS \subseteq \powerset X$ be a subset of the power set of $X$. Then there exists a unique topology $\map \tau \SS$ on $X$ such that: :$(1): \quad \SS \subseteq \map \tau \SS$. :$(2): \quad$ For any topology $\TT$ on $X$, the implication $\SS \subseteq \TT \implies \map \tau \SS \subseteq \TT$ holds.
Equivalent Conditions for Cover by Collection of Subsets
Let $X$ be a set. Then the following conditions are equivalent for a subset $\mathcal C \subseteq \mathcal P \left({X}\right)$ of the power set of $X$: :$\left({1}\right): \quad$ $\mathcal C$ is a cover for $X$. :$\left({2}\right): \quad$ $\displaystyle X = \bigcup \mathcal C$. :$\left({3}\right): \quad$ $\displaystyle \exists \mathcal S \subseteq \mathcal C: X = \bigcup \mathcal S$.
Subgroup equals Conjugate iff Normal
:$\forall g \in G: g \circ N \circ g^{-1} = N$ :$\forall g \in G: g^{-1} \circ N \circ g = N$
Subgroup is Normal iff Contains Conjugate Elements
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ : :$\forall g \in G: \paren {n \in N \iff g \circ n \circ g^{-1} \in N}$ :$\forall g \in G: \paren {n \in N \iff g^{-1} \circ n \circ g \in N}$
Subgroup is Normal iff Left Cosets are Right Cosets
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ (by definition 1) : : Every right coset of $N$ in $G$ is a left coset or equivalently: : The right coset space of $N$ in $G$ equals its left coset space.
Symmetric Group has Non-Normal Subgroup
Let $S_n$ be the (full) symmetric group on $n$ elements, where $n \ge 3$. Then $S_n$ contains at least one subgroup which is not normal.
Inclusion Mapping on Subgroup is Monomorphism
Let $\struct {G, \circ}$ be a group. Let $\struct {H, \circ {\restriction_H} }$ be a subgroup of $G$. Let $i: H \to G$ be the inclusion mapping from $H$ to $G$. Then $i$ is a group monomorphism.
Inverse of Group Isomorphism is Isomorphism
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a mapping. Then $\phi$ is an isomorphism  $\phi^{-1}: \struct {H, *} \to \struct {G, \circ}$ is also an isomorphism.
Composite of Group Homomorphisms is Homomorphism
Let: :$\struct {G_1, \circ}$ :$\struct {G_2, *}$ :$\struct {G_3, \oplus}$ be groups. Let: :$\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ :$\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$ be homomorphisms. Then the composite of $\phi$ and $\psi$ is also a homomorphism.
Composite of Group Monomorphisms is Monomorphism
Let: :$\struct {G_1, \circ}$ :$\struct {G_2, *}$ :$\struct {G_3, \oplus}$ be groups. Let: :$\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ :$\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$ be monomorphisms. Then the composite of $\phi$ and $\psi$ is also a monomorphism.
Composite of Group Epimorphisms is Epimorphism
Let: :$\struct {G_1, \circ}$ :$\struct {G_2, *}$ :$\struct {G_3, \oplus}$ be groups. Let: :$\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ :$\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$ be (group) epimorphisms. Then the composite of $\phi$ and $\psi$ is also a (group) epimorphism.
Composite of Group Isomorphisms is Isomorphism
Let $\struct {G_1, \circ}$, $\struct {G_2, *}$ and $\struct {G_3, \oplus}$ be groups. Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ and $\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$ be group isomorphisms. Then the composite of $\psi$ with $\phi$ is also a group isomorphism.
Sum of Deviations from Mean
Let $x_1, x_2, \ldots, x_n$ be real data about some quantitative variable. Let $\overline x$ be the arithmetic mean of the above data. Then:  :$\displaystyle \sum_{i \mathop = 1}^n \left({x_i - \overline x}\right) = 0$
Third Isomorphism Theorem/Groups/Corollary 2
Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$ respectively. Let $\phi: G \to H$ be a group homomorphism. Let $K$ be the kernel of $\phi$. Let $N$ be a normal subgroup of $G$. Let $q: G \to \dfrac G N$ be the quotient epimorphism from $G$ to the quotient group $\dfrac G N$. Then $N \subseteq K$  there exists a group homomorphism $\psi: \dfrac G N \to H$ such that: :$\phi = \psi \circ q$ and: :$\dfrac G K \cong \dfrac {G / N} {K / N}$
Identity of Group Direct Product
Let $\struct {G \times H, \circ}$ be the group direct product of the two groups $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$. Let $e_G$ be the identity for $\struct {G, \circ_1}$. Let $e_H$ be the identity for $\struct {H, \circ_2}$. Then $\tuple {e_G, e_H}$ is the identity for $\struct {G \times H, \circ}$.
Inverses in Group Direct Product
Let $\struct {G \times H, \circ}$ be the group direct product of the two groups $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$. Let $g^{-1}$ be an inverse of $g \in \struct {G, \circ_1}$. Let $h^{-1}$ be an inverse of $h \in \struct {H, \circ_2}$. Then $\tuple {g^{-1}, h^{-1} }$ is the inverse of $\tuple {g, h} \in \struct {G \times H, \circ}$.
Associativity of Operation in Group Direct Product
Let $\struct {G \times H, \circ}$ be the group direct product of the two groups $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$. Then the operation $\circ$ in $\struct {G \times T, \circ}$ is associative.
Anticommutativity of External Direct Product
Let $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ be algebraic structures where $S$ and $T$ both have at least two distinct elements. Let $\left({S \times T, \circ}\right)$ be their external direct product. Then $\left({S \times T, \circ}\right)$ is anticommutative iff at least one of $\left({S, \circ_1}\right)$ and $\left({T, \circ_2}\right)$ is.
Projection on Group Direct Product is Epimorphism
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups. Let $\struct {G, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$. Then: :$\pr_1$ is an epimorphism from $\struct {G, \circ}$ to $\struct {G_1, \circ_1}$ :$\pr_2$ is an epimorphism from $\struct {G, \circ}$ to $\struct {G_2, \circ_2}$ where $\pr_1$ and $\pr_2$ are the first and second projection respectively of $\struct {G, \circ}$.
Canonical Injection on Group Direct Product is Monomorphism
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identities $e_1, e_2$ respectively. Let $\struct {G_1 \times G_2, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ Then the canonical injections: :$\inj_1: \struct {G_1, \circ_1} \to \struct {G_1, \circ_1} \times \struct {G_2, \circ_2}: \forall x \in G_1: \map {\inj_1} x = \tuple {x, e_2}$ :$\inj_2: \struct {G_2, \circ_2} \to \struct {G_1, \circ_1} \times \struct {G_2, \circ_2}: \forall x \in G_2: \map {\inj_2} x = \tuple {e_1, x}$ are group monomorphisms.
Canonical Injection is Right Inverse of Projection
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identity elements $e_1$ and $e_2$ respectively. Let $\struct {G_1 \times G_2, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ Let: :$\pr_1: \struct {G_1 \times G_2, \circ} \to \struct {G_1, \circ_1}$ be the first projection from $\struct {G_1 \times G_2, \circ}$ to $\struct {G_1, \circ_1}$ :$\pr_2: \struct {G_1 \times G_2, \circ} \to \struct {G_2, \circ_2}$ be the second projection from $\struct {G_1 \times G_2, \circ}$ to $\struct {G_2, \circ_2}$. Let: :$\inj_1: \struct {G_1, \circ_1} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_1, \circ_1}$ to $\struct {G_1 \times G_2, \circ}$ :$\inj_2: \struct {G_2, \circ_2} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_2, \circ_2}$ to $\struct {G_1 \times G_2, \circ}$. Then: :$(1): \quad \pr_1 \circ \inj_1 = I_{G_1}$ :$(2): \quad \pr_2 \circ \inj_2 = I_{G_2}$ where $I_{G_1}$ and $I_{G_2}$ are the identity mappings on $G_1$ and $G_2$ respectively.
Image of Canonical Injection is Kernel of Projection
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identity elements $e_1$ and $e_2$ respectively. Let $\struct {G_1 \times G_2, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ Let: :$\pr_1: \struct {G_1 \times G_2, \circ} \to \struct {G_1, \circ_1}$ be the first projection from $\struct {G_1 \times G_2, \circ}$ to $\struct {G_1, \circ_1}$ :$\pr_2: \struct {G_1 \times G_2, \circ} \to \struct {G_2, \circ_2}$ be the second projection from $\struct {G_1 \times G_2, \circ}$ to $\struct {G_2, \circ_2}$. Let: :$\inj_1: \struct {G_1, \circ_1} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_1, \circ_1}$ to $\struct {G_1 \times G_2, \circ}$ :$\inj_2: \struct {G_2, \circ_2} \to \struct {G_1 \times G_2, \circ}$ be the canonical injection from $\struct {G_2, \circ_2}$ to $\struct {G_1 \times G_2, \circ}$. Then: :$(1): \quad \Img {\inj_1} = \map \ker {\pr_2}$ :$(2): \quad \Img {\inj_2} = \map \ker {\pr_1}$ That is: :the image of the (first) canonical injection is the kernel of the second projection :the image of the (second) canonical injection is the kernel of the first projection.
Definition:Direct Product of Group Homomorphisms
Let $G, H_1$ and $H_2$ be groups. Let $f_1: G \to H_1$ and $f_2: G \to H_2$ be group homomorphisms. Then $f_1 \times f_2: G \to H_1 \times H_2$, defined as: :$\forall g \in G: \map{\paren {f_1 \times f_2} } g = \tuple {\map {f_1} g, \map {f_2} g}$ is called the '''direct product of $f_1$ and $f_2$'''.
Direct Product of Group Homomorphisms is Homomorphism
Let $\struct {G, \circ}, \struct {H_1, *_1}$ and $\struct {H_2, *_2}$ be groups. Let $\struct {H_1 \times H_2, *}$ be the group direct product of $H_1$ and $H_2$. Let $f_1: G \to H_1$ and $f_2: G \to H_2$ be group homomorphisms. Let $f_1 \times f_2: g \to H_1 \times H_2$ be the direct product of $f_1$ and $f_2$. Then $f_1 \times f_2$ is a group homomorphism.
Projections on Direct Product of Group Homomorphisms
Let $G, H_1$ and $H_2$ be groups. Let $H_1 \times H_2$ be the group direct product of $H_1$ and $H_2$. Let $f_1: G \to H_1$ and $f_2: G \to H_2$ be group homomorphisms. Let $f_1 \times f_2: g \to H_1 \times H_2$ be the direct product of $f_1$ and $f_2$. Let: :$\pr_1: H_1 \times H_2 \to H_1$ be the first projection from $H_1 \times H_2$ to $H_1$ :$\pr_2: H_1 \times H_2 \to H_2$ be the second projection from $H_1 \times H_2$ to $H_2$. Then: :$(1) \quad \pr_1 \circ \paren {f_1 \times f_2} = f_1$ :$(2) \quad \pr_2 \circ \paren {f_1 \times f_2} = f_2$ where $\circ$ is the operation of composition of mappings.
Pointwise Addition is Associative
Let $S$ be a non-empty set. Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$. Let $f, g, h: S \to \mathbb F$ be functions. Let $f + g: S \to \mathbb F$ denote the pointwise sum of $f$ and $g$. Then: :$\paren {f + g} + h = f + \paren {g + h}$ That is, pointwise addition is associative.
Pointwise Addition on Complex-Valued Functions is Associative
Let $f, g, h: S \to \C$ be complex-valued functions. Let $f + g: S \to \C$ denote the pointwise sum of $f$ and $g$. Then: :$\paren {f + g} + h = f + \paren {g + h}$
Pointwise Addition is Commutative
Let $S$ be a non-empty set. Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$. Let $f, g: S \to \mathbb F$ be functions. Let $f + g: S \to \mathbb F$ denote the pointwise sum of $f$ and $g$. Then: :$f + g = g + f$  That is, pointwise addition is commutative.
Pointwise Multiplication is Associative
Let $S$ be a non-empty set. Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$. Let $f, g, h: S \to \mathbb F$ be functions. Let $f \times g: S \to \mathbb F$ denote the pointwise product of $f$ and $g$. Then: :$\paren {f \times g} \times h = f \times \paren {g \times h}$ That is, pointwise multiplication is associative.
Pointwise Multiplication is Commutative
Let $S$ be a non-empty set. Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$. Let $f, g, h: S \to \mathbb F$ be functions. Let $f \times g: S \to \mathbb F$ denote the pointwise product of $f$ and $g$. Then: :$f \times g = g \times f$ That is, pointwise multiplication is commutative.
Pointwise Multiplication on Real-Valued Functions is Associative
Let $f, g, h: S \to \R$ be real-valued functions. Let $f \times g: S \to \R$ denote the pointwise product of $f$ and $g$. Then: :$\left({f \times g}\right) \times h = f \times \left({g \times h}\right)$
Pointwise Multiplication on Rational-Valued Functions is Associative
Let $f, g, h: S \to \Q$ be rational-valued functions. Let $f \times g: S \to \Q$ denote the pointwise product of $f$ and $g$. Then: :$\paren {f \times g} \times h = f \times \paren {g \times h}$
Set Equation: Union
Let $A$ and $B$ be sets. Consider the set equation: :$A \cup X = B$ The solution set of this is: :$\varnothing$ if $A \nsubseteq B$ :$\left\{ {\left({B \setminus A}\right) \cup Y: Y \subseteq A}\right\}$ otherwise.
Set Equation: Intersection
Let $A$ and $B$ be sets. Consider the set equation: :$A \cap X = B$ The solution set of this is: :$\O$ if $B \nsubseteq A$ :$\set {B \cup Y: A \nsubseteq Y}$ otherwise.
Non-Abelian Group has Order Greater than 4
Let $\struct {G, \circ}$ be a non-abelian group. Then the order of $\struct {G, \circ}$ is greater than $4$.
Bayes' Theorem/General Result
Let $\set {B_1, B_2, \ldots}$ be a  partition of the event space $\Sigma$. Then, for any $B_i$ in the partition: :$\map \Pr {B_i \mid A} = \dfrac {\map \Pr {A \mid B_i} \map \Pr {B_i} } {\map \Pr A} = \dfrac {\map \Pr {A \mid B_i} \map \Pr {B_i} } {\sum_j \map \Pr {A \mid B_j} \map \Pr {B_j} }$ where $\ds \sum_j$ denotes the sum over $j$.
Topological Sum is Coproduct in Category of Topological Spaces
Let $\mathbf{Top}$ be the category of topological spaces. Let $X$ and $Y$ be topological spaces, and let $X \sqcup Y$ be their topological sum. Then $X \sqcup Y$ is the coproduct of $X$ and $Y$ in $\mathbf{Top}$.
Coproduct of Ordered Sets
Let $\mathbf{OrdSet}$ be the category of ordered sets. Let $\left({P, \preceq_1}\right)$ and $\left({Q, \preceq_2}\right)$ be ordered sets. Let $P \sqcup Q$ be the disjoint union of $P$ and $Q$. Let $\preceq$ be the ordering on $P \sqcup Q$ defined by: :$\left({x, i}\right) \preceq \left({y, j}\right)$  $i = j$ and $x \preceq_i y$ where $i = 1$ or $i = 2$ depending on whether $x,y \in P$ or $x,y \in Q$. Then $\left({P \sqcup Q, \preceq}\right)$ is the coproduct of $P$ and $Q$ in $\mathbf{OrdSet}$.
Supremum is Coproduct in Order Category
Let $\mathbf P$ be an order category with ordering $\preceq$. Let $p, q \in P_0$, and suppose they have some supremum $r = \sup \left\{{p, q}\right\}$. Then $r$ is the coproduct of $p$ and $q$ in $\mathbf P$.
Real Function is Continuous at Isolated Point
Let $A \subseteq \R$ be any subset of the real numbers. Let $f: A \to \R$ be a real function. Let $x \in A$ be an isolated point of $A$. Then $f$ is continuous at $x$.
Metric Space Continuity by Epsilon-Delta
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \in A_1$ be a point in $A_1$. Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent:
Equalizer is Monomorphism
Let $\mathbf C$ be a metacategory. Let $e: E \to C$ be the equalizer of two morphisms $f, g: C \to D$. Then $e$ is a monomorphism.
Metric Space Continuity by Open Ball
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \in A_1$ be a point in $A_1$. Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent:
Mapping from Standard Discrete Metric on Real Number Line is Continuous
Let $\R$ be the real number line. Let $\left({\R, d_1}\right)$ be the metric space such that $d_1$ be the Euclidean metric on $\R$. Let $\left({\R, d_2}\right)$ be the metric space such that $d_2$ be the standard discrete metric on $\R$. Let $f: \left({\R, d_2}\right) \to \left({\R, d_1}\right)$ be a real function. Then $f$ is $\left({d_2, d_1}\right)$-continuous on $\R$.
Properties of Affine Spaces
Let $\mathcal E$ be an affine space with difference space $V$. Let $0$ denote the zero element of $V$. Then the following hold for all $p,q,r \in \mathcal E$ and all $u,v \in V$: : $(1): \quad p - p = 0$ : $(2): \quad p + 0 = p$ : $(3): \quad p + u = p + v \iff u = v$ : $(4): \quad q - p = r - p \iff q = r$
P-Product Metric on Real Vector Space is Metric/Proof 1
Let $\R^n$ be an $n$-dimensional real vector space. Let $p \in \R_{\ge 1}$. Let $d_p: \R^n \times \R^n \to \R$ be the $p$-product metric on $\R^n$: : $\displaystyle d_p \left({x, y}\right) :=  \left({\sum_{i \mathop = 1}^n \left \vert {x_i - y_i} \right \vert^p}\right)^{\frac 1 p}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$. Then $d_p$ is a metric.