hcju/proofwikips huggingface dataset:
Inverse Relation is Left and Right Inverse iff Bijection
Let $\RR \subseteq S \times T$ be a relation on a cartesian product $S \times T$. Let: :$I_S$ be the identity mapping on $S$ :$I_T$ be the identity mapping on $T$. Let $\RR^{-1}$ be the inverse relation of $\RR$. Then $\RR$ is a bijection : :$\RR^{-1} \circ \RR = I_S$ and :$\RR \circ \RR^{-1} = I_T$ where $\circ$ denotes composition of relations.
Integer Multiples Closed under Addition
Let $n \Z$ be the set of integer multiples of $n$. Then the algebraic structure $\struct {n \Z, +}$ is closed under addition.
Integer Multiples Closed under Multiplication
Let $n \Z$ be the set of integer multiples of $n$. Then the algebraic structure $\struct {n \Z, \times}$ is closed under multiplication.
Non-Zero Integers Closed under Multiplication
The set of non-zero integers is closed under multiplication.
Two-Step Subgroup Test using Subset Product
Let $G$ be a group. Let $\O\subset H \subseteq G$ be a non-empty subset of $G$. Then $H$ is a subgroup of $G$ : :$H H \subseteq H$ :$H^{-1} \subseteq H$ where: :$H H$ is the product of $H$ with itself :$H^{-1}$ is the inverse of $H$.
One-Step Subgroup Test using Subset Product
Let $G$ be a group. Let $\varnothing \subset H \subseteq G$ be a non-empty subset of $G$. Then $H$ is a subgroup of $G$ : :$H H^{-1} \subseteq H$ where: :$H^{-1}$ is the inverse of $H$ :$H H ^{-1}$ is the product of $H$ with $H^{-1}$.
Product of Subset with Union
Let $\left({G, \circ}\right)$ be an algebraic structure. Let $X, Y, Z \subseteq G$. Then: : $X \circ \left({Y \cup Z}\right) = \left({X \circ Y}\right) \cup \left({X \circ Z}\right)$ : $\left({Y \cup Z}\right) \circ X = \left({Y \circ X}\right) \cup \left({Z \circ X}\right)$ where $X \circ Y$ denotes the subset product of $X$ and $Y$.
Hilbert's Nullstellensatz
Let $k$ be an algebraically closed field. Let $n \geq 0$ be an natural number. Let $k \left[{x_1,\ldots, x_n}\right]$ be the polynomial ring in $n$ variables over $k$. Then for every ideal $J \subseteq k \left[{x_1,\ldots, x_n}\right]$, the associated ideal of its zero-locus equals its radical: :$I \left({Z \left({J}\right)}\right) = \sqrt J$
Subset of Domain is Subset of Preimage of Image
Let $f: S \to T$ be a mapping. Then: :$A \subseteq S \implies A \subseteq \paren {f^{-1} \circ f} \sqbrk A$ where: :$f \sqbrk A$ denotes the image of $A$ under $f$ :$f^{-1} \sqbrk A$ denotes the preimage of $A$ under $f$ :$f^{-1} \circ f$ denotes composition of $f^{-1}$ and $f$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\forall A \in \powerset S: A \subseteq \map {\paren {f^\gets \circ f^\to} } A$
Clairaut's Differential Equation
'''Clairaut's differential equation''' is a first order ordinary differential equation which can be put into the form: :$y = x y' + \map f {y'}$ Its general solution is: :$y = C x + \map f C$ where $C$ is a constant.
Noether Normalization Lemma
Let $k$ be a field. Let $A$ be a non-trivial finitely generated $k$-algebra. {{explain|the above link is for Definition:Non-Trivial Ring -- we need to define a Definition:Non-Trivial Algebra}} Then there exists $n \in \N$ and a finite injective morphism of $k$-algebra: :$k \sqbrk {x_1, \dotsc, x_n} \to A$ {{DefinitionWanted|Instead of using Definition:Finite Ring Homomorphism, another page is to be generated defining the module-theory version.}}
Image of Preimage under Mapping
Let $f: S \to T$ be a mapping. Then: :$B \subseteq T \implies \paren {f \circ f^{-1} } \sqbrk B = B \cap f \sqbrk S$
Isomorphism of External Direct Products
Let: :$\struct {S_1 \times S_2, \circ}$ be the external direct product of two algebraic structures $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ :$\struct {T_1 \times T_2, *}$ be the external direct product of two algebraic structures $\struct {T_1, *_1}$ and $\struct {T_2, *_2}$ :$\phi_1$ be an isomorphism from $\struct {S_1, \circ_1}$ onto $\struct {T_1, *_1}$ :$\phi_2$ be an isomorphism from $\struct {S_2, \circ_2}$ onto $\struct {T_2, *_2}$. Then the mapping $\phi_1 \times \phi_2: \struct {S_1 \times S_2, \circ} \to \struct {T_1 \times T_2, *}$ defined as: :$\map {\paren {\phi_1 \times \phi_2} } {x, y} = \tuple {\map {\phi_1} x, \map {\phi_2} y}$ is an isomorphism from $\struct {S_1 \times S_2, \circ}$ to $\struct {T_1 \times T_2, *}$.
Dual of Lattice Ordering is Lattice Ordering
Let $\struct {S, \preccurlyeq}$ be a lattice. Let $\preccurlyeq$ be the lattice ordering on $\struct {S, \preccurlyeq}$. Then its dual ordering $\succcurlyeq$ is also a lattice ordering.
Order Isomorphism on Totally Ordered Set preserves Total Ordering
Let $\left({S, \preccurlyeq_1}\right)$ and $\left({T, \preccurlyeq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preccurlyeq_1}\right) \to \left({T, \preccurlyeq_2}\right)$ be an order isomorphism. Then $\left({S, \preccurlyeq_1}\right)$ is a totally ordered set  $\left({T, \preccurlyeq_2}\right)$ is also a totally ordered set.
Order Isomorphism on Well-Ordered Set preserves Well-Ordering
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets. Let $\phi: \struct {S_1, \preccurlyeq_1} \to \struct {S_2, \preccurlyeq_2}$ be an order isomorphism. Then $\struct {S_1, \preccurlyeq_1}$ is a well-ordered set  $\struct {S_2, \preccurlyeq_2}$ is also a well-ordered set.
Order Isomorphism on Lattice preserves Lattice Structure
Let $\left({S, \preccurlyeq_1}\right)$ and $\left({T, \preccurlyeq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preccurlyeq_1}\right) \to \left({T, \preccurlyeq_2}\right)$ be an order isomorphism. Then $\left({S, \preccurlyeq_1}\right)$ is a lattice iff $\left({T, \preccurlyeq_2}\right)$ is also a lattice.
Set of Subgroups forms Complete Lattice
Let $\struct {G, \circ}$ be a group, and let $\mathbb G$ be the set of all subgroups of $G$. Then $\struct {\mathbb G, \subseteq}$ is a complete lattice.
Infimum and Supremum of Subgroups
Let $\left({G, \circ}\right)$ be a group. Let $\mathbb G$ be the set of all subgroups of $G$. Let $\left({\mathbb G, \subseteq}\right)$ be the complete lattice formed by $\mathbb G$ and $\subseteq$. Let $H, K \in \mathbb G$.
Index Laws/Sum of Indices/Monoid
Let $\left({S, \circ}\right)$ be a monoid whose identity element is $e$. For $a \in S$, let $\circ^n a = a^n$ be defined as the $n$th power of $a$: :$a^n = \begin{cases} e & : n = 0 \\ a^x \circ a & : n = x + 1 \end{cases}$ That is: :$a^n = \underbrace{a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \circ^n \left({a}\right)$ while: :$a^0 = e$ Then: :$\forall m, n \in \N: a^{n + m} = a^n \circ a^m$
Power of Product of Commuting Elements in Semigroup equals Product of Powers
:$\forall n \in \N_{>0}: \map {\circ^n} {a \circ b} = \paren {\circ^n a} \circ \paren {\circ^n b}$
Sum of Sequences of Fifth and Seventh Powers
:$\displaystyle \sum_{i \mathop = 1}^n i^5 + \sum_{i \mathop = 1}^n i^7 = 2 \paren {\sum_{i \mathop = 1}^n i}^4$
Cardinality of Set of Strictly Increasing Mappings
Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be tosets. Let the cardinality of $S$ and $T$ be: :$\card S = m, \card T = n$ Then the number of strictly increasing mappings from $S$ to $T$ is: :$\dbinom n m = \dfrac {n!} {m! \ \paren {n - m}!}$. where $\dbinom n m$ is a binomial coefficient.
Construction of Inverse Completion/Equivalence Relation
The cross-relation $\boxtimes$ is an equivalence relation on $\left({S \times C, \oplus}\right)$.
Construction of Inverse Completion/Properties of Quotient Structure
=== Identity of Quotient Structure === {{:Construction of Inverse Completion/Identity of Quotient Structure}} === Invertible Elements in Quotient Structure === {{:Construction of Inverse Completion/Invertible Elements in Quotient Structure}} === Generator for Quotient Structure === {{:Construction of Inverse Completion/Generator for Quotient Structure}} === Quotient Structure is Inverse Completion === {{:Construction of Inverse Completion/Quotient Structure is Inverse Completion}}
Integers form Commutative Ring
The set of integers $\Z$ forms a commutative ring under addition and multiplication.
Symmetric Group is Group
Let $S$ be a set. Let $\map \Gamma S$ denote the set of all permutations on $S$. Then $\struct {\map \Gamma S, \circ}$, the symmetric group on $S$, forms a group.
Quotient Ring Defined by Ring Itself is Null Ring
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\struct {R / R, +, \circ}$ be the quotient ring defined by $R$. Then $\struct {R / R, +, \circ}$ is a null ring.
Set of Subrings forms Complete Lattice
Let $\struct {K, +, \circ}$ be a ring. Let $\mathbb K$ be the set of all subrings of $K$. Then $\struct {\mathbb K, \subseteq}$ is a complete lattice.
Set of Ideals forms Complete Lattice
Let $\struct {K, +, \circ}$ be a ring. Let $\mathbb K$ be the set of all ideals of $K$. Then $\struct {\mathbb K, \subseteq}$ is a complete lattice.
Subset Product of Abelian Subgroups
Let $\left({G, \circ}\right)$ be an abelian group. Let $H_1$ and $H_2$ be subgroups of $G$. Then $H_1 \circ H_2$ is a subgroup of $G$.
Sum of Ideals is Ideal
Let $J_1$ and $J_2$ be ideals of a ring $\struct{R, +, \circ}$. Then: : $J = J_1 + J_2$ is an ideal of $R$ where $J_1 + J_2$ is as defined in subset product.
Principal Ideal from Element in Center of Ring
Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $b \in R$ be in the center of $R$. Then: :$\left({b}\right) = R \circ b = \left\{{x \circ b: x \in R}\right\}$ where $\left({b}\right)$ is the principal ideal generated by $b$.
External Direct Sum of Rings is Ring
Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be rings. Then their (external) direct product: : $\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$ is a ring.
External Direct Product Distributivity
Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be ringoids. Let their external direct product be $\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$. Then the operation $\circ$ is distributive over $+$.
Ideal of External Direct Sum of Rings
Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be rings. Let  :$\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$  be their direct product. For each $k \in \left[{1 \,.\,.\, n}\right]$, let: :$R\,'_k = \left\{{\left({x_1, \ldots, x_n}\right) \in R: \forall j \ne k: x_j = 0}\right\}$ Then: : $\forall k \in \left[{1 \,.\,.\, n}\right]: R\,'_k$ is an ideal of $R$.
Canonical Injection from Ideal of External Direct Sum of Rings
Let $\struct {R_1, +_1, \circ_1}, \struct {R_2, +_2, \circ_2}, \dotsc, \struct {R_n, +_n, \circ_n}$ be rings. Let $\displaystyle \struct {R, +, \circ} = \prod_{k \mathop = 1}^n \struct {R_k, +_k, \circ_k}$ be their external direct product. For each $k \in \closedint 1 n$, let: :$R'_k = \set {\tuple {x_1, \dotsc, x_n} \in R: \forall j \ne k: x_j = 0}$ Let $\inj_k: R_k \to R$ be the canonical injection on the $k$th coordinate from $R_k$ into $\struct {R, +, \circ}$. Let $\pr_k: R \to R'_k$ be the projection on the $k$th coordinate of $\struct {R, +, \circ}$ onto $R'_k$. Then: :$\inj_k: R_k \to R'_k$ is an isomorphism :Its inverse is the restriction of $\pr_k$ to $R'_k$.
Conditions for Internal Ring Direct Sum
Let $\struct {R, +, \circ}$ be a ring. Let $\sequence {\struct {S_k, +, \circ} }$ be a sequence of subrings of $R$. Then $R$ is the ring direct sum of $\sequence {S_k}_{1 \mathop \le k \mathop \le n}$ : :$(1): \quad R = S_1 + S_2 + \cdots + S_n$ :$(2): \quad \sequence {\struct {S_k, +} }_{1 \mathop \le k \mathop \le n}$ is a sequence of independent subgroups of $\struct {R, +}$ :$(3): \quad \forall k \in \closedint 1 n: S_k$ is an ideal of $R$.
Structure Induced by Ring Operations is Ring
Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a set. Then $\struct {R^S, +', \circ'}$ is a ring, where $+'$ and $\circ'$ are the pointwise operations induced on $R^S$ by $+$ and $\circ$.
Division Subring Test
Let $\struct {K, +, \circ}$ be a division ring, and let $L$ be a subset of $K$. Then $\struct {L, +, \circ}$ is a division subring of $\struct {K, +, \circ}$  these all hold: :$(1) \quad L^* \ne \O$ :$(2) \quad \forall x, y \in L: x + \paren {-y} \in L$ :$(3) \quad \forall x, y \in L: x \circ y \in L$ :$(4) \quad x \in L^* \implies x^{-1} \in L^*$
Homomorphism from Integers into Ring with Unity
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let the characteristic of $R$ be $p$. For any $a \in R$, we define the mapping $g_a: \Z \to R$ from the integers into $R$ as: :$\forall n \in \Z: \map {g_a} n = n \cdot a$ Then $g_a$ is a group homomorphism from $\struct {\Z, +}$ to $\struct {R, +}$. Also: :$\ideal p \subseteq \map \ker {g_a}$ where: :$\map \ker {g_a}$ is the kernel of $g_a$; :$\ideal p$ is the principal ideal of $\Z$ generated by $p$. Also: :$p \divides n \implies n \cdot a = 0$ where $p \divides n$ denotes that $p$ is a divisor of $n$.
Properties of Prime Subfield
Let $F$ be a field. Let $K$ be the prime subfield of $F$. Then $K$ is isomorphic to either: :$\Q$, the field of rational numbers, or :$\Z_p$, the Ring of Integers Modulo $p$, where $p$ is prime.
Subgroup of Additive Group Modulo m is Ideal of Ring
Let $m \in \Z: m > 1$. Let $\struct {\Z_m, +_m}$ be the additive group of integers modulo $m$. Then every subgroup of $\struct {\Z_m, +_m}$ is an ideal of the ring of integers modulo $m$ $\struct {\Z_m, +_m, \times_m}$.
Bijection from Divisors to Subgroups of Cyclic Group
Let $G$ be a cyclic group of order $n$ generated by $a$. Let $S = \left\{{m \in \Z_{>0}: m \divides n}\right\}$ be the set of all divisors of $n$. Let $T$ be the set of all subgroups of $G$ Let $\phi: S \to T$ be the mapping defined as: :$\phi: m \to \left\langle{a^{n/m}}\right\rangle$ where  $\left\langle{a^{n/m}}\right\rangle$ is the subgroup generated by $a^{n/m}$. Then $\phi$ is a bijection.
Generator of Additive Group Modulo m iff Unit of Ring
Let $m \in \Z: m > 1$. Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$. Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$. Let $a \in \Z_m$. Then: :$a$ is a generator of $\struct {\Z_m, +_m}$   :$a$ is a unit of $\struct {\Z_m, +_m, \times_m}$
Subspace of Real Differentiable Functions
Let $\mathbb J$ be an open interval of the real number line $\R$. Let $\map \DD {\mathbb J}$ be the set of all differentiable real functions on $\mathbb J$. Then $\struct {\map \DD {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.
Subspace of Riemann Integrable Functions
Let $\mathbb J = \set {x \in \R: a \le x \le b}$ be a closed interval of the real number line $\R$. Let $\map \RR {\mathbb J}$ be the set of all Riemann integrable functions on $\mathbb J$. Then $\struct {\map \RR {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.
Basis of Vector Space of Polynomial Functions
Let $B$ be the set of all the identity functions $I^n$ on $\R^n$ where $n \in \N^*$. Then $B$ is a basis of the $\R$-vector space $\map P \R$ of all polynomial functions on $\R$.
Ordered Basis for Coordinate Plane
Let $a_1, a_2 \in \R^2$ such that $\left\{{a_1, a_2}\right\}$ forms a linearly independent set. Then $\left({a_1, a_2}\right)$ is an ordered basis for the $\R$-vector space $\R^2$. Hence the points on the plane can be uniquely identified by means of linear combinations of $a_1$ and $a_2$.
Differentiation on Polynomials is Linear Operator
Let $P \left({\R}\right)$ be the vector space of all polynomial functions on the real number line $\R$. Then the differentiation operator $D$ on $P \left({\R}\right)$ is a linear operator.
Noether's Theorem (Calculus of Variations)
Let $y_i$, $F$, $\Psi_i$, $\Phi$ be real functions. Let $x, \epsilon \in \R$. Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$ and $\mathbf \Psi = \sequence{\Psi_i}_{1 \mathop \le i \mathop \le n}$ be vectors. Let  :$\Phi = \map \Phi {x, \mathbf y, \mathbf y'; \epsilon}, \quad \Psi_i = \map {\Psi_i} {x, \mathbf y, \mathbf y'; \epsilon}$  such that: :$\map \Phi {x, \mathbf y, \mathbf y'; 0} = x, \quad \map {\Psi_i} {x, \mathbf y, \mathbf y'; 0} = y_i$ where $x, \mathbf y, \mathbf y', \epsilon$ are variables. Let: :$\ds J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$ be a functional. Let: :$X = \map \Phi {x, \mathbf y, \mathbf y'; \epsilon}, \quad \mathbf Y = \map {\mathbf \Psi} {x, \mathbf y, \mathbf y'; \epsilon}$ Suppose, $J \sqbrk {\mathbf y}$ is invariant under transformations $x \rightarrow X$ and $\mathbf y \rightarrow \mathbf Y$ for arbitrary $x_0$ and $x_1$. Then: :$\nabla_{\mathbf y'} F \cdot \boldsymbol \psi + \paren {F - \mathbf y' \cdot \nabla_{\mathbf y'} F} \phi = C$ where $C$ is a constant and: :$\map {\boldsymbol \psi} {x, \mathbf y, \mathbf y'} = \dfrac {\partial \map {\mathbf \Psi} {x, \mathbf y, \mathbf y'; \epsilon} } {\partial \epsilon} \Bigg\rvert_{\epsilon \mathop = 0}$ :$\map \phi {x, \mathbf y, \mathbf y'} = \dfrac {\partial \map \Phi {x, \mathbf y, \mathbf y'; \epsilon} } {\partial \epsilon} \Bigg\rvert_{\epsilon \mathop = 0}$
Equivalence of Definitions of Supremum of Real-Valued Function
Let $S \subseteq \R$ be a subset of the real numbers. Let $f: S \to \R$ be a real function on $S$. {{TFAE|def = Supremum of Real-Valued Function}}
Supremum does not Precede Infimum
Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ admit both a supremum $M$ and an infimum $m$. Then $m \preceq M$.
Integer Multiples form Commutative Ring
Let $n \Z$ be the set of integer multiples of $n$. Then $\struct {n \Z, +, \times}$ is a commutative ring. Unless $n = 1$, $\struct {n \Z, +, \times}$ is not a ring with unity.
Quaternions Defined by Matrices
Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\map {\mathcal M_\C} 2$: :$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} $ where $\C$ is the set of complex numbers. Then $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are related to each other in the following way: {{begin-eqn}} {{eqn | l = \mathbf i \mathbf j = - \mathbf j \mathbf i       | r = \mathbf k }} {{eqn | l = \mathbf j \mathbf k = - \mathbf k \mathbf j       | r = \mathbf i }} {{eqn | l = \mathbf k \mathbf i = - \mathbf i \mathbf k       | r = \mathbf j }} {{eqn | l = \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k       | r = -\mathbf 1 }} {{end-eqn}}
Matrix Form of Quaternion
Let $\mathbf x$ be a quaternion such that: : $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ When the quaternion basis is expressed in the form of matrices: :$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$ the general quaternion $\mathbf x$ has the form: :$\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$ or: :$\mathbf x = \begin{bmatrix} w & z \\ -\overline z & \overline w \end{bmatrix}$ where : :$w$ and $z$ are complex numbers :$\overline z$ is the complex conjugate of $z$.
Quaternion Multiplication
Let $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ be quaternions. Then their '''product''' is given by: {{begin-eqn}} {{eqn | ll=\mathbf x_1 \mathbf x_2       | l==       | o=       | r=\left({a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2}\right) \mathbf 1       | c= }} {{eqn | o=+       | r=\left({a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2}\right) \mathbf i       | c= }} {{eqn | o=+       | r=\left({a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2}\right) \mathbf j       | c= }} {{eqn | o=+       | r=\left({a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2}\right) \mathbf k       | c= }} {{end-eqn}}
Quaternion Addition forms Abelian Group
Let $\mathbb H$ be the set of quaternions. Then $\left({\mathbb H, +}\right)$, where $+$ denotes quaternion addition, is an abelian group.
Ring of Quaternions is Ring
The set $\mathbb H$ of quaternions forms a ring under the operations of addition and multiplication.
Quaternions Subring of Complex Matrix Space
The Ring of Quaternions is a subring of the matrix space $\mathcal M_\C \left({2}\right)$.
Product of Quaternion with Conjugate
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. Then their product is given by: :$\mathbf x \overline {\mathbf x} = \paren {a^2 + b^2 + c^2 + d^2} \mathbf 1 = \overline {\mathbf x} \mathbf x$
Multiplicative Inverse of Quaternion
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion such that $\mathbf x \ne \mathbf 0$. Then $\mathbf x$ has an inverse $\mathbf x^{-1}$ under the operation of quaternion multiplication: :$\mathbf x^{-1} = \lambda \overline {\mathbf x}$ where: :$\lambda = \dfrac 1 {a^2 + b^2 + c^2 + d^2} \mathbf 1$
Multiplicative Identity for Quaternions
In the set of quaternions $\mathbb H$, the element: :$\mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k$ serves as the identity element for quaternion multiplication. This element is written $\mathbf 1$.
Quaternions form Skew Field
The set $\H$ of quaternions forms a skew field under the operations of addition and multiplication.
Cancellation Law of Ring Product of Integral Domain
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $D^*$ denote $D \setminus \set {0_D}$, that is, $D$ without its zero. Let $a \in D^*$. Then: :$\forall x, y \in D: a \circ x = a \circ y \implies x = y$ That is, all elements of $D^*$ are cancellable for the ring product.
Gaussian Integers form Integral Domain
The ring of Gaussian integers $\struct {\Z \sqbrk i, +, \times}$ is an integral domain.
Set Intersection Not Cancellable
Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$. Let $S_1, S_2, T \in \mathcal P \left({S}\right)$. Suppose that $S_1 \cap T = S_2 \cap T$. Then it is not necessarily the case that $S_1 = S_2$.
Sum of Triangular Matrices
Let $\mathbf A = \left[{a}\right]_{n}, \mathbf B = \left[{b}\right]_{n}$ be square matrices of order $n$. Let $\mathbf C = \mathbf A + \mathbf B$ be the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. If $\mathbf A$ and $\mathbf B$ are upper triangular matrices, then so is $\mathbf C$. If $\mathbf A$ and $\mathbf B$ are lower triangular matrices, then so is $\mathbf C$.
Negative of Triangular Matrix
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $-\mathbf A$ be the negative of $\mathbf A$. If $\mathbf A$ is an upper triangular matrix, then so is $-\mathbf A$. If $\mathbf A$ is a lower triangular matrix, then so is $-\mathbf A$.
Natural Numbers form Subsemiring of Integers
The semiring of natural numbers $\struct {\N, +, \times}$ forms a subsemiring of the ring of integers $\struct {\Z, +, \times}$.
Integers form Subdomain of Rationals
The integral domain of integers $\left({\Z, +, \times}\right)$ forms a subdomain of the field of rational numbers.
Real Numbers form Subfield of Complex Numbers
The field of real numbers $\struct {\R, +, \times}$ forms a subfield of the field of complex numbers $\struct {\C, +, \times}$.
Sum of All Ring Products is Additive Subgroup
Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({S, +}\right)$ and $\left({T, +}\right)$ be additive subgroups of $\left({R, +, \circ}\right)$. Let $S + T$ be defined as subset product. Let $S T$ be defined as: :$\displaystyle S T = \left\{{\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \left[{1 \,.\,.\ n}\right]}\right\}$ Then both $S + T$ and $S T$ are additive subgroups of $\left({R, +, \circ}\right)$.