hcju/proofwikips huggingface dataset:
Definition:Order Embedding/Definition 4
Let $T' = \Img S$ be the image of $S$ under $\phi$. $\phi$ is an '''order embedding of $S$ into $T$''' : :the restriction of $\phi$ to $S \times T'$ is an order isomorphism between $\struct {S, \preceq_1}$ and $\struct {T', \preceq_2 \restriction_{T' \times T'} }$.
Definition:Argumentum ad Passiones
An '''argumentum ad passiones''' is a logical argument that, rather than prove or present evidence for a claim, attempts to generate support for an idea by instilling either: : an emotional attachment to the argument being presented : a feeling of embarrassment at holding a position which is subject to ridicule or : a feeling of distaste for the argument being refuted. Such arguments are commonly seen along with argumentum ad hominem.
Definition:Order Embedding/Definition 3
$\phi$ is an '''order embedding of $S$ into $T$'''  both of the following conditions hold: :$(1): \quad \phi$ is an injection :$(2): \quad \forall x, y \in S: x \prec_1 y \iff \map \phi x \prec_2 \map \phi y$
Definition:Null Polynomial/Ring
Let $\left({R, +, \times}\right)$ be a ring. The zero $0_R$ of $R$ can be considered as being the '''null polynomial over $R$''' of any arbitrary element $x$ of $R$.
Definition:Null Polynomial/Polynomial Form
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial form over $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$. For all $i = 1, 2, \ldots, r$, let $a_i = 0$. Then $f$ is the '''null polynomial''' in the indeterminates $\left\{{X_j: j \in J}\right\}$.
Definition:Polynomial/Real Numbers
A '''polynomial (in $\R$)''' is an expression of the form: :$\displaystyle \map P x = \sum_{j \mathop = 0}^n \paren {a_j x^j} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{n - 1} x^{n - 1} + a_n x^n$ where: :$x \in \R$ :$a_0, \ldots a_n \in \mathbb k$ where $\mathbb k$ is one of the standard number sets $\Z, \Q, \R$.
Definition:Polynomial/Complex Numbers
A '''polynomial (in $\C$)''' is an expression of the form: :$\displaystyle P \left({z}\right) = \sum_{j \mathop = 0}^n \left({a_j z^j}\right) = a_0 + a_1 z + a_2 z^2 + \cdots + a_{n-1} z^{n-1} + a_n z^n$ where: :$z \in \C$ :$a_0, \ldots a_n \in \mathbb k$ where $\mathbb k$ is one of the standard number sets $\Z, \Q, \R, \C$.
Definition:Degree of Polynomial/Ring
{{questionable|ill-defined}} Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({S, +, \circ}\right)$ be a subring of $R$. Let $x \in R$. Let $\displaystyle P = \sum_{j \mathop = 0}^n \left({r_j \circ x^j}\right) = r_0 + r_1 \circ x + \cdots + r_n \circ x^n$ be a polynomial in the element $x$ over $S$ such that $r_n \ne 0$. Then the '''degree of $P$''' is $n$. The '''degree of $P$''' can be denoted $\deg \left({P}\right)$ or $\partial P$.
Definition:Degree of Polynomial/Integral Domain
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $X \in R$ be transcendental over $D$. Let $\displaystyle f = \sum_{j \mathop = 0}^n \paren {r_j \circ X^j} = r_0 + r_1 X + \cdots + r_n X^n$ be a polynomial over $D$ in $X$ such that $r_n \ne 0$. Then the '''degree of $f$''' is $n$. The '''degree of $f$''' is denoted on  by $\map \deg f$.
Definition:Degree of Polynomial/Sequence/Field
Let $f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ be a polynomial over a field $F$. The '''degree of $f$''' is defined as the largest $n \in \Z$ such that $a_n \ne 0$.
Definition:Antisymmetric Quotient
Let $\struct {S, \precsim}$ be a preordered set. Let $\sim$ be the equivalence relation on $S$ induced by $\precsim$. Let $S / {\sim}$ be the quotient set of $S$ by $\sim$. Let $\preceq$ be the relation on $S / {\sim}$ defined by letting $P \preceq Q$ : :$\exists p \in P: \exists q \in Q: p \precsim q$ Then $\struct {S / {\sim}, \preceq}$ is the '''antisymmetric quotient''' of $\struct {S, \precsim}$.
Definition:Fuzzy Set
Let $S$ be a set. Let $\left [{0 \,.\,.\, 1}\right]$ denotes the closed unit interval. Let $\mu: S \to \left [{0 \,.\,.\, 1}\right]$ be a mapping. Then $\left({S, \mu}\right)$ is a '''fuzzy set'''.
Definition:Fuzzy Set/Domain
$S$ is the '''domain''' of $\left({S, \mu}\right)$.
Definition:Fuzzy Set/Membership Function
$\mu$ is the '''membership function''' of $\left({S, \mu}\right)$.
Definition:Fuzzy Intersection
Let $\textbf A = \left({A, \mu_A}\right)$ and $\textbf B = \left({B, \mu_B}\right)$ be fuzzy sets. Then the '''(fuzzy) intersection''' of $\textbf A$ and $\textbf B$ is denoted $\textbf A \cap \textbf B$. It is defined to be the fuzzy set such that: * The domain of $\textbf A \cap \textbf B$ is $A \cap B$. * The membership function of $\textbf A \cap \textbf B$ is: ::$\forall x \in A \cap B: \mu_{A \cap B}(x) = \operatorname{min}\left({\mu_A(x), \mu_B(x)}\right)$ where $\operatorname{min}$ is the min operation.
Definition:Addition of Polynomials
Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({S, +, \circ}\right)$ be a subring of $R$. For arbitrary $x \in R$, let $S \left[{x}\right]$ be the set $S \left[{x}\right]$ be the set of polynomials in $x$ over $S$. Let  $p, q \in S \left[{x}\right]$ be polynomials in $x$ over $S$: : $\displaystyle p = \sum_{k \mathop = 0}^m a_k \circ x^k$ : $\displaystyle q = \sum_{k \mathop = 0}^n b_k \circ x^k$ where: : $(1): \quad a_k, b_k \in S$ for all $k$ : $(2): \quad m, n \in \Z_{\ge 0}$. The operation '''polynomial addition''' is defined as: :$\displaystyle p + q := \sum_{k \mathop = 0}^{\max \left({m, n}\right)} \left({a_k + b_k}\right) x^k$ where: :$\forall k \in \Z: k > m \implies a_k = 0$ :$\forall k \in \Z: k > n \implies b_k = 0$ The expression $p + q$ is known as the '''sum''' of $p$ and $q$. === Polynomial Forms === {{:Definition:Addition of Polynomials/Polynomial Forms}} === Polynomials as Sequences === {{:Definition:Addition of Polynomials/Sequence}}
Definition:Zero Divisor/Commutative Ring
Let $\struct {R, +, \circ}$ be a commutative ring. A '''zero divisor (in $R$)''' is an element $x \in R$ such that: : $\exists y \in R^*: x \circ y = 0_R$ where $R^*$ is defined as $R \setminus \set {0_R}$. The expression: :'''$x$ is a zero divisor''' can be written: : $x \divides 0_R$
Definition:Leading Coefficient of Polynomial/Polynomial Form
Let $R$ be a commutative ring with unity. Let $f = a_0 + a_1 X + \cdots + a_{r-1} X^{r-1} + a_r X^r$ be a polynomial form in the single indeterminate $X$ over $R$. Then the ring element $a_r$ is called the '''leading coefficient''' of $f$.
Definition:Degree of Polynomial/Field
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $\struct {K, +, \times}$ be a subfield of $F$. Let $x \in F$. Let $\displaystyle f = \sum_{j \mathop = 0}^n \paren {a_j x^j} = a_0 + a_1 x + \cdots + a_n x^n$ be a polynomial over $K$ in $x$ such that $a_n \ne 0$. Then the '''degree of $f$''' is $n$. The '''degree of $f$''' can be denoted $\map \deg f$ or $\partial f$.
Definition:Degree of Polynomial/Sequence
=== Ring === {{:Definition:Degree of Polynomial/Sequence/Ring}} === Field === {{:Definition:Degree of Polynomial/Sequence/Field}}
Definition:Degree of Polynomial/Sequence/Ring
Let $f = \left \langle {a_k}\right \rangle = \left({a_0, a_1, a_2, \ldots}\right)$ be a polynomial over a ring $R$. The '''degree of $f$''' is defined as the largest $n \in \Z$ such that $a_n \ne 0$.
Definition:Degree of Polynomial/Polynomial Form
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial in the indeterminates $\family {X_j: j \in J}$ for some multiindices $k_1, \ldots, k_r$. Let $f$ '''not''' be the null polynomial. Let $k = \family {k_j}_{j \mathop \in J}$ be a multiindex. Let $\ds \size k = \sum_{j \mathop \in J} k_j \ge 0$ be the degree of the monomial $\mathbf X^k$. The '''degree of $f$''' is the supremum: :$\ds \map \deg f = \max \set {\size {k_r}: i = 1, \ldots, r}$
Definition:Coefficient of Polynomial/Polynomial Form
Let $R$ be a commutative ring with unity. Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$. The ring elements $a_1, \ldots, a_r$ are the '''coefficients''' of $f$.
Definition:Monic Polynomial/Polynomial Form
Let $R$ be a commutative ring with unity $1_R$. Let $f = a_0 + a_1 X + \cdots + a_{r-1} X^{r-1} + a_r X^r$ be a polynomial from in the single indeterminate $X$ over $R$. Then $f$ is '''monic''' if the leading coefficient of $f$ is $1_R$.
Definition:Degree of Polynomial/Null Polynomial/Integral Domain
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. For arbitrary $x \in R$, let $D \sqbrk x$ be the ring of polynomials in $x$ over $D$. The null polynomial $0_R \in D \sqbrk x$ does ''not'' have a degree.
Definition:Inflationary Mapping/Subset
Let $C$ be a class. Let $f: C \to C$ be a mapping from $C$ to $C$. Then $f$ is '''inflationary''' : :$x \in C \implies x \subseteq \map f x$ That is,  for each $x \in C$, $x$ is a subset of $\map f x$.
Definition:Subclass
Let $A$ and $B$ be classes. Then $A$ is a '''subclass''' of $B$, and we write $A \subseteq B$, : :$\forall x: \paren {x \in A \implies x \in B}$ where $x \in A$ denotes that $x$ is an element of $A$.
Definition:Inductive Class
Let $A$ be a class. Then $A$ is '''inductive''' : {{begin-axiom}} {{axiom | n = 1         | lc= $A$ contains the empty set:         | q =          | m = \quad \O \in A }} {{axiom | n = 2         | lc= $A$ is closed under the successor mapping:         | q = \forall x         | m = \paren {x \in A \implies x^+ \in A}         | rc= where $x^+$ is the successor of $x$ }} {{axiom | rc= That is, where $x^+ = x \cup \set x$ }} {{end-axiom}}
Definition:Polynomial Function/Real
Let $S \subset \R$ be a subset of the real numbers. === Definition 1 === {{Definition:Polynomial Function/Real/Definition 1}} === Definition 2 === {{Definition:Polynomial Function/Real/Definition 2}}
Definition:Polynomial
=== Real Numbers === {{Definition:Polynomial/Real Numbers}} === Complex Numbers === {{Definition:Polynomial/Complex Numbers}}
Definition:Polynomial Function/Complex
Let $S \subset \C$ be a subset of the complex numbers. === Definition 1 === {{Definition:Polynomial Function/Complex/Definition 1}} === Definition 2 === {{Definition:Polynomial Function/Complex/Definition 2}}
Definition:Argumentum ad Verecundiam
'''Argumentum ad verecundiam''', or '''argument from authority''', is a logical argument that infers something is true on the basis that a knowledgeable and well-reputed person has said it. An '''argument from authority''' may be valid or invalid, depending on the circumstances. If the trustworthy person is also an expert in the field, and is free to speak their mind about the matter, then the argument has some validity. Otherwise, it generally does not. That said, '''argumentum ad verecundiam''' can never serve as a (mathematical) proof, since no authority could ever be infallible. Thus, in the context of , such an argument should be considered fallacious.
Definition:Eisenstein Integer
An '''Eisenstein integer''' is a complex number of the form :$a + b \omega$ where $a$ and $b$ are both integers and: :$\omega = e^{2 \pi i / 3} = \dfrac 1 2 \paren {i \sqrt 3 - 1}$ that is, the (complex) cube roots of unity. The set of all '''Eisenstein integers''' can be denoted $\Z \left[{\omega}\right]$: :$\Z \sqbrk \omega = \set {a + b \omega: a, b \in \Z}$
Definition:Eisenstein Prime
Let $\Z \sqbrk \omega$ be the ring of Eisenstein integers. Let $\alpha \in \Z \sqbrk \omega$ be an Eisenstein integer. Then $\alpha$ is an '''Eisenstein prime''' if $\alpha$ is prime in $\Z \sqbrk \omega$. {{NamedforDef|Ferdinand Gotthold Max Eisenstein|cat = Eisenstein}}
Definition:Degree of Polynomial/Null Polynomial/Sequence
Let $f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ be a polynomial over a field $F$. Let $0_F$ be the zero of $F$. Let $a_0 = a_1 = a_2 = \ldots = 0_F$. Then $f$ is a '''null polynomial over $F$'''.
Definition:Monic Polynomial/Sequence
Let $f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ be a polynomial over a field $F$. Then $f$ is a '''monic polynomial'''  its leading coefficient $a_n$ is $1$.
Definition:Addition of Polynomials/Sequence
Let: :$f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ and: :$g = \sequence {b_k} = \tuple {b_0, b_1, b_2, \ldots}$ be polynomials over a field $F$. Then the operation of '''(polynomial) addition''' is defined as: :$f + g := \tuple {a_0 + b_0, a_1 + b_1, a_2 + b_2, \ldots}$
Definition:Multiplication of Polynomials/Sequence
Let: : $f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ and: : $g = \sequence {b_k} = \tuple {b_0, b_1, b_2, \ldots}$ be polynomials over a field $F$. Then the operation of '''(polynomial) multiplication''' is defined as: :$f g := \tuple {c_0, c_1, c_2, \ldots}$ where $\displaystyle c_i = \sum_{j \mathop + k \mathop = i} a_j b_k$
Definition:Standard Affine Space
Let $n \geq 0$ be an integer. Let $k$ be a field. The '''standard affine space of dimension $n$''' over $k$ is the vector space $k^n$ together with the standard affine structure.
Definition:Null Polynomial/Sequence
Let $f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ be a polynomial over a field $F$. Let $0_F$ be the zero of $F$. Let $a_0 = a_1 = a_2 = \ldots = 0_F$. Then $f$ is known as the '''null polynomial'''.
Definition:Irreducible Polynomial
{{Definition:Irreducible Polynomial/Definition 1}}
Definition:Operations on Polynomial Ring of Sequences
{{begin-axiom}} {{axiom | n = 1         | lc= '''Ring Addition:'''         | ml= \sequence {r_0, r_1, r_2, \ldots} \oplus \sequence {s_0, s_1, s_2, \ldots}         | mo= =         | mr= \sequence {r_0 + s_0, r_1 + s_1, r_2 + s_2, \ldots}         | c =  }} {{axiom | n = 2         | lc= '''Ring Negative:'''         | ml= -\sequence {r_0, r_1, r_2, \ldots}         | mo= =         | mr= \sequence {-r_0, -r_1, -r_2, \ldots}         | c =  }} {{axiom | n = 3         | lc= '''Ring Product:'''         | ml= \sequence {r_0, r_1, r_2, \ldots} \odot \sequence {s_0, s_1, s_2, \ldots}         | mo= =         | mr= \sequence {t_0, t_1, t_2, \ldots}         | rc= where $\displaystyle t_i = \sum_{j \mathop + k \mathop = i} r_j \circ s_k$ }} {{end-axiom}}
Definition:Constant Polynomial
Let $R$ be a commutative ring with unity. Let $P \in R \left[{x}\right]$ be a polynomial in one variable over $R$. === Definition 1 === {{Definition:Constant Polynomial/Definition 1}} === Definition 2 === {{Definition:Constant Polynomial/Definition 2}} === Definition 3 === {{Definition:Constant Polynomial/Definition 3}}
Definition:Greatest Set by Set Inclusion/Class Theory
Let $A$ be a class. Then a set $M$ is the '''greatest element''' of $A$ (with respect to the subset relation) : :$(1): \quad M \in A$ :$(2): \quad \forall S: \paren {S \in A \implies S \subseteq M}$
Definition:Relation of Set Inclusion
The '''relation of set inclusion''', written $\subseteq$, is the class of all ordered pairs $\left({x, y}\right)$ such that $x$ is a subset of $y$.
Definition:Substitution
'''Substitution''' is a word that denotes the process of replacement of a term in an expression or equation with another which has the same value.  * Formal Systems: ** Definition:Substitution (Formal Systems) *** Definition:Substitution for Well-Formed Part *** Definition:Substitution for Letter **** Definition:Alphabetic Substitution *** Definition:Substitution of Term *** Definition:Substitution for Metasymbol * Mathematical Logic: ** Definition:Substitution (Mathematical Logic)
Definition:Well-Formed Part/Proper Well-Formed Part
Let $\mathbf B$ be a well-formed part of $\mathbf A$. Then $\mathbf B$ is a '''proper well-formed part of $\mathbf A$'''  $\mathbf B$ is not equal to $\mathbf A$.
Definition:Logical Not/Truth Function
The logical not connective defines the truth function $f^\neg$ as follows: {{begin-eqn}} {{eqn | l = \map {f^\neg} \F       | r = \T }} {{eqn | l = \map {f^\neg} \T       | r = \F }} {{end-eqn}}
Definition:Conjunction/Truth Function
The conjunction connective defines the truth function $f^\land$ as follows: {{begin-eqn}} {{eqn | l = \map {f^\land} {F, F}       | r = F }} {{eqn | l = \map {f^\land} {F, T}       | r = F }} {{eqn | l = \map {f^\land} {T, F}       | r = F }} {{eqn | l = \map {f^\land} {T, T}       | r = T }} {{end-eqn}}
Definition:Disjunction/Truth Function
The disjunction connective defines the truth function $f^\lor$ as follows: {{begin-eqn}} {{eqn | l = f^\lor \left({F, F}\right)       | r = F }} {{eqn | l = f^\lor \left({F, T}\right)       | r = T }} {{eqn | l = f^\lor \left({T, F}\right)       | r = T }} {{eqn | l = f^\lor \left({T, T}\right)       | r = T }} {{end-eqn}}
Definition:Conditional/Truth Function
The conditional connective defines the truth function $f^\to$ as follows: {{begin-eqn}} {{eqn | l = f^\to \left({F, F}\right)       | r = T }} {{eqn | l = f^\to \left({F, T}\right)       | r = T }} {{eqn | l = f^\to \left({T, F}\right)       | r = F }} {{eqn | l = f^\to \left({T, T}\right)       | r = T }} {{end-eqn}}
Definition:Biconditional/Truth Function
The biconditional connective defines the truth function $f^\leftrightarrow$ as follows: {{begin-eqn}} {{eqn | l = f^\leftrightarrow \left({F, F}\right)       | r = T }} {{eqn | l = f^\leftrightarrow \left({F, T}\right)       | r = F }} {{eqn | l = f^\leftrightarrow \left({T, F}\right)       | r = F }} {{eqn | l = f^\leftrightarrow \left({T, T}\right)       | r = T }} {{end-eqn}}
Definition:Proper Relational Structure
Let $A$ be a set or class. Let $\mathcal R$ be a relation on $A$. Then $(A, \mathcal R)$ is a '''proper relational structure''' : : For each $a \in A$, the preimage $\mathcal R^{-1} \left({a}\right)$ of $a$ under $\mathcal R$ is a set (or small class).
Definition:Transitive Closure (Set Theory)/Definition 2
Let $x$ be a set. For each natural number $n \in \N_{\ge 0}$ let: : $\bigcup^n x = \underbrace{\bigcup \bigcup \cdots \bigcup}_n x$ Then the '''transitive closure''' of $x$ is the union of the sets: :$\left\{ {x}\right\}, x, \bigcup x, \bigcup^2 x, \dots, \bigcup^n x, \dots$
Definition:Transitive Closure (Set Theory)/Definition 1
Let $x$ be a set. Then the '''transitive closure''' of $x$ is the smallest transitive superset of $x$.
Definition:Positively Totally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a totally ordered semigroup. Then $\left({S, \circ, \preceq}\right)$ is a '''positively totally ordered semigroup''' iff for all $a, b \in S$: : $a \preceq a \circ b$ : $b \preceq a \circ b$
Definition:Right Naturally Totally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a positively totally ordered semigroup. Then $\left({S, \circ, \preceq}\right)$ is a '''right naturally totally ordered semigroup''' iff for all $a, b \in S$: : $a < b$ implies that for some $x \in S$, $b = a \circ x$.
Definition:Left Naturally Totally Ordered Semigroup
Let $\struct {S, \circ, \preceq}$ be a positively totally ordered semigroup. Then $\struct {S, \circ, \preceq}$ is a '''left naturally totally ordered semigroup'''  for all $a, b \in S$: :$a < b$ implies that for some $y \in S$, $b = x \circ a$.
Definition:Naturally Totally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a right naturally totally ordered semigroup and a left naturally totally ordered semigroup. Then $\left({S, \circ, \preceq}\right)$ is a '''naturally totally ordered semigroup'''.
Definition:Ordinal/Definition 2
Let $A$ be a set. Then $A$ is an '''ordinal'''  $A$ is: : transitive : epsilon-connected, that is: :: $\forall x, y \in A: x \ne y \implies x \in y \lor y \in x$ : well-founded
Definition:Strongly Well-Founded Relation
Let $A$ be a class. Let $\RR$ be a relation on $A$. Then $A$ is '''strongly well-founded''' : :whenever $B$ is a non-empty subclass of $A$, $B$ has an $\RR$-minimal element.
Definition:Well-Founded Class
Let $A$ be a class. Then $A$ is '''well-founded''' iff for every non-empty subclass $B \subseteq A$: : $\exists x \in B: x \cap B = \varnothing$ {{MissingLinks|non-empty class}}
Definition:Top (Logic)/Truth Table
The characteristic truth table of the top constant $\top$ of propositional logic is as follows: :$\begin{array}{|c|} \hline \top \\ \hline \T \\ \hline \end{array}$
Definition:Bottom (Logic)/Truth Table
The characteristic truth table of the bottom constant $\bot$ of propositional logic is as follows: :$\begin{array}{|c|} \hline \bot \\ \hline \F \\ \hline \end{array}$
Definition:Polygon/Height
The '''height''' of a polygon is the length of a perpendicular from the base to the vertex most distant from the base.  :''{{Definition:Euclid's Definitions - Book VI/4 - Height}}'' {{EuclidDefRefNocat|VI|4|Height}}
Definition:Linear Measure/Height
'''Height''', like depth, is used as a term for linear measure in a dimension perpendicular to both length and breadth. However, whereas depth has connotations of '''down''', '''height''' is used for distances '''up''' from the plane.
Definition:Linear Measure/Breadth
'''Breadth''' is linear measure in a dimension perpendicular to length. In the context of a two-dimensional geometric figure, the '''breadth''' is in the plane of that figure. In a three-dimensional figure, the choice of which direction is referred to as '''breadth''' is often arbitrary.
Definition:Dimension (Measurement)/Fundamental Dimensions
The SI-recommended '''fundamental dimensions''' are: :$\mathsf M$: mass :$\mathsf L$: length  :$\mathsf T$: time :$\Theta$: temperature :$\mathsf I$: electric current :$\mathsf N$: amount of substance :$\mathsf J$: luminous intensity
Definition:Dimension (Measurement)/Units
Compare with units of measurement. This concept of '''dimension''' is more abstract than that of units, which are standard quantities of the particular dimension in question. === Examples === * Displacement has dimension $\mathsf L$. * Velocity has dimension $\mathsf {L T}^{-1}$ (change in displacement, that is length traveled, per unit of time). * Acceleration has dimension $\mathsf {L T}^{-2}$ (change in velocity per unit of time). * Force has dimension $\mathsf {M L T}^{-2}$ (mass times acceleration, from Newton's Second Law of Motion).
Definition:Linear Measure/Length
'''Length''' is linear measure taken in a particular direction. Usually, in multi-dimensional figures, the dimension in which the linear measure is greatest is referred to as '''length'''. It is the most widely used term for linear measure, as it is the standard term used when only one dimension is under consideration. '''Length''' is the fundamental notion of Euclidean geometry, never defined but regarded as an intuitive concept at the basis of every geometrical theorem.
Definition:Linear Measure/Depth
'''Depth''' is linear measure in a dimension perpendicular to both length and breadth. The choice of '''depth''' is often arbitrary, although in two-dimensional diagrams of three-dimensional figures, '''depth''' is usually imagined as being the dimension perpendicular to the plane the figure is drawn in.
Definition:Set/Implicit Set Definition/Infinite Set
If there is no end to the list of elements in the set, the ellipsis can be left open: :$S = \set {1, 2, 3, \ldots}$ which is taken to mean: :'''$S = $ the set containing $1, 2, 3, $ and so on for ever.'''
Definition:Endorelation/General Definition
An $n$-ary relation $\RR$ on a cartesian space $S^n$ is an '''$n$-ary endorelation on $S$''': :$\RR = \struct {S, S, \ldots, S, R}$ where $R \subseteq S^n$.
Definition:Well-Ordering/Definition 1
The ordering $\preceq$ is a '''well-ordering''' on $S$  ''every'' non-empty subset of $S$ has a smallest element under $\preceq$: :$\forall T \subseteq S, T \ne \O: \exists a \in T: \forall x \in T: a \preceq x$
Definition:Well-Ordering/Definition 2
The ordering $\preceq$ is a '''well-ordering''' on $S$  $\preceq$ is a well-founded total ordering.