Transfinite Induction/Principle 2

Let $A$ be a class satisfying the following conditions: * $\varnothing \in A$ * $\forall x \in A: x^+ \in A$ * If $y$ is a limit ordinal, then $\left({ \forall x < y: x \in A }\right) \implies y \in A$ where $x^+$ denotes the successor of $x$. Then $\operatorname{On} \subseteq A$.


Transfinite Induction/Schema 2

Let $\phi \left({x}\right)$ be a property satisfying the following conditions: :$(1): \quad \phi \left({\varnothing}\right)$ is true :$(2): \quad$ If $x$ is an ordinal, then $\phi \left({x}\right) \implies \phi \left({x^+}\right)$ :$(3): \quad$ If $y$ is a limit ordinal, then $\left({\forall x < y: \phi \left({x}\right)}\right) \implies \phi \left({y}\right)$ where $x^+$ denotes the successor of $x$. Then, $\phi \left({x}\right)$ is true for all ordinals $x$.


Transfinite Recursion/Theorem 2

Let $\Dom x$ denote the domain of $x$. Let $\Img x$ denote the image of the mapping $x$. {{explain|We infer that $x$ is a mapping, but what is its context?}} Let $G$ be a class of ordered pairs $\tuple {x, y}$ satisfying at least one of the following conditions: :$(1): \quad x = \O$ and $y = a$ {{explain|What is $a$?}} :$(2): \quad \exists \beta: \Dom x = \beta^+$ and $y = \map H {\map x {\bigcup \Dom x} }$ {{explain|What is $H$?}} :$(3): \quad \Dom x$ is a limit ordinal and $y = \bigcup \Rng x$. {{explain|Is this invoking well-founded recursion?}} Let $\map F \alpha = \map G {F \restriction \alpha}$ for all ordinals $\alpha$. Then: :$F$ is a mapping and the domain of $F$ is the ordinals, $\On$. :$\map F \O = a$ :$\map F {\beta^+} = \map H {\map F \beta}$ :For limit ordinals $\beta$, $\displaystyle \map F \beta = \bigcup_{\gamma \mathop \in \beta} \map F \gamma$ :$F$ is unique. ::That is, if there is another function $A$ satisfying the above three properties, then $A = F$.


Transfinite Recursion/Corollary

Let $x$ be an ordinal. Let $G$ be a mapping There exists a unique mapping $f$ that satisfies the following properties: : The domain of $f$ is $x$ : $\forall y \in x: f \left({y}\right) = G \left({f \restriction y}\right)$


Inverse of Product/Monoid/General Result

Let $\left({S, \circ}\right)$ be a monoid whose identity is $e$. Let $a_1, a_2, \ldots, a_n \in S$ be invertible for $\circ$, with inverses $a_1^{-1}, a_2^{-1}, \ldots, a_n^{-1}$. Then $a_1 \circ a_2 \circ \cdots \circ a_n$ is invertible for $\circ$, and: :$\forall n \in \N_{> 0}: \left({a_1 \circ a_2 \circ \cdots \circ a_n}\right)^{-1} = a_n^{-1} \circ \cdots \circ a_2^{-1} \circ a_1^{-1}$


Inverse of Group Product

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $a, b \in G$, with inverses $a^{-1}, b^{-1}$. Then: :$\paren {a \circ b}^{-1} = b^{-1} \circ a^{-1}$


Inverse of Group Product/General Result

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $a_1, a_2, \ldots, a_n \in G$, with inverses $a_1^{-1}, a_2^{-1}, \ldots, a_n^{-1}$. Then: : $\paren {a_1 \circ a_2 \circ \cdots \circ a_n}^{-1} = a_n^{-1} \circ \cdots \circ a_2^{-1} \circ a_1^{-1}$


Well-Ordered Transitive Subset is Equal or Equal to Initial Segment

Let $\left({\prec, A}\right)$ be a well-ordered set. For every $x \in A$, let every $\prec$-initial segment $A_x$ be a set. Let $B$ be a subclass of $A$ such that :$\forall x \in A: \forall y \in B: \left({x \prec y \implies x \in B}\right)$. That is, $B$ must be $\prec$-transitive. Then: :$A = B$ or: :$\exists x \in A: B = A_x$


Inverse of Inverse/General Algebraic Structure

Let $\struct {S, \circ}$ be an algebraic structure with an identity element $e$. Let $x \in S$ be invertible, and let $y$ be an inverse of $x$. Then $x$ is also an inverse of $y$.


Inverse of Group Inverse

Let $\struct {G, \circ}$ be a group. Let $g \in G$, with inverse $g^{-1}$. Then: :$\paren {g^{-1} }^{-1} = g$


Condition for Injective Mapping on Ordinals

Let $F$ be a mapping satisfying the following properties: :$(1): \quad$ The domain of $F$ is $\On$, the ordinal class :$(2): \quad$ For all ordinals $x$, $\map F x = \map G {F \restriction x}$ :$(3): \quad$ For all ordinals $x$, $\map G {F \restriction x} \in \paren {A \setminus \Img x}$ where $\Img x$ is the image of $x$ under $F$. Let $\Img F$ denote the image of $F$. Then the following properties hold: :$(1): \quad \Img F \subseteq A$ :$(2): \quad F$ is injective :$(3): \quad A$ is a proper class. Note that only the third property of $F$ is the most important. For any function $G$, a function $F$ can be constructed satisfying the first two using transfinite recursion. {{explain|Exactly which of the above statements is the "third" is unclear. This statement needs to be made precise. It also needs to be explained. Also, it needs to be put in a separate section as it is peripheral to the statement of the theorem.}}


Maximal Injective Mapping from Ordinals to a Set

Let $F$ be a mapping satisfying the following properties: {{explain|What is $G$?}} : The domain of $F$ is $\operatorname{On}$, the ordinal class : For all ordinals $x$, $F \left({x}\right) = G \left({F \restriction x}\right)$. : For all ordinals $x$, if $(A \setminus \operatorname{Im} \left({x}\right) ) \ne \varnothing$, then $G  \left({F \restriction x}\right) \in (A \setminus \operatorname{Im} \left({x}\right))$ where $\operatorname{Im} \left({x}\right)$ is the image of the subset $x$ under $F$. : $A$ is a set. Then there exists an ordinal $y$ satisfying the following properties: : $\forall x \in y: \left({A \setminus \operatorname{Im} \left({x}\right)}\right) \ne \varnothing$ : $\operatorname {Im} \left({y}\right) = A$ : $F \restriction y$ is an injective mapping. Note that the first third and fourth properties of $F$ are the most important.  For any mapping $G$, a mapping $F$ can be constructed satisfying the first two properties using transfinite recursion.


Power of Element/Semigroup

Let $\struct {S, \oplus}$ be a magma. Let $a \in S$. Let $n \in \N_{>0}$. Let $\tuple {a_1, a_2, \ldots, a_n}$ be the ordered $n$-tuple defined by $a_k = a$ for each $k \in \N_n$. Then: :$\displaystyle \bigoplus_{k \mathop = 1}^n a_k = \oplus^n a$ where: :$\displaystyle \bigoplus_{k \mathop = 1}^n a_k$ is the composite of $\tuple {a_1, a_2, \ldots, a_n}$ for $\oplus$ :$\oplus^n a$ is the $n$th power of $a$ under $\oplus$.


Order Isomorphism between Ordinals and Proper Class

Let $\left({A, \prec}\right)$ be a strict well-ordering. Let $A$ be a proper class. Let the initial segment of $x$ be a set for every $x \in A$. Then we may make the following definitions: Set $G$ equal to the collection of ordered pairs $\left({x, y}\right)$ such that: :$y \in \left({A \setminus \operatorname{Im} \left({x}\right)}\right)$ :$\left({A \setminus \operatorname{Im}\left({x}\right)}\right) \cap A_y = \varnothing$ Use transfinite recursion to construct a mapping $F$ such that: : The domain of $F$ is $\operatorname{On}$ : For all ordinals $x$, $F \left({x}\right) = G \left({F {\restriction_x} }\right)$ Then $F: \operatorname{On} \to A$ is an order isomorphism between $\left({\operatorname{On}, \in}\right)$ and $\left({A, \prec}\right)$.


Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping

Let $S$ be a set. Let $\left({S, \prec}\right)$ be a strict well-ordering. Then there exists a unique ordinal $x$ and unique mapping $f$ such that $f: x \to S$ is an order isomorphism.


Unique Isomorphism between Ordinal Subset and Unique Ordinal

Let $\operatorname{On}$ be the class of ordinals. Let $S \subset \operatorname{On}$ where $S$ is a set. Then there exists a unique mapping $\phi$ and a unique ordinal $x$ such that $\phi : x \to S$ is an order isomorphism.


Lexicographic Order forms Well-Ordering on Ordered Pairs of Ordinals

The Lexicographic Order $\operatorname{Le}$ is a strict well-ordering on $\left({\operatorname{On} \times \operatorname{On}}\right)$.


Canonical Order Well-Orders Ordered Pairs of Ordinals

The canonical order, $R_0$ strictly well-orders the ordered pairs of ordinal numbers.


Initial Segment of Canonical Order is Set

Let $R_0$ denote the canonical ordering of $\left({\operatorname{On} \times \operatorname{On} }\right)$. Then, for all $\left({x, y}\right) \in \left({\operatorname{On} \times \operatorname{On} }\right)$, the $R_0$-initial segment is a set. {{explain|The $R_0$-initial segment of what?}}


Ordinal Addition is Closed

Let $\operatorname{On}$ be the class of all ordinals. Then: :$\forall x, y \in \operatorname{On}: \left({x + y}\right) \in \operatorname{On}$ That is: the sum $x+y$ is an ordinal.


Union of Ordinals is Ordinal

Let $y$ be a set. Let $F \left({z}\right)$ be a mapping such that: :$\displaystyle \forall z \in y: F \left({z}\right) \in \operatorname{On}$ Then: :$\displaystyle \bigcup_{z \mathop \in y} F \left({z}\right) \in \operatorname{On}$


Ordinal Addition by Zero

Let $x$ be an ordinal. Let $\varnothing$ be the zero ordinal. Then: :$x + \varnothing = x = \varnothing + x$ where $+$ denotes ordinal addition.


Union of Limit Ordinal

Let $x$ be a limit ordinal. Then: :$\displaystyle x = \bigcup x$ :$\displaystyle x = \bigcup_{y \mathop \in x} y$


Substitutivity of Equality

Let $x$ and $y$ be sets. Let $\map P x$ be a well-formed formula of the language of set theory. Let $\map P y$ be the same proposition $\map P x$ with some (not necessarily all) free instances of $x$ replaced with free instances of $y$. Let $=$ denote set equality. :$x = y \implies \paren {\map P x \iff \map P y}$


Membership is Left Compatible with Ordinal Addition

Let $x$, $y$, and $z$ be ordinals. Let $<$ denote membership $\in$, since $\in$ is a strict well-ordering on the ordinals. Then: :$x < y \implies \paren {z + x} < \paren {z + y}$


Ordinal Addition is Left Cancellable

Let $x, y, z$ be ordinals. Then: :$\left({z + x}\right) = \left({z + y}\right) \implies x = y$ That is, ordinal addition is left cancellable.


Supremum Inequality for Ordinals

Let $A \subseteq \operatorname{On}$ and $B \subseteq \operatorname{On}$ be ordinals. Then: :$\displaystyle \forall x \in A: \exists y \in B: x \le y \implies \bigcup A \le \bigcup B$


Subset is Right Compatible with Ordinal Addition

Let $x, y, z$ be ordinals. Then: :$x \le y \implies \left({x + z}\right) \le \left({y + z}\right)$


Ordinal Subtraction when Possible is Unique

Let $x$ and $y$ be ordinals such that $x \le y$. Then there exists a unique ordinal $z$ such that $\paren {x + z} = y$. That is: :$x \le y \implies \exists! z \in \On: \paren {x + z} = y$


Equality of Successors

Let $x$ and $y$ be ordinals. Let $x^+$ denote the successor set of $x$. Then, $x = y \iff x^+ = y^+$


Finite Ordinal Plus Transfinite Ordinal

Let $n$ be a finite ordinal. Let $x$ be a transfinite ordinal. Then: : $n + x = x$


Class is Transitive iff Union is Subset

A class $A$ is transitive {{iff}}: :$\displaystyle \bigcup A \subseteq A$


Union of Successor Ordinal

Let $x$ be an ordinal. Let $x^+$ denote the successor of $x$. Then: :$\displaystyle \map \bigcup {x^+} = x$


Union Distributes over Union

Set union is distributive over itself: :$\forall A, B, C: \paren {A \cup B} \cup \paren {A \cup C} = A \cup B \cup C = \paren {A \cup C} \cup \paren {B \cup C}$ where $A, B, C$ are sets.


Subset of Empty Set

Let $A$ be a class. Then: : $A$ is a subset of the empty set $\O$ {{iff}}; : $A$ is equal to the empty set: :$\displaystyle A \subseteq \O \iff A = \O$


Ordinal is Subset of Successor

Let $x$ and $y$ be ordinals. Let $x^+$ denote the successor of $x$. Then: : $x \subseteq y^+ \iff \left({x \subseteq y \lor x = y^+}\right)$


Indexed Union Equality

Let $A$, $B_x$, and $C_x$ be classes. Then: :$\displaystyle \forall x \in A: B_x = C_x \implies \bigcup_{x \mathop \in A} B_x = \bigcup_{x \mathop \in A} C_x$ {{MissingLinks|$\bigcup_{x \mathop \in A} B_x$ and $\bigcup_{x \mathop \in A} C_x$}}


Indexed Union Subset

Let $A$, $B_x$ and $C_x$ be classes. {{explain|It can be inferred from the context, but the meaning of the subscript on $B_x$ and $C_x$ needs to be explained.}} Then: :$\displaystyle \forall x \in A: B_x \subseteq C_x \implies \bigcup_{x \mathop \in A} B_x \subseteq \bigcup_{x \mathop \in A} C_x$ {{MissingLinks|$\bigcup_{x \mathop \in A} B_x$ and $\bigcup_{x \mathop \in A} C_x$}}


Ordinal is Less than Sum

Let $x$ and $y$ be ordinals. Then: :$x \le \left({x + y}\right)$ :$x \le \left({y + x}\right)$


Limit Ordinals Preserved Under Ordinal Addition

Let $x$ and $y$ be ordinals such that $x$ is a limit ordinal. Then $\left({y + x}\right)$ is a limit ordinal. That is, letting $K_{II}$ denote the class of all limit ordinals: :$\forall x \in K_{II}: \left({y + x}\right) \in K_{II}$


Successor in Limit Ordinal

Suppose that $x$ is a limit ordinal and that $y \in x$. Then $y^+ \in x$ where $y^+$ denotes the successor set of $y$: :$\forall y \in x: y^+ \in x$


Successor is Less than Successor

Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$. Then, $x \in y \iff x^+ \in y^+$.


Ordinal Addition is Associative

Ordinal addition is associative, i.e.: :$\left({x + y}\right) + z = x + \left({y + z}\right)$ holds for all ordinals $x$, $y$ and $z$.


Derivative of Composite Function/Corollary

:$\dfrac {\d y} {\d x} = \dfrac {\paren {\dfrac {\d y} {\d u} } } {\paren {\dfrac {\d x} {\d u} } }$ for $\dfrac {\d x} {\d u} \ne 0$.


Product of Subset with Intersection/Corollary

Let $\struct {G, \circ}$ be a group. Let $X, Y, Z \subseteq G$ such that $X$ is a singleton. Then: :$X \circ \paren {Y \cap Z} = \paren {X \circ Y} \cap \paren {X \circ Z}$ :$\paren {Y \cap Z} \circ X = \paren {Y \circ X} \cap \paren {Z \circ X}$ where $X \circ Y$ denotes the subset product of $X$ and $Y$.


Single Instruction URM Programs/Zero Function

The zero function $\Zero: \N \to \N$, defined as: :$\forall n \in \N: \map \Zero n = 0$


Single Instruction URM Programs/Successor Function

The successor function $\Succ: \N \to \N$, defined as: :$\forall n \in \N: \map \Succ n = n + 1$


Single Instruction URM Programs/Projection Function

The projection functions $\pr_j^k: \N^k \to \N$, defined as: :$\forall j \in \closedint 1 k: \forall \tuple {n_1, n_2, \ldots, n_k} \in \N^k: \map {\pr_j^k} {n_1, n_2, \ldots, n_k} = n_j$


Antiassociative Operation has no Idempotent Elements

Let $\struct {S, \circ}$ be an algebraic structure. Let $\circ$ be antiassociative on $S$. Then no element of $S$ is idempotent under $ \circ$. That is: :$\forall x \in S: x \circ x \ne x$


Antiassociative Operation is not Commutative

Let $\left({S, \circ}\right)$ be an algebraic structure. Let $\circ$ be antiassociative on $S$. Then $\circ$ is not commutative on $S$.


Antiassociative Structure of Finite Order

Let $n \in \N$ such that $n > 2$. Then there exists an algebraic structure $\left({S, \circ}\right)$ of order $n$ such that $\circ$ is antiassociative on $S$.


Example:Antiassociative Structure

Let $\left({\R_{>0}, \circ}\right)$ be an algebraic structure where $\R_{>0}$ denotes the strictly positive real numbers. Define: :$\forall x, y \in \R_{>0}: x \circ y = x y + y$ Then $\circ$ is antiassociative on $\R_{>0}$: :$\forall x, y, z \in \R_{>0}: \left({x \circ y}\right) \circ z \ne x \circ \left({y \circ z}\right)$


Reciprocal Sequence is Strictly Decreasing

The reciprocal sequence: :$\sequence {\operatorname {recip} }: \N_{>0} \to \R$: $n \mapsto \dfrac 1 n$ is strictly decreasing.


Two-Step Vector Subspace Test

Let $V$ be a vector space over a division ring $K$. Let $U \subseteq V$ be a non-empty subset of $V$ such that: :$(1): \qquad \forall u \in U, \lambda \in K: \lambda u \in U$ :$(2): \qquad \forall u, v \in U: u + v \in U$ Then $U$ is a subspace of $V$.


Reciprocal Function is Strictly Decreasing

The reciprocal function: :$\operatorname{recip}: \R \setminus \set 0 \to \R$, $x \mapsto \dfrac 1 x$ is strictly decreasing: :on the open interval $\openint 0 \to$ :on the open interval $\openint \gets 0$


Coefficients of Polynomial Product

Let $J$ be a set. Let $p_1, \ldots p_n$ be polynomial forms in the indeterminates $\set {X_j : j \in J}$ over a commutative ring $R$. Suppose that for each $i$ with $1 \le i \le n$, we have, for appropriate $a_{i, k} \in R$: :$p_i = \displaystyle \sum_{k \mathop \in Z} a_{i, k} X^k$ where $Z$ comprises the multiindices of natural numbers over $J$. Then: :$\displaystyle \prod_{i \mathop = 1}^n p_i = \displaystyle \sum_{k \mathop \in Z} b_k X^k$ where: :$\displaystyle b_k := \sum_{k_1 + \cdots + k_n \mathop = k} \paren {\prod_{i \mathop = 1}^n a_{i, k_i} }$


Subset Product of Subgroups/Necessary Condition

Let $\left({G, \circ}\right)$ be a group. Let $H, K$ be subgroups of $G$. Let $H \circ K$ be a subgroup of $G$. Then $H$ and $K$ are permutable. That is: :$H \circ K = K \circ H$ where $H \circ K$ denotes subset product.


Subset Product of Subgroups/Sufficient Condition

Let $\struct {G, \circ}$ be a group. Let $H, K$ be subgroups of $G$. Let $H$ and $K$ be permutable subgroups of $G$. That is, suppose: :$H \circ K = K \circ H$ where $H \circ K$ denotes subset product. Then $H \circ K$ is a subgroup of $G$.


Characteristics of Eulerian Graph/Necessary Condition

Let $G$ be a finite (undirected) graph. Let $G$ be Eulerian. Then $G$ is connected and each vertex of $G$ is even. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph.


Characteristics of Eulerian Graph/Sufficient Condition/Proof 1

Let $G$ be a finite (undirected) graph which is connected Let each vertex of $G$ be even. Then $G$ is an Eulerian graph. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph.


Characteristics of Eulerian Graph/Sufficient Condition/Proof 2

Let $G$ be a finite (undirected) graph which is connected. Let each vertex of $G$ be even. Then $G$ is an Eulerian graph. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph but does not include directed graph.


Characteristics of Eulerian Graph/Sufficient Condition


Characterization of Pre-Measures

Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Let $\varnothing \in \mathcal S$. Denote $\overline \R_{\ge 0}$ for the set of positive extended real numbers. A mapping $\mu: \mathcal S \to \overline \R_{\ge 0}$ is a pre-measure {{iff}}: :$(1):\quad \mu \left({\varnothing}\right) = 0$ :$(2):\quad \mu$ is finitely additive :$(3):\quad$ For every increasing sequence $\left({E_n}\right)_{n \in \N}$ in $\mathcal S$, if $E_n \uparrow E$ for some $E \in \mathcal S$, then: ::$\mu \left({E}\right) = \displaystyle \lim_{n \to \infty} \mu \left({E_n}\right)$ where $E_n \uparrow E$ denotes limit of increasing sequence of sets. Alternatively, and equivalently, $(3)$ may be replaced by either of: :$(3'):\quad$ For every decreasing sequence $\left({E_n}\right)_{n \in \N}$ in $\mathcal S$ for which $\mu \left({E_1}\right)$ is finite, if $E_n \downarrow E$ for some $E \in \mathcal S$, then: ::$\mu \left({E}\right) = \displaystyle \lim_{n \to \infty} \mu \left({E_n}\right)$ :$(3''):\quad$ For every decreasing sequence $\left({E_n}\right)_{n \in \N}$ in $\mathcal S$ for which $\mu \left({E_1}\right)$ is finite, if $E_n \downarrow \varnothing$, then: ::$\displaystyle \lim_{n \to \infty} \mu \left({E_n}\right) = 0$ where $E_n \downarrow E$ denotes limit of decreasing sequence of sets.


Power Set is Magma of Sets

Let $X$ be a set. Let $\family {\phi_i}: i \in I$ be an indexed family of mappings. Then $\powerset X$, the power set of $X$, is a magma of sets for $\family {\phi_i}: i \in I$ on $X$.


Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets

Let $X$ be a set, and let $\Phi := \left\{{\phi_i: i \in I}\right\}$ be a collection of partial mappings with codomain $\mathcal P \left({X}\right)$, the power set of $X$. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then the magma of sets generated by $\mathcal G$ exists and is unique.


Intersection of Magmas of Sets is Magma of Sets

Let $X$ be a set. Let $\Phi := \set {\phi_i: i \in I}$ be a collection of partial mappings with codomain $\powerset X$, the power set of $X$. Let $\mathcal S_j$ be a magma of sets for $\Phi$, for each $j \in J$, for some index set $J$. Then: :$\mathcal S := \displaystyle \bigcap_{j \mathop \in J} \mathcal S_j$ is also a magma of sets for $\Phi$.


Sigma-Algebra as Magma of Sets

The concept of $\sigma$-algebra is an instance of a magma of sets.


Ordinal Multiplication is Closed

Let $x$ and $y$ be ordinals. Let $\operatorname{On}$ denote the ordinal class. :$\left({x \cdot y}\right) \in \operatorname{On}$


Natural Number Multiplication is Closed

Let $m$ and $n$ be natural numbers. Then: :$m \times n \in \N$ where $\times$ denotes natural number multiplication.


Ordinal Multiplication by Zero

Let $x$ be an ordinal. {{begin-eqn}} {{eqn | l = \left({x \cdot \varnothing}\right)       | r = \varnothing       | c =  }} {{eqn | l = \left({\varnothing \cdot x}\right)       | r = \varnothing       | c =  }} {{end-eqn}}


Ordinal Multiplication by One

Let $x$ be an ordinal. Let $1$ denote the successor of $\varnothing$. {{begin-eqn}} {{eqn|l = \left({x \cdot 1}\right)      |r = x      |c =  }} {{eqn|l = \left({1 \cdot x}\right)      |r = x      |c =  }} {{end-eqn}}


Ordinal Addition by One

Let $x$ be an ordinal. Let $x^+$ denote the successor of $x$. Let $1$ denote (ordinal) one, the successor of the zero ordinal $\O$. Then: :$x + 1 = x^+$ where $+$ denotes ordinal addition.


Closed Set Measurable in Borel Sigma-Algebra

Let $\left({S, \tau}\right)$ be a topological space, and let $\mathcal B \left({\tau}\right)$ be the associated Borel $\sigma$-algebra. Let $C$ be a closed set in $S$. Then $C$ is $\mathcal B \left({\tau}\right)$-measurable.


Euler Phi Function of Product with Prime/Corollary

:$d \divides n \implies \map \phi d \divides \map \phi n$ where $d \divides n$ denotes that $d$ is a divisor of $n$.


Commutative B-Algebra Induces Abelian Group

Let $\left({X, \circ }\right)$ be a commutative $B$-algebra. Let $*$ be the binary operation on $X$ defined as: :$\forall a, b \in X: a * b := a \circ \left({0 \circ b}\right)$ Then the algebraic structure $\left({X, *}\right)$ is an abelian group such that: :$\forall x \in X: 0 \circ x$ is the inverse element of $x$ under $*$. That is: :$\forall a, b \in X: a * b^{-1} := a \circ b$


Membership is Left Compatible with Ordinal Multiplication

Let $x$, $y$, and $z$ be ordinals. Then: :$\left({ x < y \land z > 0 }\right) \iff \left({ z \cdot x }\right) < \left({ z \cdot y }\right)$


Ordinal Multiplication is Left Cancellable

Let $x, y, z$ be ordinals. Let $z \ne 0$. Then: :$\displaystyle \left({z \cdot x}\right) = \left({z \cdot y}\right) \implies x = y$ That is, ordinal multiplication is left cancellable.


Subset is Right Compatible with Ordinal Multiplication

Let $x, y, z$ be ordinals. Then: :$x \le y \implies \left({x \cdot z}\right) \le \left({y \cdot z}\right)$


Ordinals have No Zero Divisors

Let $x$ and $y$ be ordinals. Then: :$\paren {x \cdot y} = 0 \iff \paren {x = 0 \lor y = 0}$


Absolute Value of Simple Function is Simple Function/Proof 2

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \R$ be a simple function. Then $\left\vert{f}\right\vert: X \to \R$, the absolute value of $f$, is also a simple function.


Absolute Value of Simple Function is Simple Function/Proof 1


Convergence in Norm Implies Convergence in Measure

Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \R, p \ge 1$. Let $\sequence {f_n}_{n \mathop \in \N}, f_n : X \to \R$ be a sequence of $p$-integrable functions. Also, let $f: X \to \R$ be a $p$-integrable function. Suppose that $f_n$ converges in norm to $f$ (in the $p$-norm). Then $f_n$ converges in measure to $f$ (in $\mu$). That is: :$\displaystyle \operatorname {\mathcal L^{\textit p}-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f \implies \operatorname {\mu-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f$


Pointwise Convergence Implies Convergence in Measure

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $\sequence {f_n}_{n \mathop \in \N}, f_n : X \to \R$ be a sequence of measurable functions. Also, let $f: X \to \R$ be a measurable function. Suppose that $f$ is the pointwise limit of the $f_n$ $\mu$-almost everywhere. Then $f_n$ converges in measure to $f$ (in $\mu$). That is: :$\displaystyle \lim_{n \mathop \to \infty} \map {f_n} x \stackrel {a.e.} {=} \map f x \implies \operatorname {\mu-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f$


Convergence in Sigma-Finite Measure

Let $\left({X, \Sigma, \mu}\right)$ be a $\sigma$-finite measure space. Let $\left({f_n}\right)_{n \in \N}, f_n: X \to \R$ be a sequence of measurable functions. Also, let $f, g: X \to \R$ be measurable functions. Suppose that $f_n$ converges in measure to both $f$ and $g$ (in $\mu$). Then $f$ and $g$ are equal $\mu$-almost everywhere.


Vitali's Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a $\sigma$-finite measure space, and let $p \in \R, p \ge 1$. Let $\left({f_n}\right)_{n \in \N}, f_n: X \to \R$ be a sequence of $p$-integrable functions. Also, let $f: X \to \R$ be a measurable function. Suppose that $\displaystyle \operatorname{\mu-\!\lim\,} \limits_{n \to \infty} f_n = f$, i.e. $f_n$ converges in measure to $f$. Then the following are equivalent: :$(1): \quad \displaystyle \lim_{n \to \infty} \left\Vert{f_n - f}\right\Vert_p = 0$, where $\left\Vert{\cdot}\right\Vert_p$ is the $p$-seminorm (i.e. $\displaystyle \operatorname{\mathcal L^{\textit p}-\!\lim\,} \limits_{n \to \infty} f_n = f$) :$(2): \left({\left\vert{f_n}\right\vert^p}\right)_{n \in \N}$ is a uniformly integrable collection :$(3): \displaystyle \lim_{n \to \infty} \int \left\vert{f_n}\right\vert^p \, \mathrm d \mu = \int \left\vert{f}\right\vert^p \, \mathrm d \mu$ If $\left({X, \Sigma, \mu}\right)$ is not $\sigma$-finite, $(1)$ and $(3)$ must be replaced by, respectively: :$(1'): \quad \left({f_n}\right)_{n \in \N}$ converges in $\mathcal L^p$ :$(3'): \quad \displaystyle \lim_{n \to \infty} \int \left\vert{f_n}\right\vert^p \, \mathrm d \mu$ exists in $\R$


Trivial Subgroup is Subgroup

Let $\struct {G, \circ}$ be a group whose identity is $e$. Then the trivial subgroup $\struct {\set e, \circ}$ is indeed a subgroup of $\struct {G, \circ}$.


Trivial Subgroup is Normal

Let $\struct {G, \circ}$ be a group whose identity is $e$. Then the trivial subgroup $\struct {\set e, \circ}$ of $G$ is a normal subgroup in $G$.


Group is Normal in Itself

Let $\struct {G, \circ}$ be a group. Then $\struct {G, \circ}$ is a normal subgroup of itself.


Relativisation is Standard Model

Let $P$ be a well-formed formula. Let $A$ be a finite set such that $x \in A$ {{iff}} $x$ is a free variable in $P$. Then: :$\displaystyle A \subseteq B \implies \left({B \models P \iff P^B}\right)$ {{explain|Definition of $P^B$}}


Induction on Well-Formed Formulas

Let $\mathcal F$ be a formal language with a bottom-up grammar. Let $\Phi$ be a proposition about the well-formed formulas of $\mathcal F$. Suppose that $\Phi$ is true for all letters of $\mathcal F$. Suppose further that every rule of formation preserves $\Phi$, i.e. when fed well-formed formulas satisfying $\Phi$, it yields new well-formed formulas satisfying $\Phi$. Then all well-formed formulas of $\mathcal F$ satisfy $\Phi$.


Substitution of Elements

Let $a$, $b$, and $x$ be sets. :$a = b \implies \left({a \in x \iff b \in x}\right)$


Limit Ordinals Preserved Under Ordinal Multiplication

Let $x$ and $y$ be ordinals. Let $x$ be non-empty. Let $y$ be a limit ordinal. It follows that the ordinal product $\left({x \times y}\right)$ is a limit ordinal.


Ordinal is Less than Ordinal times Limit

Let $y$ be a limit ordinal. Let $x$ and $z$ be ordinals. Then: :$z < \left({x \times y}\right) \iff \exists w < y: z < \left({x \times w}\right)$


Epsilon Induction

Let $A$ be a class. Let $\Bbb U$ denote the universe. :$\left({\forall x: \left({x \subseteq A \implies x \in A}\right)}\right) \implies A = \Bbb U$


Class Equality is Reflexive

Let $A$ be a class. Then: :$A = A$ where $=$ denotes class equality.


Class Equality is Transitive

Let $A$, $B$, and $C$ be classes. Let $=$ denote class equality. Then :$\left({ A = B \land B = C }\right) \implies B = A$


Substitutivity of Class Equality

Let $A$ and $B$ be classes. Let $P \left({A}\right)$ be a well-formed formula of the language of set theory. Let $P \left({B}\right)$ be the same proposition $P \left({A}\right)$ with all instances of $A$ replaced with instances of $B$. Let $=$ denote class equality. :$A = B \implies \left({ P \left({A}\right) \iff P \left({B}\right) }\right)$


Class Equal to All its Elements

Let $A$ be a class. Then: :$A = \left\{{x : x \in A }\right\}$


Class Member of Class Builder

Let $A$ be a class. Let $x$ be a set. Let $P \left({x}\right)$ be a well-formed formula in the language of set theory. Let $P \left({A}\right)$ denote the formula $P\left({x}\right)$ with all free instances of $x$ replaced with instances of $A$. Let $\left\{{x: P \left({x}\right)}\right\}$ be a class specified using class builder notation. Then: :$A \in \left\{{x : P \left({x}\right)}\right\} \iff \left({\exists x: x = A \land P \left({A}\right)}\right)$


Class Equality Extension of Set Equality

Let $=_1$ denote set equality. Let $=_2$ denote class equality. Let $x$ and $y$ be sets. Then $x =_1 y$ iff $x =_2 y$.


B-Algebra Induced by Group Induced by B-Algebra

Let $\left({S, *}\right)$ be a $B$-algebra. Let $\left({S, \circ}\right)$ be the group described on $B$-Algebra Induces Group. Let $\left({S, *'}\right)$ be the $B$-algebra described on Group Induces $B$-Algebra. Then $\left({S, *'}\right) = \left({S, *}\right)$.


Union of Doubleton

Let $\set {x, y}$ be a doubleton. Then: :$\displaystyle \bigcup \set {x, y} = x \cup y$


Union of Small Classes is Small

Let $x$ and $y$ be small classes. Then $x \cup y$ is also small.


Set Difference is Set

Let $x$ be a small class. Let $A$ be a class. Let $\map \MM B$ denote that $B$ is small. Then:  :$\map \MM {x \setminus A}$


Nonempty Class has Members

Let $A$ be a class. Then: :$A \ne \O \iff \exists x: x \in A$


Class is Not Element of Itself

Let $A$ be a class. Then $A$ is not an element of itself: :$A \notin A$


Category of Relations is Category

Let $\mathbf{Rel}$ be the category of relations. Then $\mathbf{Rel}$ is a metacategory.


Identity Functor is Functor

Let $\mathbf C$ be a metacategory. Let $\operatorname{id}_{\mathbf C}: \mathbf C \to \mathbf C$ be the identity functor on $\mathbf C$. Then $\operatorname{id}_{\mathbf C}$ is a functor.


Category of Categories is Category

Let $\mathbf{Cat}$ be the category of categories. Then $\mathbf{Cat}$ is a metacategory.


Identity Functor is Left Identity

Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor, and let $\operatorname{id}_{\mathbf D}$ be the identity functor on $\mathbf D$. Then the composite functor $\operatorname{id}_{\mathbf D} F$ satisfies: :$\operatorname{id}_{\mathbf D} F = F$


Identity Functor is Right Identity

Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor, and let $\operatorname{id}_{\mathbf C}$ be the identity functor on $\mathbf C$. Then the composite functor $F \operatorname{id}_{\mathbf C}$ satisfies: :$F \operatorname{id}_{\mathbf C} = F$


Composition of Functors is Associative

Let $\mathbf A$, $\mathbf B$, $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf A \to \mathbf B$, $G: \mathbf B \to \mathbf C$ and $H: \mathbf C \to \mathbf D$ be functors. Composition of functors is associative: :$H \paren {G F} = \paren {H G} F$


Abel-Ruffini Theorem

There is no general algebraic solution for determining all the roots of a polynomial of degree $5$ or higher.


Cartesian Product is Small

Let $a$ and $b$ be small classes. Then their Cartesian product $\left({a \times b}\right)$ is small: :$\mathscr M \left({a \times b}\right)$


Preorder Category is Category

Let $\struct {S, \precsim}$ be a preordered set. Let $\mathbf S$ be its associated preorder category. Then $\mathbf S$ is a category.


Category Induces Preorder

Let $\mathbf S$ be a category with set of objects $S$. Then the binary relation $\precsim$ defined by: :$\forall a, b \in S: a \precsim b \iff \exists f: a \to b$ is a preorder on $S$.


Preorder Induced by Preorder Category

Let $\left({S, \precsim}\right)$ be a preordered set. Let $\mathbf S$ be its associated preorder category. Let $\precsim'$ be the preorder induced by $\mathbf S$ as on Category Induces Preorder. Then $\precsim'$ is the same as $\precsim$.


Functor between Order Categories

Let $\struct {S, \preceq}$ and $\struct {T, \preceq'}$ be ordered sets. Let $\mathbf S$ and $\mathbf T$ be their associated order categories, respectively. Let $F: \mathbf S \to \mathbf T$ be a functor. Then its object functor $F: S \to T$ is a monotone mapping.


Discrete Category on Set is Discrete Category

Let $S$ be a set. Let $\mathbf{Dis} \left({S}\right)$ be the discrete category on $S$. Then $\mathbf{Dis} \left({S}\right)$ determines a unique (up to isomorphism discrete category $\mathbf{Dis} \left({S}\right)$ whose objects precisely comprise $S$. {{expand|That is, $\mathbf{Dis}: \mathbf{Set} \to \mathbf{Cat}$ is a functor (with left inverse $\mathbf{ob}$) }}


Image of Small Class under Mapping is Small

Let $A$ be a mapping. Let $a$ be a small class. Then, the image of $a$ under $A$ is small. {{improve|$A$ being a class variable, I strongly suggest a different letter to denote the mapping}}


Inverse of Small Relation is Small

Let $a$ be a small class. Let $a$ also be a relation. Then the inverse relation of $a$ is small.


Functor between Monoid Categories

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be monoids. Let $\mathbf S$ and $\mathbf T$ be the associated monoid categories. Let $F: \mathbf S \to \mathbf T$ be a functor. Then the morphism functor $F_1$ of $F$ is a monoid homomorphism.


Cayley's Theorem (Category Theory)

Let $\mathbf C$ be a small category. Denote with $\mathbf{Set}$ the category of sets. Then there exists a category $\mathbf D$, subject to: :$(1): \quad $ The objects of $\mathbf D$ are sets. :$(2): \quad $ The morphisms of $\mathbf D$ are mappings. :$(3): \quad \mathbf C \cong \mathbf D$, i.e. $\mathbf C$ and $\mathbf D$ are isomorphic. That is, $\mathbf C$ is isomorphic to a subcategory of $\mathbf{Set}$.


Domain of Small Relation is Small

Let $a$ be a small class. Let $a$ also be a relation. Then the domain of $a$ is small.


Range of Small Relation is Small

Let $a$ be a small class. Let $a$ also be a relation. Then the range of $a$ is small.


Definition:Opposite Group

Let $\struct {G, \circ}$ be a group. We define a new operation $*$ on $G$ by: :$\forall a, b \in G: a * b = b \circ a$ The algebraic structure $\struct {G, *}$ is called the '''opposite group''' to $G$.


Center of Opposite Group

Let $\struct {G, \circ}$ be a group. Let $\struct {G, *}$ be the opposite group to $G$. Let $\map Z {G, \circ}$ and $\map Z {G, *}$ be the centers of $\struct {G, \circ}$ and $\struct {G, *}$, respectively. Then: :$\map Z {G, \circ} = \map Z {G, *}$


Opposite Group of Opposite Group

Let $\left({G, \circ}\right)$ be a group. Let $\left({G, *}\right)$ be the opposite group to $\left({G, \circ}\right)$. Let $\left({G, \circ'}\right)$ be the opposite group to $\left({G, *}\right)$. Then: :$\left({G, \circ}\right) = \left({G, \circ'}\right)$


Slice Category is Category

Let $\mathbf C$ be a metacategory. Let $C \in \mathbf C_0$ be a object of $\mathbf C$. Let $\mathbf C / C$ be the slice category of $\mathbf C$ over $C$. Then $\mathbf C / C$ is a metacategory.


Monoid Category is Category

Let $\left({S, \circ}\right)$ be a monoid with identity $e_S$. Let $\mathbf S$ be the associated monoid category. Then $\mathbf S$ is a category.


Dual Category is Category

Let $\mathbf C$ be a metacategory. Let $\mathbf C^{\text{op}}$ be its dual category. Then $\mathbf C^{\text{op}}$ is also a metacategory.


Signum Function on Integers is Extension of Signum on Natural Numbers

Let $\sgn_\Z: \Z \to \set {-1, 0, 1}$ be the signum function on the integers. Let $\sgn_\N: \N \to \set {0, 1}$ be the signum function on the natural numbers. Then $\sgn_\Z: \Z \to \Z$ is an extension of $\sgn_\N: \N \to \N$.


Cartesian Product with Proper Class is Proper Class

Let $A$ be a proper class. Let $B$ be a class which is not empty. Then the Cartesian product $\paren {A \times B}$ is a proper class.


Uniqueness Condition for Relation Value

Let $\mathcal R$ be a relation. Let $\left({x, y}\right) \in \mathcal R$. Let: :$\exists ! y: \left({x, y}\right) \in \mathcal R$ Then: :$\mathcal R \left({x}\right) = y$ where $\mathcal R \left({x}\right)$ denotes the image of $\mathcal R$ at $x$. If $y$ is not unique, then: :$\mathcal R \left({x}\right) = \varnothing$


Value of Relation is Small

The value of a relation is always a small class.


Mapping whose Domain is Small Class is Small

Let $F$ be a mapping. Let the domain of $F$ be a small class. Then, $F$ is a small class.


Restriction of Mapping to Small Class is Small

Let $F$ be a mapping. Let $A$ be a small class. Then the restriction $F {\restriction_A}$ is a small class.


Preimage of Singleton

Let $\RR$ be a relation. Let $\map {\RR^{-1} } t$ denote the preimage of $t$ under $\RR$. Let $\RR^{-1} \sqbrk {\set t}$ denote the preimage of $\set t$ under $\RR$. Then: :$\RR^{-1} \sqbrk {\set t} = \map {\RR^{-1} } t$


Order Isomorphism Preserves Minimal Elements

Let $A_1$ and $A_2$ be classes. Let $\prec_1$ and $\prec_2$ be relations. Let $\phi : A_1 \to A_2$ create an order isomorphism between $\left({ A_1 , \prec_1 }\right)$ and $\left({ A_2 , \prec_2 }\right)$. Let $B \subseteq A_1$. Then $\phi$ maps the $\prec_1$-minimal elements (for strict orderings) of $B$ to the $\prec_2$-minimal element of $\phi \left({B}\right)$.


Order Isomorphism Preserves Initial Segments

Let $A_1$ and $A_2$ be classes. Let $\prec_1$ and $\prec_2$ be strict orderings. Let $\phi : A_1 \to A_2$ create an order isomorphism between $\left({A_1, \prec_1}\right)$ and $\left({A_2, \prec_2}\right)$. Suppose $x \in A_1$. Then $\phi$ maps the $\prec_1$-initial segment of $x$ to the $\prec_2$-initial segment of $\phi \left({x}\right)$.


Alternating Group is Set of Even Permutations

Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ be the alternating group on $n$ letters. Then $A_n$ consists of the set of even permutations of $S_n$.


Induced Relation Generates Order Isomorphism

Let $\left({ A_1 , \preceq_1 }\right)$ be an ordered set. Let $\phi : A_1 \to A_2$ be a bijection. Let: : $\preceq_2 \mathop{:=} \left\{ \left({\phi\left({x}\right), \phi\left({y}\right) }\right): x \in A_1 \land y \in A_1 \land x \mathop{\preceq_1} y \right\}$ Then $\phi : \left({ A_1 , \preceq_1 }\right) \to \left({ A_2 , \preceq_2 }\right)$ is an order isomorphism.


Ordinal Multiplication is Left Distributive

Let $x$, $y$, and $z$ be ordinals. Let $\times$ denote ordinal multiplication. Let $+$ denote ordinal addition. Then: :$x \times \left({ y + z }\right) = \left({ x \times y }\right) + \left({ x \times z }\right)$


Ordinal Multiplication is Associative

Let $x, y, z$ be ordinals. Let $\times$ denote ordinal multiplication. Then: :$x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$


Division Theorem for Ordinals

Let $x$ and $y$ be ordinals. Let $0$ denote the zero ordinal. Suppose $y \ne 0$. Then there exist unique ordinals $z$ and $w$ such that: :$x = \paren {y \times z} + w$ and $w < y$.


Finite Ordinal Times Ordinal

Let $m$ and $n$ be finite ordinals. Let $m \ne 0$, where $0$ is the zero ordinal. Let $x$ be a limit ordinal. Then: :$m \times \left({ x + n }\right) = x + \left({ m \times n }\right)$ {{expand|Via Cantor normal form, all ordinals are of the form $x+n$}}


Factorization of Limit Ordinals

Let $x$ be a limit ordinal. Then: :$x = ( \omega \times y )$ for some $y \in \operatorname{On}$ where $\omega$ is the minimal infinite successor set.


Ordinal Exponentiation is Closed

Let $x$ and $y$ be ordinals. Then: :$x^y \in \operatorname{On}$ That is, ordinal exponentiation is closed.


Exponent Base of One

Let $x$ be an ordinal. Then: :$1^x = 1$


Exponent Not Equal to Zero

Let $x$ and $y$ be ordinals. Let $x \ne 0$. Then: :$x^y \ne 0$


Membership is Left Compatible with Ordinal Exponentiation

Let $x$, $y$, and $z$ be ordinals. Suppose $1 < z$. Then: :$x < y \iff z^x < z^y$


Conjugacy Action on Group Elements is Group Action

The conjugacy action on $G$: : $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$ is a  group action on itself.


Conjugacy Action on Subgroups is Group Action

Let $X$ be the set of all subgroups of $G$. For any $H \le G$ and for any $g \in G$, the conjugacy action: :$g * H := g \circ H \circ g^{-1}$ is a group action.


Slice Category of Order Category

Let $\mathbf P$ be a order category, and denote its ordering by $\preceq$. Let $p \in \mathbf P_0$ be an object of $\mathbf P$. Then: :$\mathbf P \mathbin / p \cong p^\preceq$ where: :$\mathbf P \mathbin / p$ is the slice of $\mathbf P$ over $p$ :$p^\preceq$ is the order category defined by the weak lower closure of $p$.


Category of Pointed Sets as Coslice Category

Let $\mathbf{Set}_*$ be the category of pointed sets. Let $\mathbf{Set}$ be the category of sets. Let $1 := \left\{{*}\right\}$ be any singleton. Then: :$\mathbf{Set}_* \cong 1 \mathbin / \mathbf{Set}$ where $1 \mathbin / \mathbf{Set}$ denotes the coslice of $\mathbf{Set}$ under $1$ and $\cong$ signifies isomorphic categories.


Subset is Right Compatible with Ordinal Exponentiation

Let $x, y, z$ be ordinals. Then: :$x \le y \implies x^z \le y^z$


Condition for Membership is Right Compatible with Ordinal Exponentiation

Let $x, y, z$ be ordinals. Let $z$ be the successor of some ordinal $w$. Then: :$x < y \iff x^z < y^z$


Lower Bound for Ordinal Exponentiation

Let $x$ and $y$ be ordinals. Let $x$ be greater than $1$, where $1$ denotes the successor of the zero ordinal. Then: :$y \le x^y$


Unique Ordinal Exponentiation Inequality

Let $x$ and $y$ be ordinals. Let $x > 1$ and $y > 0$. Then there exists a unique ordinal $z$ such that: :$x^z ≤ y$ and $y < x^{z^+}$


Limit Ordinals Closed under Ordinal Exponentiation

Let $x$ and $y$ be ordinals. Let $y$ be a limit ordinal. Let $x^y$ denote ordinal exponentiation. Then: :If $x > 1$, then $x^y$ is a limit ordinal. :If $x \ne \varnothing$, then $y^x$ is a limit ordinal.


Successor of Element of Ordinal is Subset

Let $x$ and $y$ be ordinals. Then: :$x \in y \iff x^+ \subseteq y$


Ordinal is Less than Ordinal to Limit Power

Let $x$, $y$, and $z$ be ordinals. Let $z$ be a limit ordinal. Then: :$x < y^z \iff \exists w \in z: x < y^w$


Ordinal Sum of Powers

Let $x$, $y$, and $z$ be ordinals. Then: :$x^y \times x^z = x^{y + z}$


Kleene Closure is Monoid

Let $S$ be a set, and let $S^*$ be its Kleene closure. Let $*$ denote concatenation of ordered tuples. Then $\struct {S^*, *}$ is a monoid.


Ordinal Power of Power

Let $x$, $y$, and $z$ be ordinals. Then: :$\left({x^y}\right)^z = x^{y \mathop \times z}$


Upper Bound of Ordinal Sum

Let $x$ and $y$ be ordinals. Suppose $x > 1$. Let $\langle a_n \rangle$ be a finite sequence of ordinals such that: :$a_n < x$ for all $n$ Let $\left\langle{b_n}\right\rangle$ be a strictly decreasing finite sequence of ordinals such that: :$b_n < y$ for all $n$ Then: :$\displaystyle \sum_{i \mathop = 1}^n x^{b_i} a_i < x^y$


General Associative Law for Ordinal Sum

Let $x$ be a finite ordinal. Let $\left\langle{a_i}\right\rangle$ be a sequence of ordinals. Then: :$\displaystyle \sum_{i \mathop = 1}^{x + 1} a_i = a_1 + \sum_{i \mathop = 1}^x a_{i + 1}$


Power Sets of Equinumerous Sets are Equinumerous

Let $x$ and $y$ be sets such that $x \sim y$. Then: :$\powerset x \sim \powerset y$


General Associative Law for Ordinal Sum/Proof 1


General Associative Law for Ordinal Sum/Proof 2


Basis Representation Theorem for Ordinals

Let $x$ and $y$ be ordinals. Let $x > 1$ and $y > 0$. Then there exist unique finite sequences of ordinals: :$\sequence {a_i}, \sequence {b_i}$ both of unique length $n$ such that: :$(1): \quad \sequence {a_i}$ is a strictly decreasing sequence for $1 \le i \le n$ :$(2): \quad 0 < b_i < x$ for all $1 \le i \le n$ :$(3): \quad \ds y = \sum_{i \mathop = 1}^n x^{a_i} b_i$


Ordinal Multiplication via Cantor Normal Form/Limit Base

Let $x$ and $y$ be ordinals. Let $x$ be a limit ordinal. Let $y > 0$. Let $\left\langle{a_i}\right\rangle$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$. Let $\left\langle{b_i}\right\rangle$ be a sequence of finite ordinals. Then: :$\displaystyle \sum_{i \mathop = 1}^n \left({x^{a_i} b_i}\right) \times x^y = x^{a_1 \mathop + y}$


Ordinal Exponentiation of Terms

Let $x, y, z$ be ordinals. Let $n$ be a finite ordinal. Let $x$ be a limit ordinal. Let $y, z, n$ all be greater than $0$. Then: :$\left({x^y \times n}\right)^z = x^{y \mathop \times z} \times n$ if $z$ is not a limit ordinal :$\left({x^y \times n }\right)^z = x^{y \mathop \times z}$ if $z$ is a limit ordinal


Prime Triplet is Unique

Let $n \in \Z$ be an integer such that $n > 3$. Then $n$, $n + 2$, $n + 4$ cannot all be prime. That is, the only prime triplet is $\set {3, 5, 7}$.


Ordinal Exponentiation via Cantor Normal Form/Limit Exponents

Let $x$ and $y$ be ordinals. Let $x$ and $y$ be limit ordinals. Let $\sequence {a_i}$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$. Let $\sequence {b_i}$ be a sequence of natural numbers. Then: :$\displaystyle \paren {\sum_{i \mathop = 1}^n x^{a_i} \times b_i}^y = x^{a_1 \mathop \times y}$


Inverse Relation Functor is Contravariant Functor

Let $\mathbf{Rel}$ be the category of relations. Let $C: \mathbf{Rel} \to \mathbf{Rel}$ be the inverse relation functor. Then $C$ is a contravariant functor.


Injection iff Monomorphism in Category of Sets

Let $\mathbf{Set}$ be the category of sets. Let $f: X \to Y$ be a morphism in $\mathbf{Set}$, i.e. a mapping. Then $f$ is an injection iff it is a monomorphism.


Morphism in Preorder Category is Monic

Let $\mathbf P$ be a preorder category. Let $f \in \mathbf P_1$ be a morphism. Then $f$ is monic.


Surjection iff Epimorphism in Category of Sets

Let $\mathbf{Set}$ be the category of sets. Let $f: X \to Y$ be a morphism in $\mathbf{Set}$, i.e. a mapping. Then $f$ is a surjection iff it is an epimorphism.


Inclusion of Natural Numbers in Integers is Epimorphism

Let $\mathbf{Mon}$ be the category of monoids. Let $\left({\N, +}\right)$ denote the monoid of natural numbers as on Natural Numbers under Addition form Commutative Monoid. Let $\left({\Z, +}\right)$ denote the monoid of integers as on additive group of integers. Denote with $\iota: \N \to \Z$ the inclusion mapping. Then $\iota: \N \to \Z$ is an epimorphism in $\mathbf{Mon}$.


Isomorphism (Category Theory) is Monic

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be an isomorphism. Then $f: C \rightarrowtail D$ is monic.


Stabilizer of Element under Conjugacy Action is Centralizer

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*$ be the conjugacy action on $G$ defined by the rule: : $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$ Let $x \in G$. Then the stabilizer of $x$ under this conjugacy action is: :$\Stab x = \map {C_G} x$ where $\map {C_G} x$ is the centralizer of $x$ in $G$.


Stabilizer of Subset Product Action on Power Set

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\powerset G$ be the power set of $\struct {G, \circ}$. Let $*: G \times \powerset G \to \powerset G$ be the subset product action on $\powerset G$ defined as: :$\forall g \in G: \forall S \in \powerset G: g * S = g \circ S$ where $g \circ S$ is the subset product $\set g \circ S$. Then the stabilizer of $S$ in $\powerset G$ is the set: :$\Stab S = S$


Stabilizer of Coset Action on Set of Subgroups

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\powerset G$ denote the power set of $G$. Let $\HH \subseteq \powerset G$ denote the set of subgroups of $G$. Let $*$ be the subset product action on $\HH \subseteq \powerset G$ defined as: :$\forall g \in G: \forall H \in \HH: g * H = g \circ H$ where $g \circ H$ is the (left) coset of $g$ by $H$. Then the stabilizer of $H$ in $\powerset G$ is $H$ itself: :$\Stab H = H$


Isomorphism (Category Theory) is Epic

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be an isomorphism. Then $f: C \twoheadrightarrow D$ is epic.


Split Epimorphism is Epic

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a split epimorphism. Then $f: C \twoheadrightarrow D$ is epic.


Split Monomorphism is Monic

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a split monomorphism. Then $f: C \rightarrowtail D$ is monic.


Relational Closure Exists for Set-Like Relation

Let $A$ be a class. Let $\prec$ be a relation on $A$. Furthermore, let ${\prec^{-1}}\left({a}\right)$ be a small class for each $a \in A$. Let $S$ be a small class that is a subset of the class $A$. Let $G$ be a mapping such that: :$G \left({ x }\right) = A \cap \left({ {\prec^{-1}} \left({ x }\right) }\right)$ Let $F$ be defined using the Principle of Recursive Definition: :$F \left({0}\right) = S$ :$F \left({n^+}\right) = F \left({n}\right) \cup G \left({F \left({n}\right)}\right)$ Let $\displaystyle T = \bigcup_{n \mathop \in \omega} F\left({n}\right)$. Then: :$(1): \quad T$ is a set and satisfies: :::::$\forall x \in A: \forall y \in T: \left({ x \prec y \implies x \in T }\right)$ :::::In other words, $T$ is $\prec$-transitive. :$(2): \quad S \subseteq T$ :$(3): \quad$ If $R$ is $\prec$-transitive and $S \subseteq R$, then $T \subseteq R$. That is, given any set $S$, there is an explicit construction for its relational closure.


Stabilizer of Conjugacy Action on Subgroup is Normalizer

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $X$ be the set of all subgroups of $G$. Let $*$ be the conjugacy action on $H$: :$\forall g \in G, H \in X: g * H = g \circ H \circ g^{-1}$ Then the stabilizer of $H$ in $\powerset G$ is given by: :$\Stab H = \map {N_G} H$ where $\map {N_G} H$ is the normalizer of $H$ in $G$.


Well-Founded Proper Relational Structure Determines Minimal Elements

Let $A$ and $B$ be classes. Let $\left({ A, \prec }\right)$ be a proper relational structure. Let $\prec$ be a foundational relation. Suppose $B \subset A$ and $B \ne \varnothing$. Then $B$ has a $\prec$-minimal element.


Initial Object is Unique

Let $\mathbf C$ be a metacategory. Let $0$ and $0'$ be two initial objects of $\mathbf C$. Then there is a unique isomorphism $u: 0 \to 0'$. Hence, initial objects are unique up to unique isomorphism.


Terminal Object is Unique

Let $\mathbf C$ be a metacategory. Let $1$ and $1'$ be two terminal objects of $\mathbf C$. Then there is a unique isomorphism $u: 1 \to 1'$. Hence, terminal objects are unique up to unique isomorphism.


Empty Set is Initial Object

Let $\mathbf{Set}$ be the category of sets. Then the empty set $\O$ is an initial object of $\mathbf{Set}$.


Singleton is Terminal Object of Category of Sets

Let $\mathbf{Set}$ be the category of sets. Let $S = \left\{{x}\right\}$ be any singleton set. Then $S$ is a terminal object of $\mathbf{Set}$.


Zero (Category) is Initial Object

Let $\mathbf{Cat}$ be the category of categories. Let $\mathbf 0$ be the zero category. Then $\mathbf 0$ is an initial object of $\mathbf{Cat}$.


One (Category) is Terminal Object

Let $\mathbf{Cat}$ be the category of categories. Let $\mathbf 1$ be the category one. Then $\mathbf 1$ is a terminal object of $\mathbf{Cat}$.


Empty Mapping is Mapping

For each set $T$, the empty mapping, where the domain is the empty set, is a mapping.


Empty Mapping is Unique

For each set $T$ there exists exactly one empty mapping $f: \O \to T$ whose domain is the empty set.


Null Relation is Mapping iff Domain is Empty Set

Let $S$ and $T$ be sets. The null relation $\RR = \O \subseteq S \times T$ is a mapping {{iff}} $S = \O$.


Orbit of Element of Group Acting on Itself is Group

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*$ be the group action of $\struct {G, \circ}$ on itself by the rule: :$\forall g, h \in G: g * h = g \circ h$ Then the orbit of an element $x \in G$ is given by: :$\Orb x = G$


Orbit of Element under Conjugacy Action is Conjugacy Class

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*$ be the conjugacy group action on $G$: : $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$ Let $x \in G$. Then the orbit of $x$ under this group action is: :$\Orb x = C_x$ where $C_x$ is the conjugacy class of $x$.


Axiom of Foundation (Strong Form)

Let $B$ be a class. Suppose $B$ is non-empty. Then $B$ has a $\in$-minimal element.


Trivial Group is Terminal Object of Category of Groups

Let $\mathbf{Grp}$ be the category of groups. Let $\left\{{e}\right\}$ be the trivial group. Then $\left\{{e}\right\}$ is a terminal object of $\mathbf{Grp}$.


Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $X$ be the set of all subgroups of $G$. Let $*$ be the conjugacy action on $H$ defined as: : $\forall g \in G, H \in X: g * H = g \circ H \circ g^{-1}$ Then the orbit $\Orb H$ of $H$ in $\powerset G$ is the set of subgroups of $G$ conjugate to $H$.


Closure for Finite Collection of Relations and Operations

Let $\RR_1, \RR_2, \ldots \RR_n$ be relations. Let $\SS_1, \SS_2, \ldots \SS_m$ be operations. Let $T$ be a small class. Let the image of $\RR_i$ over any small class $x$ be small classes for $1 \le i \le n$. Let the image of $\SS_i$ over any Cartesian product $x \times x$ be small classes for $1 \le i \le m$. Then there exists a small class $X$ such that: :$(1): \quad$ $T \subseteq X$ :$(2): \quad$ Each $\RR_i$ is closed with respect to $X$.  Each $\SS_i$ is closed with respect to $X \times X$. :$(3): \quad$ $X$ is the smallest small class satisfying conditions $(1)$ and $(2)$ above. If $Y$ satisfies conditions $(1)$ and $(2)$, then $X \subseteq Y$.


Well-Founded Recursion

Let $A$ be a class. Let $\mathcal R$ be a relation that is foundational on $A$. Let every $\mathcal R$-initial segment of any element $x$ of $A$ be a small class. Let $K$ be a class of mappings $f$ that satisfy: :$(1): \quad$ The domain of $f$ is some subset $y \subseteq A$ such that $y$ is transitive with respect to $\mathcal R$. :$(2): \quad \forall x \in y: f \left({x}\right) = G \left({F \restriction \mathcal R^{-1} x}\right)$ where $F \restriction R^{-1} x$ denotes the restriction of $F$ to the $\mathcal R$-initial segment of $x$. {{explain|What is $G$? And should $F$ not be introduced before it is used, instead of after?}} Let $F = \bigcup K$, the union of $K$. Then: : $F$ is a mapping with domain $A$ : $\forall x \in A: F \left({x}\right) = G \left({F \restriction \mathcal R^{-1} x}\right)$ : $F$ is unique, in the sense that if another mapping $A$ has the above two properties, then $A = F$.


Von Neumann Hierarchy is Supertransitive

Let $V$ denote the Von Neumann Hierarchy. Let $x$ be an ordinal. Then $\map V x$ is supertransitive.


Von Neumann Hierarchy Comparison

Let $x$ and $y$ be ordinals such that $x < y$. Then: :$\map V x \in \map V y$ :$\map V x \subset \map V y$ {{explain|$\map V x$ etc.}}


Every Set in Von Neumann Universe

Let $S$ be a small class. Then $S$ is well-founded. {{explain|The connection between "well-founded" and "in von Neumann universe".}}


Ordinal Equal to Rank

Let $x$ be an ordinal. Let $S$ be a small class. Let $V \left({ x }\right)$ denote the von Neumann hierarchy on the ordinal $x$. Then $x$ equals the rank of $S$ iff $S \in V \left({x+1}\right) \land S \notin V \left({x}\right)$.


Tau of Prime Number

Let $p \in \Z_{> 0}$. Then $p$ is a prime number {{iff}}: :$\map \tau p = 2$ where $\map \tau p$ denotes the tau function of $p$.


Join is Commutative

Let $\left({S, \vee, \preceq}\right)$ be a join semilattice. Then $\vee$ is commutative.


Meet is Commutative

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Then $\wedge$ is commutative.


Join Succeeds Operands

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b \in S$ admit a join $a \vee b \in S$. Then: :$a \preceq a \vee b$ :$b \preceq a \vee b$ i.e., $a \vee b$ succeeds its operands $a$ and $b$.


Meet Precedes Operands

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b \in S$ admit a meet $a \wedge b \in S$. Then: :$a \wedge b \preceq a$ :$a \wedge b \preceq b$ i.e., $a \wedge b$ precedes its operands $a$ and $b$.


Meet is Associative

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Then $\wedge$ is associative.


Poset Elements Equal iff Equal Weak Upper Closure

Let $\left({S, \preccurlyeq}\right)$ be an ordered set. Let $s, t \in S$. Then $s = t$ {{iff}}: :$s^\succcurlyeq = t^\succcurlyeq$ where $s^\succcurlyeq$ denotes weak upper closure of $s$. That is, {{iff}}, for all $r \in S$: :$s \preccurlyeq r \iff t \preccurlyeq r$


Join is Associative

Let $\struct {S, \vee, \preceq}$ be a join semilattice. Then $\vee$ is associative.


Prime iff Equal to Product

Let $p \in \Z$ be an integer such that $p \ne 0$ and $p \ne \pm 1$. Then $p$ is prime {{iff}}: :$\forall a, b \in \Z: p = ab \implies p = \pm a \lor p = \pm b$


Multiple of Divisor Divides Multiple

Let $a, b, c \in \Z$. Let: :$a \divides b$ where $\divides$ denotes divisibility. Then: :$a c \divides b c$


Divisor Relation is Transitive

The divisibility relation is a transitive relation on $\Z$, the set of integers. That is: :$\forall x, y, z \in \Z: x \divides y \land y \divides z \implies x \divides z$


Polynomial Forms over Field form Integral Domain/Formulation 1

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental in $F$. Let $F \sqbrk X$ be the ring of polynomial forms in $X$ over $F$. Then $F \sqbrk X$ is an integral domain.


Sum of Indices of Real Number/Positive Integers

Let $n, m \in \Z_{\ge 0}$ be positive integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: : $r^{n + m} = r^n \times r^m$


Product of Indices of Real Number/Positive Integers

Let $n, m \in \Z_{\ge 0}$ be positive integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$\paren {r^n}^m = r^{n m}$


Smallest Element is Initial Object

Let $\mathbf P$ be an order category. Suppose the objects $\mathbf P_0$ of $\mathbf P$, considered as an ordered set, have a smallest element $p$. Then $p$ is an initial object of $\mathbf P$.


Ordinal is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy

Let $x$ be an ordinal. Let $S$ be a small class. Let $V \left({ x }\right)$ denote the von Neumann hierarchy on the ordinal $x$. Then $x$ is a subset of the rank of $S$ {{iff}} $S \notin V \left({x}\right)$.


Membership Rank Inequality

Let $S$ and $T$ be sets. Let $\operatorname{rank} \left({ S }\right)$ denote the rank of $S$. Then: :$S \in T \implies \operatorname{rank} \left({S}\right) < \operatorname{rank} \left({ T }\right)$


Greatest Element is Terminal Object

Let $\mathbf P$ be an order category. Let $p$ be the greatest element of the objects $\mathbf P_0$ of $\mathbf P$, considered as a ordered set. Then $p$ is a terminal object of $\mathbf P$.


Rank of Set Determined by Members

Let $S$ be a set. Let $\operatorname{rank} \left({ S }\right)$ denote the rank of $S$. Then: :$\operatorname{rank} \left({ S }\right) = \bigcap \left\{ x \in \operatorname{On} : \forall y \in S: \operatorname{rank} \left({ y }\right) < x \right\}$


Bounded Rank implies Small Class

Let $S$ be a class. Suppose the rank, denoted $\operatorname{rank} \left({x}\right)$, of each $x \in S$ is bounded above by some ordinal $y$. {{MissingLinks|rank}} Then $S$ is a small class.


Count of Subsets with Even Cardinality

Let $S$ be a set whose cardinality is $n$. Then the number of subsets of $S$ whose cardinality is even is $2^{n-1}$.


Count of Subsets with Odd Cardinality

Let $S$ be a set whose cardinality is $n$. Then the number of subsets of $S$ whose cardinality is odd is $2^{n-1}$.


Well-Founded Relation is Strongly Well-Founded

Let $A$ be a class. Let $\prec$ be a foundational relation on $A$. Let $B$ be a nonempty class such that $B \subseteq A$. Then $B$ has a $\prec$-minimal element.


Category of Ordered Sets has Enough Constants

Let $\mathbf{OrdSet}$ be the category of ordered sets. Then $\mathbf{OrdSet}$ has enough constants.


Cardinal Number is Ordinal

Let $S$ be a set such that $S \sim x$ for some ordinal $x$. Let $\left|{S}\right|$ denote the cardinal number of $S$. Then: :$\left|{S}\right| \in \operatorname{On}$ where $\operatorname{On}$ denotes the class of ordinals.


Condition for Set Equivalent to Cardinal Number

Let $S$ be a set. Let $\card S$ denote the cardinal number of $S$. That is, let $\card S$ be the intersection of all ordinals equivalent to $S$. {{refactor|This following statement needs to go somewhere else as it's a non-sequitur in this context.}} Note that in the absence of the Axiom of Choice, $\card S$ may be the class of all sets. Then the following are equivalent: : $(1): \quad S \sim \card S$ : $(2): \quad \exists x \in \On: S \sim x$ : $(3): \quad \exists x \in \On: \exists y: \paren {y \subseteq x \land S \sim y}$ where $A \sim B$ means that there is a bijection from $A$ onto $B$, and the quantification over $y$ is unbounded (so $y$ may be any set).


Cardinal Number Equivalence or Equal to Universe

Let $S$ be a set. Let $\left\vert{S}\right\vert$ denote the cardinal number of $S$. Let $\mathbb U$ denote the universal class. Then: :$S \sim \left\vert{S}\right\vert \lor \left\vert{S}\right\vert = \mathbb U$


Ordinal Number Equivalent to Cardinal Number

Let $x$ be an ordinal. Let $\card x$ denote the cardinal number of $x$. Then: :$x \sim \card x$ where $\sim$ denotes set equivalence.


Cardinal Number Less than Ordinal

Let $S$ be a set. Let $\left|{S}\right|$ denote the cardinal number of $S$. Let $x$ be an ordinal such that $S \sim x$. Then: :$\left|{S}\right| \le x$


Equivalent Sets have Equal Cardinal Numbers

Let $S$ and $T$ be sets. Let $\card S$ denote the cardinal number of $S$. Then: :$S \sim T \implies \card S = \card T$


Condition for Set Union Equivalent to Associated Cardinal Number

Let $S$ and $T$ be sets. Let $\left|{S}\right|$ denote the cardinal number of $S$. Let $\sim$ denote set equivalence. Then: :$S \cup T \sim \left|{S \cup T}\right| \iff S \sim \left|{S}\right| \land T \sim \left|{T}\right|$


Cardinal of Cardinal Equal to Cardinal

Let $S$ be a set such that $S$ is equivalent to its cardinal. If the axiom of choice holds, then this condition holds for any set. Then: :$\left|{ \left({ \left|{S}\right| }\right) }\right| = \left|{S}\right|$ where $\left|{S}\right|$ denotes the cardinal number of $S$.


Cardinal of Finite Ordinal

Let $n$ be a finite ordinal. Let $\left|{n}\right|$ denote the cardinal number of $n$. Then: :$\left|{n}\right| = n$


Finite Ordinal is equal to Natural Number

Let $n$ be an element of the minimal infinite successor set. Let $x$ be an ordinal. Then: :$n \sim x \implies n = x$


Product (Category Theory) is Unique

Let $\mathbf C$ be a metacategory. Let $A$ and $B$ be objects of $\mathbf C$. Let $A \times B$ and $A \times' B$ both be products of $A$ and $B$. Then there is a unique isomorphism $u: A \times B \to A \times' B$. That is, products are unique up to unique isomorphism. {{expand|cover extended theorem (for general def)}}


Subset implies Cardinal Inequality

Let $S$ and $T$ be sets such that $S \subseteq T$. Furthermore, let: :$T \sim \card T$ where $\card T$ denotes the cardinality of $T$. Then: :$\card S \le \card T$


Subset of Ordinal implies Cardinal Inequality

Let $S$ be a set. Let $x$ be an ordinal such that $S \subseteq x$. Then: :$\left|{S}\right| \le \left|{x}\right|$ where $\left|{S}\right|$ denotes the cardinality of $S$.


Set Less than Cardinal Product

Let $S$ and $T$ be sets. Let $T$ be nonempty. Suppose that $S \times T \sim \left|{ S \times T }\right|$. Then: :$\left|{ S }\right| \le \left|{ S \times T }\right|$


Injection implies Cardinal Inequality

Let $S$ and $T$ be sets. Let $f: S \to T$ be an injection. Let $\card T$ denote the cardinal number of $T$. Let: :$T \sim \card T$ where $\sim$ denotes set equivalence Then: :$\card S \le \card T$


Set is Equivalent to Image under Injection

Let $S$ and $T$ be sets. Let $f: S \to T$ be an injection. Then the image of $S$ under $f$ is equivalent to $S$.


Cardinality of Image of Mapping not greater than Cardinality of Domain

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let $\card S$ denote the cardinal number of $S$. Let $S \sim \card S$. Then: :$\card {\Img f} \le \card S$


Injection iff Cardinal Inequality

Let $\card T$ denote the cardinal number of $T$. Let $S$ and $T$ be sets such that $S \sim \card S$ and $T \sim \card T$. Then: :$\card S \le \card T $ {{iff}} there exists an injection $f: S \to T$.


Surjection iff Cardinal Inequality

Let $S$ and $T$ be sets such that $S \sim \card S$ and $T \sim \card T$. Furthermore, let $S$ be non-empty. Then: :$0 < \card T \le \card S$ {{iff}} there exists a surjection $f: S \to T$.


Right Identity while exists Right Inverse for All is Identity

Let $\struct  {S, \circ}$ be a semigroup with a right identity $e_R$ such that: :$\forall x \in S: \exists x_R: x \circ x_R = e_R$ That is, every element of $S$ has a right inverse with respect to the right identity. Then $e_R$ is also a left identity, that is, is an identity.


Left Identity while exists Left Inverse for All is Identity

Let $\struct {S, \circ}$ be a semigroup with a left identity $e_L$ such that: :$\forall x \in S: \exists x_L: x_L \circ x = e_L$ That is, every element of $S$ has a left inverse with respect to the left identity. Then $e_L$ is also a right identity, that is, is an identity.


Right Inverse for All is Left Inverse

Let $\struct {S, \circ}$ be a semigroup with a right identity $e_R$ such that: :$\forall x \in S: \exists x_R: x \circ x_R = e_R$ That is, every element of $S$ has a right inverse with respect to the right identity. Then $x_R \circ x = e_R$, that is, $x_R$ is also a left inverse with respect to the right identity.


Exponential on Complex Plane is Group Homomorphism

Let $\struct {\C, +}$ be the additive group of complex numbers. Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers. Let $\exp: \struct {\C, +} \to \struct {\C_{\ne 0}, \times}$ be the mapping: :$x \mapsto \map \exp x$ where $\exp$ is the complex exponential function. Then $\exp$ is a group homomorphism.


Group Direct Product is Product in Category of Groups

Let $\mathbf{Grp}$ be the category of groups. Let $G$ and $H$ be groups, and let $G \times H$ be their direct product. Then $G \times H$ is a binary product of $G$ and $H$ in $\mathbf{Grp}$.


Product Category is Product in Category of Categories

Let $\mathbf{Cat}$ be the category of categories. Let $\mathbf C$ and $\mathbf D$ be small categories, and let $\mathbf C \times \mathbf D$ be their product category. Then $\mathbf C \times \mathbf D$ is a binary product of $\mathbf C$ and $\mathbf D$ in $\mathbf{Cat}$.


Infimum is Product in Order Category

Let $\mathbf P$ be an order category whose ordering is $\preceq$. Let $p,q \in \mathbf P_0$, and suppose that they have some infimum $r = \inf \left\{{p, q}\right\}$. Then $r$ is a binary product of $p$ and $q$ in $\mathbf P$.


Cardinal of Union Less than Cardinal of Cartesian Product

Let $S$ and $T$ be sets that are equivalent to their cardinal numbers. Let $\card S$ denote the cardinal number of $S$. Let $\card S > 1$ and $\card T > 1$. Then: :$\card {S \cup T} \le \card {S \times T}$


Cardinal Product Equinumerous to Ordinal Product

Let $S$ and $T$ be sets that are equivalent to their cardinal numbers. Let $\left|{ S }\right|$ denote the cardinal number of $S$. Let $\cdot$ denote ordinal multiplication and let $\times$ denote the Cartesian product. Then: :$S \times T \sim \left|{ S }\right| \cdot \left|{ T }\right|$


Product of Finite Sets is Finite

Let $S$ and $T$ be finite sets. Then $S \times T$ is a finite set.


Union of Finite Sets is Finite

Let $S$ and $T$ be finite sets. Then $S \cup T$ is a finite set.


Ordinal is Finite iff Natural Number

Let $x$ be an ordinal. Then $x$ is a finite set {{iff}} $x$ is an element of the minimal infinite successor set.


Cardinal Inequality implies Ordinal Inequality

Let $T$ be a set. Let $\left|{T}\right|$ denote the cardinal number of $T$. Let $x$ be an ordinal. Then: :$x < \left|{T}\right| \iff \left|{x}\right| < \left|{T}\right|$


Cardinal Number Plus One Less than Cardinal Product

Let $x$ be an ordinal such that $x > 1$. Then: :$\left|{ x + 1 }\right| \le \left|{ x \times x }\right|$ Where $\times$ denotes the Cartesian product.


Non-Finite Cardinal is equal to Cardinal Product

Let $\omega$ denote the minimal infinite successor set. Let $x$ be an ordinal such that $x \ge \omega$. Then: :$\left|{x}\right| = \left|{ x\times x}\right|$ where $\times$ denotes the Cartesian product.


Cartesian Product Preserves Cardinality

Let $R$, $S$, and $T$ be sets. Suppose that $S$ is equivalent to $T$. Then: :$R \times S \sim R \times T$ :$S \times R \sim T \times R$


Product of Composite Morphisms

Let $\mathbf C$ be a metacategory. Let $f \times f': A \times A' \to B \times B'$ and $g \times g': B \times B' \to C \times C'$ be two composable products of morphisms in $\mathbf C$. Then: :$\left({g \circ f}\right) \times \left({g' \circ f'}\right) = \left({g \times g'}\right) \circ \left({f \times f'}\right)$ where $\times$ signifies product of morphisms.


Cardinal Product Equal to Maximum

Let $S$ and $T$ be sets that are equinumerous to their cardinal number. Let $\left|{S}\right|$ denote the cardinal number of $S$. Suppose $S$ is infinite. Suppose $T > 0$. Then: :$\left|{S \times T}\right| = \max\left({\left|{S}\right|, \left|{T}\right| }\right)$


Cardinal of Union Equal to Maximum

Let $S$ and $T$ be sets that are equinumerous to their cardinal number. Let $\left|{ S }\right|$ denote the cardinal number of $S$. Suppose $S$ is infinite. Then: :$\left|{ S \cup T }\right| = \max\left({ \left|{ S }\right| , \left|{ T }\right| }\right)$


Cardinal Class is Subset of Ordinal Class

Let $\mathcal N$ denote the class of all cardinal numbers. Let $\operatorname{On}$ denote the class of all ordinals. Then: :$\mathcal N \subseteq \operatorname{On}$


Cardinal of Cardinal Equal to Cardinal/Corollary

Let $\mathcal N$ denote the class of all cardinal numbers. Let $x$ be an ordinal. Then: :$x \in \mathcal N \iff x = \left|{ x }\right|$


Cardinal Equal to Collection of All Dominated Ordinals

Let $S$ be a set. Let $\preccurlyeq$ denote the dominance relation. Let $\On$ denote the class of all ordinals. Let $x = \set {y \in \On: y \preccurlyeq S}$ Then: :$(1): \quad x$ is an element of the class of cardinals. :$(2): \quad$ There is no injection $f$ such that $f : x \to S$


Cardinal Class is Proper Class

The class $\mathcal N$ of all cardinal numbers is a proper class.


Infinite Cardinal Class is Proper Class

The infinite cardinal class $\mathcal N’$ is a proper class.


Ordinal in Aleph iff Cardinal in Aleph

Let $x$ and $y$ be ordinals. Then: :$x \in \aleph_y \iff \left|{ x }\right| \in \aleph_y$ where $\aleph$ denotes the aleph mapping.


Aleph Product is Aleph

Let $x$ be an ordinal. Then: :$\left|{\aleph_x \times \aleph_x}\right| = \aleph_x$ where $\aleph$ denotes the aleph mapping.


Aleph is Infinite Cardinal

Let $x$ be an ordinal. Then $\aleph_x$ is an infinite cardinal where $\aleph$ denotes the aleph mapping.


Surjection from Aleph to Ordinal

Let $x$ and $y$ be ordinals. Suppose that: :$0 < y < \aleph_{x+1}$ Then there is a surjection: :$f : \aleph_x \to y$


Ordinal Less than Successor Aleph

Let $x$ and $y$ be ordinals. Then: :$y < \aleph_{x+1} \iff y < \aleph_x \lor y \sim \aleph_x$


Aleph is Infinite

Let $x$ be an ordinal. :$\aleph_x \ge \omega$ where: :$\aleph$ denotes the aleph mapping :$\omega$ denotes the minimal infinite successor set.


Aleph-Null

Let $\omega$ denote the minimal infinite successor set. :$\omega = \aleph_0$ where $\aleph$ denotes the aleph mapping.


Minimal Infinite Successor Set is Infinite Cardinal

$\omega$, the minimal infinite successor set, is an element of the infinite cardinal class $\mathcal N’$.


Set of All Mappings is Small Class

Let $S$ and $T$ be small classes. It follows that the set of all mappings $S^T$ is a small class.


Covariant Hom Functor is Functor

Let $\mathbf{Set}$ be the category of sets. Let $\mathbf C$ be a locally small category. Let $C \in \mathbf C_0$ be an object of $\mathbf C$. Let $\hom_\mathbf C \paren {C, \cdot}: \mathbf C \to \mathbf{Set}$ be the covariant representable functor based at $C$. Then $\hom_\mathbf C \paren {C, \cdot}$ is a functor.


Inverse Morphism is Unique

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be an isomorphism of $\mathbf C$. Then $f$ admits a unique inverse morphism $g: D \to C$.


Cofinal Ordinal Relation is Reflexive

Let $x$ be an ordinal. Then $x$ is cofinal to itself. That is: :$\operatorname{cof} \left({x, x}\right)$


Cofinal Ordinal Relation is Transitive

Let $x$, $y$, and $z$ be ordinals. Let $\operatorname{cof}$ denote the cofinal relation. Then: :$\operatorname{cof} \left({ x,y }\right) \land \operatorname{cof} \left({ y,z }\right) \implies \operatorname{cof} \left({ x,z }\right)$


Cofinal to Zero iff Ordinal is Zero

Let $x$ be an ordinal. Let $\operatorname{cof}$ denote the cofinal relation. Let $0$ denote the zero ordinal. Then the following are equivalent: :$\operatorname{cof} \left({ x,0 }\right)$ :$\operatorname{cof} \left({ 0,x }\right)$ :$x = 0$


Condition for Cofinal Nonlimit Ordinals

Let $x$ and $y$ be nonlimit ordinals. Let $\operatorname{cof}$ denote the cofinal relation. Let $\le$ denote the subset relation. {{explain|This statement could be worded a little more carefully. It is, from examining the link, clear that the subset relation and the ordering relation are the same thing, but it grates to see $\le$ being defined as the subset relation. I'm not saying it ''is'' wrong, because it's not wrong, but it ''looks'' uncomfortable.}} Furthermore, let $x$ and $y$ satisfy the condition: :$0 < x \le y$ Then: :$\map {\operatorname{cof}} { y,x }$


Cofinal Limit Ordinals

Let $x$ and $y$ be ordinals. Let $\operatorname{cof}$ denote the cofinal relation. Let $K_{II}$ denote the class of all limit ordinals. Then: :$\operatorname{cof} \left({ x,y }\right) \implies \left({ x \in K_{II} \iff y \in K_{II} }\right)$


Matrix is Invertible iff Determinant has Multiplicative Inverse

Let $R$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$. Then $\mathbf A$ is invertible {{iff}} its determinant is invertible in $R$. If $R$ is one of the standard number fields $\Q$, $\R$ or $\C$, this translates into: :$\mathbf A$ is invertible {{iff}} its determinant is non-zero.


Prime Groups of Same Order are Isomorphic

Two prime groups of the same order are isomorphic to each other.


Infinite Cyclic Group is Unique up to Isomorphism

All infinite cyclic groups are isomorphic. That is, up to isomorphism, there is only one infinite cyclic group.


Intersection of Subgroups is Subgroup/General Result

Let $\mathbb S$ be a set of subgroups of $\struct {G, \circ}$, where $\mathbb S \ne \O$. Then the intersection $\displaystyle \bigcap \mathbb S$ of the elements of $\mathbb S$ is itself a subgroup of $G$. Also, $\displaystyle \bigcap \mathbb S$ is the largest subgroup of $\struct {G, \circ}$ contained in each element of $\mathbb S$.


Union of Subgroups/Corollary 2

Let $H \vee K$ be the join of $H$ and $K$. Then $H \vee K = H \cup K$ {{iff}} $H \subseteq K$ or $K \subseteq H$.


Existence of Unique Subgroup Generated by Subset/Singleton Generator

Let $a \in G$. Then $H = \gen a = \set {a^n: n \in \Z}$ is the unique smallest subgroup of $G$ such that $a \in H$. That is: : $K \le G: a \in K \implies H \subseteq K$


Join of Subgroups is Group Generated by Union

Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $S$ be the set of words of $H \cup K$. Then $S$ is a subgroup of $K$ such that: :$S = \gen {H \cup K} = H \vee K$ where $H \vee K$ denotes the join of $H$ and $K$.


Order of Subgroup Product/Corollary

:$\size {H \vee K} \ge \dfrac {\order H \order K} {\order {H \cap K} }$ or :$\dfrac {\size {H \vee K} } {\order H} \ge \dfrac {\order K} {\order {H \cap K} }$ where $H \vee K$ denotes join and $\order H$ denotes the order of $H$.


Homomorphic Image of Group Element is Coset

Let $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ be a group homomorphism. Let $\ker \left({\phi}\right)$ be the kernel of $\phi$. Let $h \in H$. Then $\operatorname{Im}^{-1} \left({h}\right)$ is either the empty set or a coset of $\ker \left({\phi}\right)$.


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 $\struct {S, \tau}$ {{iff}} 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) {{iff}}: : $\forall g \in G: g \circ N \circ g^{-1} \subseteq N$ : $\forall g \in G: g^{-1} \circ N \circ g \subseteq N$


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) {{iff}}: :$\forall g \in G: N \subseteq g \circ N \circ g^{-1}$ :$\forall g \in G: N \subseteq g^{-1} \circ N \circ g$


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 $G$ {{iff}}: :$\forall g \in G: \paren {n \in N \iff g \circ n \circ g^{-1} \in N}$ :$\forall g \in G: \paren {n \in N \iff g^{-1} \circ n \circ g \in N}$


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) {{iff}}: : 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 {{iff}} $\phi^{-1}: \struct {H, *} \to \struct {G, \circ}$ 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 $N \subseteq K$ {{iff}} 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: :$f + g = g + f$ </onlyinclude> 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: :$\left({x, i}\right) \preceq \left({y, j}\right)$ {{iff}} $i = j$ 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 $\mathbf P$.