hcju/proofwikips huggingface dataset:
Definition:Isolated Point (Topology)/Subset
{{:Definition:Isolated Point (Topology)/Subset/Definition 1}}
Definition:Isolated Point (Topology)/Space
$x \in S$ is an '''isolated point of $T$''' : :$\exists U \in \tau: U = \set x$ That is,  there exists an open set of $T$ containing no points of $S$ other than $x$.
Definition:Integer Reciprocal Space
Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$. Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ That is: :$A := \set {1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \ldots}$ Then $\struct {A, \tau_d}$ is the '''integer reciprocal space'''.
Definition:Exponential Distribution
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $X$ has the '''exponential distribution with parameter $\beta$''' : :$\map X \Omega = \R_{\ge 0}$ :$\map \Pr {X < x} = 1 - e^{-\frac x \beta}$ where $0 < \beta$.
Definition:Expectation/Continuous
Let $X$ be a continuous random variable over the probability space $\struct {\Omega, \Sigma, \Pr}$. Let $F = \map \Pr {X < x}$ be the cumulative probability function of $X$. The '''expectation of $X$''' is written $\expect X$, and is defined over the probability measure as: :$\expect X := \displaystyle \int_{x \mathop \in \Omega} x \rd F$ whenever the integral is absolutely convergent, i.e. when: :$\displaystyle \int_{x \mathop \in \Omega} \size x \rd F < \infty$
Definition:Logical Implication
In a valid argument, the premises '''logically imply''' the conclusion. If the truth of one statement $p$ can be shown in an argument directly to cause the meaning of another statement $q$ to be true, then $q$ follows from $p$ by '''logical implication'''. We may say: :'''$p$, therefore $q$''' and write $p \vdash q$. :'''$q$, because $p$''' and write $q \dashv p$. In symbolic logic, the concept of '''logical consequence''' occurs in the form of semantic consequence and provable consequence. In the context of proofs of a conventional mathematical nature on , the notation: :$p \leadsto q$ is preferred, where $\leadsto$ can be read as '''leads to'''. === Semantic Consequence === {{:Definition:Semantic Consequence}} === Provable Consequence === {{:Definition:Provable Consequence}}
Definition:Semantic Consequence/Boolean Interpretations
<section notitle="true" name="tc"> Let $\mathcal F$ be a collection of WFFs of propositional logic. Then a WFF $\mathbf A$ is a '''semantic consequence''' of $\mathcal F$ iff: :$v \models_{\mathrm{BI}} \mathcal F$ implies $v \models_{\mathrm{BI}} \mathbf A$ where $\models_{\mathrm{BI}}$ is the models relation. </section> {{transclude:Definition:Semantic Consequence/Boolean Interpretations/Single Formula   |section = tc   |title = Semantic Consequence of Single Formula   |header = 3   |link = true   |increase = 1 }} === Notation === That $\mathbf A$ is a '''semantic consequence''' of $\mathcal F$ is denoted as: :$\mathcal F \models_{\mathbf{BI}} \mathbf A$
Definition:Union of Adjacent Open Intervals
Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$. Let $a, b, c \in \R$ where $a < b < c$. Let $A$ be the union of the two open intervals: :$A := \openint a b \cup \openint b c$ Then $\struct {A, \tau_d}$ is the '''union of adjacent open intervals'''.
Definition:Logical Equivalence
If two statements $p$ and $q$ are such that: :$p \vdash q$, that is: $p$ therefore $q$ :$q \vdash p$, that is: $q$ therefore $p$ then $p$ and $q$ are said to be '''(logically) equivalent'''. That is: :$p \dashv \vdash q$ means: :$p \vdash q$ and $q \vdash p$. Note that because the conclusion of an argument is a single statement, there can be only one statement on either side of the $\dashv \vdash$ sign. In symbolic logic, the notion of '''logical equivalence''' occurs in the form of provable equivalence and semantic equivalence. === Provable Equivalence === {{:Definition:Provable Equivalence}} === Semantic Equivalence === {{:Definition:Semantic Equivalence}}
Definition:Provable Equivalence
Let $\mathcal P$ be a proof system for a formal language $\mathcal L$. Let $\phi, \psi$ be $\mathcal L$-WFFs. Then $\phi$ and $\psi$ are '''$\mathscr P$-provably equivalent''' : :$\phi \vdash_{\mathscr P} \psi$ and $\psi \vdash_{\mathscr P} \phi$ that is,  they are $\mathscr P$-provable consequences of one another. The '''provable equivalence''' of $\phi$ and $\psi$ can be denoted by: :$\phi \dashv \vdash_{\mathscr P} \psi$
Definition:Semantic Equivalence
Let $\mathscr M$ be a formal semantics for a formal language $\mathcal L$. Let $\phi, \psi$ be $\mathcal L$-WFFs. Then $\phi$ and $\psi$ are $\mathscr M$-'''semantically equivalent''' : :$\phi \models_{\mathscr M} \psi$ and $\psi \models_{\mathscr M} \phi$ that is, iff they are $\mathscr M$-semantic consequences of one another.
Definition:Semantic Equivalence/Boolean Interpretations
<section notitle="true" name="tc"> Let $\mathbf A, \mathbf B$ be WFFs of propositional logic. === Definition 1 === {{:Definition:Semantic Equivalence/Boolean Interpretations/Definition 1}} === Definition 2 === {{:Definition:Semantic Equivalence/Boolean Interpretations/Definition 2}} === Definition 3 === {{:Definition:Semantic Equivalence/Boolean Interpretations/Definition 3}} </section>
Definition:Semantic Consequence/Boolean Interpretations/Single Formula
<section notitle="true" name="tc"> Let $\mathbf A, \mathbf B$ be WFFs of propositional logic. === Definition 1 === {{:Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 1}} === Definition 2 === {{:Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 2}} </section> === Notation === That $\mathbf A$ is a '''consequence of $\mathbf B$ for boolean interpretations''' can be denoted as: :$\mathbf B \models_{\mathrm{BI}} \mathbf A$
Definition:Semantic Equivalence/Boolean Interpretations/Definition 1
Then $\mathbf A$ and $\mathbf B$ are '''equivalent for boolean interpretations''' : :$\mathbf A \models_{\mathrm{BI}} \mathbf B$ and $\mathbf B \models_{\mathrm{BI}} \mathbf A$ that is, each is a semantic consequence of the other. That is to say, $\mathbf A$ and $\mathbf B$ are '''equivalent''' : :$v \left({\mathbf A}\right) = T$  $v \left({\mathbf B}\right) = T$ for all boolean interpretations $v$.
Definition:Semantic Equivalence/Boolean Interpretations/Definition 2
Then $\mathbf A$ and $\mathbf B$ are '''equivalent for boolean interpretations''' : :$\map v {\mathbf A} = \map v {\mathbf B}$ for all boolean interpretations $v$.
Definition:Semantic Equivalence/Boolean Interpretations/Definition 3
Then $\mathbf A$ and $\mathbf B$ are '''equivalent for boolean interpretations''' : :$\mathbf A \iff \mathbf B$ is a tautology where $\iff$ is the biconditional connective.
Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 1
Then $\mathbf A$ is a '''semantic consequence''' of $\mathbf B$ iff: :$v \models_{\mathrm{BI}} \mathbf B$ implies $v \models_{\mathrm{BI}} \mathbf A$ for all boolean interpretations $v$. Here, $\models_{\mathrm{BI}}$ is the models relation.
Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 2
Then $\mathbf A$ is a '''semantic consequence''' of $\mathbf B$ : :$\mathbf A \implies \mathbf B$ is a tautology where $\implies$ is the conditional connective.
Definition:Trace (Linear Algebra)/Matrix
Let $A = \sqbrk a_n$ be a square matrix of order $n$. The '''trace''' of $A$ is: :$\ds \map \tr A = \sum_{i \mathop = 1}^n a_{ii}$
Definition:Trace (Linear Algebra)/Linear Operator
Let $V$ be a vector space. Let $A: V \to V$ be a linear operator of $V$. The '''trace''' of $A$ is the trace of the matrix of $A$ with respect to some basis.
Definition:P-Element
Let $\left({G, \circ}\right)$ be a group. Let $p$ be a prime number. Let $x \in G$ be an element of $G$. Then $x$ is a '''$p$-element of $G$'''  its order is a power of $p$.
Definition:P-Group
A '''$p$-group''' is a group whose elements are all $p$-elements.
Definition:Elementary Abelian P-Group
Let $G$ be an abelian group. Then $G$ is an '''elementary abelian $p$-group'''  all its non-trivial elements are of order $p$.
Definition:Abelian Group/Definition 1
An '''abelian group''' is a group $G$ where: : $\forall a, b \in G: a b = b a$ That is, every element in $G$ commutes with every other element in $G$.
Definition:Abelian Group/Definition 2
An '''abelian group''' is a group $G$ : : $G = \map Z G$ where $\map Z G$ is the center of $G$.
Definition:Normal Subset/Definition 6
:$\map {N_G} S = G$ where $\map {N_G} S$ denotes the normalizer of $S$ in $G$.
Definition:Normal Subset/Definition 7
:$\forall g \in G: g \circ S \subseteq S \circ g$ or: :$\forall g \in G: S \circ g \subseteq g \circ S$
Definition:Normal Subgroup/Notation
The statement that $N$ is a normal subgroup of $G$ is represented symbolically as $N \lhd G$. A normal subgroup is often represented by the letter $N$, as opposed to $H$ (which is used for a general subgroup which may or may not be normal). To use the notation introduced in the definition of the conjugate: :$N \lhd G \iff \forall g \in G: N^g = N$
Definition:Refinement of Normal Series/Proper Refinement
A '''proper refinement''' of a normal series is a refinement which is not equal to the original normal series. That is, it contains extra (normal) subgroups which are not present in the original normal series.
Definition:Characteristic Subset
Let $G$ be a group. Let $S \subseteq G$ be a subset of $G$ such that: :$\forall \phi \in \Aut G: \map \phi S = S$ where $\Aut G$ is the automorphism group of $G$. Then $S$ is  '''characteristic (in $G$)''', or '''a characteristic subset of $G$'''.
Definition:Minimal Subgroup
Let $G$ be a group. Let $M \le G$ be a non-trivial subgroup of $G$. Then $M$ is a '''minimal subgroup of $G$''' : :For every subgroup $H$ of $G$, $H \subseteq M$ means $H = M$ or $H = \set e$. That is, if there is no subgroup of $M$, except $M$ and $\set e$ itself, which is a subset of $M$.
Definition:Asymptotics
Suppose the task to evaluate a certain number is sufficiently unwieldy that it becomes impractical to calculate it directly. In such cases it is frequently useful to have a completely different method to calculate an approximation to that number. Let the accuracy of this approximation improve as the number of operations involved in its calculation increases. The field of mathematics concerned with studying such techniques is called '''asymptotics'''.
Definition:Primitive Triple
Let $x, y, z \in \Z_{>0}$ be (strictly) positive integers. Let $A, B, C \in \Z_{>0}$ be (strictly) positive integers that satisfy: :$A^x + B^y = C^z$ Let $\left({A, B, C}\right)$ be pairwise coprime. Then $\left({A, B, C}\right)$ is a '''primitive triple'''.
Definition:Pythagorean Triple/Primitive/Canonical Form
Let $\tuple {x, y, z}$ be a primitive Pythagorean triple. The convention for representing $\tuple {x, y, z}$ as a (primitive) Pythagorean triple is that $x$ is the even element, while $y$ and $z$ are both odd. This is the '''canonical form''' of a Pythagorean triple.
Definition:Pythagorean Triple/Primitive
Let $\tuple {x, y, z}$ be a Pythagorean triple such that $x \perp y$ (that is, $x$ and $y$ are coprime). Then $\tuple {x, y, z}$ is a '''primitive Pythagorean triple'''.
Definition:Perfect Set/Definition 1
A '''perfect set''' of a topological space $T = \left({S, \tau}\right)$ is a subset $H \subseteq S$ such that: : $H = H'$ where $H'$ is the derived set of $H$. That is, where: : every point of $H$ is a limit point of $H$ and : every limit point of $H$ is a point of $H$.
Definition:Perfect Set/Definition 2
A '''perfect set''' of a topological space $T = \left({S, \tau}\right)$ is a subset $H \subseteq S$ such that: : $H$ is a closed set of $T$ : $H$ has no isolated points.
Definition:Perfect Set/Definition 3
A '''perfect set''' of a topological space $T = \struct {S, \tau}$ is a subset $H \subseteq S$ such that: :$H$ is dense-in-itself. :$H$ contains all its limit points.
Definition:Conjugate Symmetric Mapping
Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$ Let $\innerprod \cdot \cdot: V \times V \to \mathbb F$ be a mapping. Then $\innerprod \cdot \cdot: V \times V \to \mathbb F$ is '''conjugate symmetric''' : :$\forall x, y \in V: \quad \innerprod x y = \overline {\innerprod y x}$ where $\overline {\innerprod y x}$ denotes the complex conjugate of $\innerprod x y$.
Definition:Symmetric Mapping (Linear Algebra)
Let $\R$ be the field of real numbers. Let $\F$ be a subfield of $\R$. Let $V$ be a vector space over $\F$ Let $\innerprod \cdot \cdot: V \times V \to \mathbb F$ be a mapping. Then $\innerprod \cdot \cdot: V \times V \to \mathbb F$ is '''symmetric''' : :$\forall x, y \in V: \innerprod x y = \innerprod y x$
Definition:Non-Negative Definite Mapping
Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$ Let $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ be a mapping. Then $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ is '''non-negative definite''' : :$\forall x \in V: \quad \left \langle {x, x} \right \rangle \in \R_{\geq 0}$ That is, the image of $\left \langle {x, x} \right \rangle$ is always a non-negative real number.
Definition:Positiveness
Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$ Let $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ be a mapping. Then $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ is '''positive''' iff: :$\forall x \in V: \quad \left \langle {x, x} \right \rangle = 0 \implies x = \mathbf 0_V$ where $\mathbf 0_V$ denotes the zero vector of $V$.
Definition:Prime Number/Definition 4
A '''prime number''' $p$ is an integer greater than $1$ that has no positive integer divisors other than $1$ and $p$.
Definition:Partial Subtraction
'''Partial subtraction''' is the operation $\dot - : \N \times \N \to \N$ defined on the set of natural numbers $\N$ as follows: :$\forall a, b \in \N: a \mathop {\dot -} b = \begin{cases} 0 & : a < b \\ a - b & : a \ge b \end{cases}$ where $a - b$ denotes natural number subtraction.
Definition:Algebraically Closed Field/Definition 1
:The only algebraic field extension of $K$ is $K$ itself.
Definition:Algebraically Closed Field/Definition 2
:Every irreducible polynomial $f$ over $K$ has degree $1$.
Definition:Algebraically Closed Field/Definition 3
:Every polynomial $f$ over $K$ of strictly positive degree has a root in $K$.
Definition:Hilbert 23
The '''Hilbert 23''' is a list of $23$ at-the-time unsolved problems in mathematics published by {{AuthorRef|David Hilbert}} during $1900$. They are as follows: === 1: The Continuum Hypothesis === {{:Definition:Hilbert 23/1}} === 2: Consistency of Axioms of Mathematics === {{:Definition:Hilbert 23/2}} === 3: Finite Dissection of Polyhedra === {{:Definition:Hilbert 23/3}} === 4: Construction of all Metrics where Lines are Geodesics === {{:Definition:Hilbert 23/4}} === 5: Whether Continuous Groups are Differential Groups === {{:Definition:Hilbert 23/5}} === 6: Axiomatize all of Physics === {{:Definition:Hilbert 23/6}} === 7: The Gelfond-Schneider Theorem === {{:Definition:Hilbert 23/7}} === 8a: The Riemann Hypothesis === {{:Definition:Hilbert 23/8a}} === 8b: The Goldbach Conjecture === {{:Definition:Hilbert 23/8b}} === 8c: The Twin Prime Conjecture === {{:Definition:Hilbert 23/8c}} === 9: General Reciprocity Theorem in Algebraic Number Field === {{:Definition:Hilbert 23/9}} === 10: Algorithm to determine whether Polynomial Diophantine Equation has Integer Solution === {{:Definition:Hilbert 23/10}} === 11: Quadratic Forms with Algebraic Numerical Coefficients === {{:Definition:Hilbert 23/11}} === 12: Extension of Kronecker-Weber Theorem to any base Number Field === {{:Definition:Hilbert 23/12}} === 13: Solution of 7th Degree Equations using Two Parameter Functions === {{:Definition:Hilbert 23/13}} === 14: Proof of Finiteness of certain Complete Systems of Functions === {{:Definition:Hilbert 23/14}} === 15: Rigorous foundation of Schubert's Enumerative Calculus === {{:Definition:Hilbert 23/15}} === 16: Topology of Algebraic Curves and Surfaces === {{:Definition:Hilbert 23/16}} === 17: Definite Rational Function as Quotient of Sums of Squares === {{:Definition:Hilbert 23/17}} === 18a: Existence of Non-Regular Space-Filling Polyhedron === {{:Definition:Hilbert 23/18a}} === 18b: Kepler's Conjecture (Densest Sphere Packing) === {{:Definition:Hilbert 23/18b}} === 19: Solutions of Lagrangian are Analytic === {{:Definition:Hilbert 23/19}} === 20: Existence of Solutions of Variational Problems with certain Boundary Conditions === {{:Definition:Hilbert 23/20}} === 21: Existence of Linear Differential Equation with prescribed Monodromic Group === {{:Definition:Hilbert 23/21}} === 22: Uniformization of Analytic Relations by means of Automorphic Functions === {{:Definition:Hilbert 23/22}} === 23: Further Development of the Calculus of Variations === {{:Definition:Hilbert 23/23}}
Definition:Millennium Problems
The '''Millennium problems''' are a collection of seven mathematical problems stated by the {{WP|Clay_Mathematics_Institute|Clay Mathematics Institute}} on $24$ May $2000$. Each carries a prize of $\$1 \, 000 \, 000$ (US dollars). Six of them remain unsolved. They are as follows: # P versus NP # The Hodge Conjecture # The Poincaré Conjecture (the only one of the seven to be solved) # The Riemann Hypothesis # Yang-Mills Existence and Mass Gap # Navier-Stokes Existence and Smoothness # The Birch and Swinnerton-Dyer Conjecture
Definition:Almost Perfect Number
Let $n \in \Z_{\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Almost Perfect Number/Definition 1}} === Definition 2 === {{:Definition:Almost Perfect Number/Definition 2}} === Definition 3 === {{:Definition:Almost Perfect Number/Definition 3}} Thus an '''almost perfect number''' can be described as "deficient, but only just".
Definition:Quasiperfect Number
Let $n \in \Z_{\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Quasiperfect Number/Definition 1}} === Definition 2 === {{:Definition:Quasiperfect Number/Definition 2}} === Definition 3 === {{:Definition:Quasiperfect Number/Definition 3}}
Definition:Abundant Number/Definition 1
Let $A \left({n}\right)$ denote the abundance of $n$. $n$ is '''abundant'''  $A \left({n}\right) > 0$.
Definition:Abundant Number/Definition 2
Let $\sigma \left({n}\right)$ be the sigma function of $n$. $n$ is '''abundant'''  $\dfrac {\sigma \left({n}\right)} n > 2$.
Definition:Deficient Number/Definition 1
Let $A \left({n}\right)$ denote the abundance of $n$. $n$ is '''deficient'''  $A \left({n}\right) < 0$.
Definition:Deficient Number/Definition 2
Let $\sigma \left({n}\right)$ be the sigma function of $n$. $n$ is '''deficient''' : :$\dfrac {\sigma \left({n}\right)} n < 2$
Definition:Antisymmetric Matrix
Let $R$ be a ring. Let $\mathbf A$ be a square matrix over $R$. $\mathbf A$ is '''antisymmetric''' : :$\mathbf A = - \mathbf A^\intercal$ where $\mathbf A^\intercal$ is the transpose of $\mathbf A$.
Definition:Orthogonal Matrix
Let $R$ be a ring with unity. Let $\mathbf Q$ be an invertible square matrix over $R$. Then $\mathbf Q$ is '''orthogonal''' : :$\mathbf Q^{-1} = \mathbf Q^\intercal$ where: :$\mathbf Q^{-1}$ is the inverse of $\mathbf Q$ :$\mathbf Q^\intercal$ is the transpose of $\mathbf Q$
Definition:Proper Orthogonal Matrix
Let $\mathbf Q$ be an orthogonal matrix. Then $\mathbf Q$ is a '''proper orthogonal matrix''' : :$\map \det {\mathbf Q} = 1$ where $\map \det {\mathbf Q}$ is the determinant of $\mathbf Q$.
Definition:Einstein Summation Convention
The '''Einstein summation convention''' is a notational device used in the manipulation of matrices and vectors, in particular square matrices in the context of physics and applied mathematics. If the same index occurs twice in a given expression involving matrices, then summation over that index is automatically assumed. Thus the summation sign can be omitted, and expressions can be written more compactly.
Definition:Trace (Linear Algebra)/Matrix/Einstein Summation Convention
The trace of $A$, using the Einstein summation convention, is: :$\map \tr A = a_{ii}$
Definition:Determinant/Matrix/Order 3/Einstein Summation Convention
The determinant of a square matrix of order $3$ $\mathbf A$ can be expressed using the Einstein summation convention as: :$\map \det {\mathbf A} = \dfrac 1 6 \map \sgn {i, j, k} \map \sgn {r, s, t} a_{i r} a_{j s} a_{k t}$ Note that there are $6$ indices which appear twice, and so $6$ summations are assumed.
Definition:Matrix/Square Matrix/Order
Let $\mathbf A$ be an $n \times n$ square matrix. That is, let $\mathbf A$ have $n$ rows (and by definition $n$ columns). Then the '''order''' of $\mathbf A$ is defined as being $n$.
Definition:Matrix Product (Conventional)/Einstein Summation Convention
The matrix product of $\mathbf A$ and $\mathbf B$ can be expressed using the Einstein summation convention as: Then: :$c_{i j} := a_{i k} \circ b_{k j}$ The index which appears twice in the expressions on the  is the entry $k$, which is the one summated over.
Definition:Dot Product/Definition 1
The '''dot product''' of $\mathbf a$ and $\mathbf b$ is defined as: :$\ds \mathbf a \cdot \mathbf b = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = \sum_{i \mathop = 1}^n a_i b_i$ If the vectors are represented as column matrices: :$\mathbf a = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix} , \mathbf b = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}$ we can express the dot product as: :$\mathbf a \cdot \mathbf b = \mathbf a^\intercal \mathbf b$ where: :$\mathbf a^\intercal = \begin{bmatrix} a_1 & a_2 & \cdots & a_n \end{bmatrix}$ is the transpose of $\mathbf a$ :the operation between the matrices is the matrix product.
Definition:Dot Product/Definition 2
The '''dot product''' of $\mathbf a$ and $\mathbf b$ is defined as: :$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \cos \angle \mathbf a, \mathbf b$ where: :$\norm {\mathbf a}$ denotes the length of $\mathbf a$ :$\angle \mathbf a, \mathbf b$ is the angle between $\mathbf a$ and $\mathbf b$, taken to be between $0$ and $\pi$.
Definition:Left Module Axioms
{{begin-axiom}} {{axiom | n = \text M 1         | lc= Scalar Multiplication (Left) Distributes over Module Addition         | q = \forall \lambda \in R: \forall x, y \in G         | ml= \lambda \circ \paren {x +_G y}         | mo= =         | mr= \paren {\lambda \circ x} +_G \paren {\lambda \circ y} }} {{axiom | n = \text M 2         | lc= Scalar Multiplication (Right) Distributes over Scalar Addition         | q = \forall \lambda, \mu \in R: \forall x \in G         | ml= \paren {\lambda +_R \mu} \circ x         | mo= =         | mr= \paren {\lambda \circ x} +_G \paren {\mu \circ x} }} {{axiom | n = \text M 3         | lc= Associativity of Scalar Multiplication         | q = \forall \lambda, \mu \in R: \forall x \in G         | ml= \paren {\lambda \times_R \mu} \circ x         | mo= =         | mr= \lambda \circ \paren {\mu \circ x} }} {{end-axiom}}
Definition:Unitary Module Axioms
{{begin-axiom}} {{axiom | n = \text {UM} 1         | q = \forall \lambda \in R: \forall x, y \in G         | m = \lambda \circ \paren {x +_G y} = \paren {\lambda \circ x} +_G \paren {\lambda \circ y} }} {{axiom | n = \text {UM} 2         | q = \forall \lambda, \mu \in R: \forall x \in G         | m = \paren {\lambda +_R \mu} \circ x = \paren {\lambda \circ x} +_G \paren {\mu \circ x} }} {{axiom | n = \text {UM} 3         | q = \forall \lambda, \mu \in R: \forall x \in G         | m = \paren {\lambda \times_R \mu} \circ x = \lambda \circ \paren {\mu \circ x} }} {{axiom | n = \text {UM} 4         | q = \forall x \in G         | m = 1_R \circ x = x }} {{end-axiom}}
Definition:Simultaneous Equations/Solution
An ordered $n$-tuple $\tuple {x_1, x_2, \ldots, x_n}$ which satisfies each of the equations in a system of $m$ simultaneous equations in $n$ variables is called '''a solution''' of the system.