Definition:Series/General

Let $\left({S, \circ}\right)$ be a semigroup. Let $\left \langle {a_n} \right \rangle$ be a sequence in $S$. Informally, a '''series''' is what results when an infinite product is taken of $\left \langle {a_n} \right \rangle$: :$\displaystyle s := \sum_{n \mathop = 1}^\infty a_n = a_1 \circ a_2 \circ a_3 \circ \cdots$ Formally, a '''series''' is a sequence in $S$.


Definition:Series/Number Field

The '''series''' is what results when $\sequence {a_n}$ is summed to infinity: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$


Definition:Series/Tail

Let $N \in \N$. The expression $\displaystyle \sum_{n \mathop = N}^\infty a_n$ is known as a '''tail''' of the '''series''' $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.


Definition:Conditional/Semantics of Conditional

$p \implies q$ can be stated thus: * '''''If'' $p$ is true ''then'' $q$ is true.''' * '''$q$ is true ''if'' $p$ is true.''' * '''(The truth of) $p$ ''implies'' (the truth of) $q$.''' * '''(The truth of) $q$ ''is implied by'' (the truth of) $p$.''' * '''$q$ ''follows from'' $p$.''' * '''$p$ is true ''only if'' $q$ is true.''' The latter one may need some explanation. $p$ can be either true or false, as can $q$. But if $q$ is false, and $p \implies q$, then $p$ can not be true. Therefore, $p$ can be true ''only if'' $q$ is also true, which leads us to our assertion. * '''$p$ is true ''therefore'' $q$ is true.''' * '''$p$ is true ''entails'' that $q$ is true.''' * '''$q$ is true ''because'' $p$ is true.''' * '''$p$ ''may'' be true ''unless'' $q$ is false.''' * '''''Given that'' $p$ is true, $q$ is true.''' * '''$q$ is true ''whenever'' $p$ is true.''' * '''$q$ is true ''provided that'' $p$ is true.''' * '''$q$ is true ''in case'' $p$ is true.''' * '''$q$ is true ''assuming that'' $p$ is true.''' * '''$q$ is true ''on the condition that'' $p$ is true.'''


Definition:Conditional/Language of Conditional/Weak

In a conditional $p \implies q$, the statement $q$ is '''weaker''' than $p$.


Definition:Conditional/Language of Conditional/Strong

In a conditional $p \implies q$, the statement $p$ is '''stronger''' than $q$.


Definition:Conditional/Superimplicant

In a conditional $p \implies q$, the statement $p$ is '''superimplicant''' to $q$.


Definition:Conditional/Subimplicant

In a conditional $p \implies q$, the statement $q$ is '''subimplicant''' to $p$.


The conditional has been discussed at great length throughout the ages, and a whole language has evolved around it. For now, here are a few definitions: === Weak === {{:Definition:Conditional/Language of Conditional/Weak}} === Strong === {{:Definition:Conditional/Language of Conditional/Strong}} Thus we have the notion of certain theorems having a weak and a strong version. === Superimplicant === {{:Definition:Conditional/Superimplicant}} === Subimplicant === {{:Definition:Conditional/Subimplicant}} === Antecedent === {{:Definition:Conditional/Antecedent}} === Consequent === {{:Definition:Conditional/Consequent}} === Necessary Condition === {{:Definition:Conditional/Necessary Condition}} === Sufficient Condition === {{:Definition:Conditional/Sufficient Condition}}


Definition:Independent Statements

Let $p$ and $q$ be statements. Let it be the case that: :$(1): \quad p$ and $q$ are not contrary :$(2): \quad p$ and $q$ are not subcontrary :$(3): \quad p$ is not superimplicant to $q$ :$(4): \quad p$ is not subimplicant to $q$ :$(5): \quad p$ and $q$ are not equivalent :$(6): \quad p$ and $q$ are not contradictory. Then $p$ and $q$ are '''independent statements'''.


Definition:Primitive (Calculus)/Complex

Let $F: D \to \C$ be a complex function which is complex-differentiable on a connected domain $D$. Let $f: D \to \C$ be a continuous complex function. Let: :$\forall z \in D: \map {F'} z = \map f z$ where $F'$ denotes the derivative of $F$ with respect to $z$. Then $F$ is '''a primitive of $f$''', and is denoted: :$\displaystyle F = \int \map f z \rd z$


Definition:Primitive (Calculus)/Real

Let $F$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$. Let $f$ be a real function which is continuous on the open interval $\openint a b$. Let: :$\forall x \in \openint a b: \map {F'} x = \map f x$ where $F'$ denotes the derivative of $F$ with respect to $x$. Then $F$ is '''a primitive of $f$''', and is denoted: :$\displaystyle F = \int \map f x \rd x$


Definition:Primitive (Calculus)/Arbitrary Constant

From the language in which it is couched, it is apparent that the primitive of a function may not be unique, otherwise we would be referring to $F$ as '''''the'' primitive''' of $f$. This point is made apparent in Primitives which Differ by Constant: if a function has a primitive, there is an infinite number of them, all differing by a constant. That is, if $F$ is a primitive for $f$, then so is $F + C$, where $C$ is a constant. This constant is known as an '''arbitrary constant'''.


Definition:Primitive (Calculus)/Indefinite Integral

From the Fundamental Theorem of Calculus, it is apparent that to find the value of a definite integral for a function between two points, one can find the value of the primitive of the function at those points and subtract one from the other. Thus arises the notation: :$\displaystyle \int \map f x \rd x = \map F x + C$ where $C$ is the arbitrary constant. In this context, the expression $\displaystyle \int \map f x \rd x$ is known as the '''indefinite integral''' of $f$.


Definition:Circular Relation

$\mathcal R$ is '''circular''' {{iff}}: :$\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({z, x}\right) \in \mathcal R$


Definition:Exponential Function/Complex/Sum of Series

The '''exponential function''' can be defined as a (complex) power series: {{begin-eqn}} {{eqn | l = \exp z       | r = \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}       | c =  }} {{eqn | r = 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots       | c =  }} {{end-eqn}}


Definition:Exponential Function/Complex/Real Functions

The '''exponential function''' can be defined by the real exponential, sine and cosine functions: :$\exp z := e^x \paren {\cos y + i \sin y}$ where $z = x + i y$ with $x, y \in \R$. Here, $e^x$ denotes the real exponential function, which must be defined first.


Definition:Exponential Function/Complex/Limit of Sequence

The '''exponential function''' can be defined as a limit of a sequence: :$\displaystyle \exp z := \lim_{n \to \infty} \left({1 + \dfrac z n}\right)^n$


Definition:Exponential Function/Complex/Differential Equation

The '''exponential function''' can be defined as the unique particular solution $y = \map f z$ to the first order ODE: :$\dfrac {\d y} {\d z} = y$ satisfying the initial condition $\map f 0 = 1$. That is, the defining property of $\exp$ is that it is its own derivative.


Definition:Exponential Function/Complex

For all definitions of the '''complex exponential function''': : The domain of $\exp$ is $\C$. : The image of $\exp$ is $\C \setminus \left\{ {0}\right\}$, as shown in Image of Complex Exponential Function. For $z \in \C$, the complex number $\exp z$ is called the '''exponential of $z$'''. === As a Sum of a Series === {{:Definition:Exponential Function/Complex/Sum of Series}} === By Real Functions === {{:Definition:Exponential Function/Complex/Real Functions}} === As a Limit of a Sequence === {{:Definition:Exponential Function/Complex/Limit of Sequence}} === As the Solution of a Differential Equation === {{:Definition:Exponential Function/Complex/Differential Equation}}


Definition:Topological Division Ring

Let $\left({R, +, \circ}\right)$ be a division ring with zero $0_R$. Let $\tau$ be a topology on $R$. Let the mapping $\phi: R \setminus \left\{{0_R}\right\} \to R$ be defined as: :$\phi \left({x}\right) = x^{-1}$ for each $x \in R \setminus \left\{{0_R}\right\}$ Then $\left({R,+,\circ,\tau}\right)$ is a '''topological division ring''' {{iff}}: :$(1): \quad \left({R, +, \circ, \tau}\right)$ is a topological ring :$(2): \quad \phi$ is a $\tau'$-$\tau$-continuous mapping, where $\tau'$ is the $\tau$-relative subspace topology on $R\setminus\left\{{0_R}\right\}$.


Definition:Injective on Objects

Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ is said to be '''injective on objects''' iff for all objects $C_1, C_2$ of $\mathbf C$: :$F C_1 = F C_2$ implies $C_1 = C_2$


Definition:Surjective on Objects

Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ is said to be '''surjective on objects''' iff: :For every object $D$ of $\mathbf D$, there is an object $C$ of $\mathbf C$ such that $F C = D$.


Definition:Injective on Morphisms

Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ is said to be '''injective on morphisms''' iff for all morphisms $f, g$ of $\mathbf C$: :$F f = F g$ implies $f = g$ Note that it is '''not''' required that $f$ and $g$ have equal domains or codomains.


Definition:Surjective on Morphisms

Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ is said to be '''surjective on morphisms''' iff: :For every morphism $g$ of $\mathbf D$, there is a morphism $f$ of $\mathbf C$ such that $F f = g$


Definition:Full Functor

Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a covariant functor. Then $F$ is '''full''' iff for all objects $C_1, C_2$ of $\mathbf C$: :$F: \operatorname{Hom}_{\mathbf C} \left({C_1, C_2}\right) \to \operatorname{Hom}_{\mathbf D} \left({F C_1, F C_2}\right), \ f \mapsto F f$ is a surjection. Here $\operatorname{Hom}$ signifies a hom class.


Definition:Faithful Functor

Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ is '''faithful''' iff for all objects $C_1, C_2$ of $\mathbf C$: :$F: \operatorname{Hom}_{\mathbf C} \left({C_1, C_2}\right) \to \operatorname{Hom}_{\mathbf D} \left({F C_1, F C_2}\right), \ f \mapsto F f$ is an injection. Here $\operatorname{Hom}$ signifies a hom class.


Definition:Full Subcategory

Let $\mathbf D$ be a metacategory, and let $\mathbf C$ be a subcategory of $\mathbf D$. Then $\mathbf C$ is called a '''full subcategory''' iff the inclusion functor $i: \mathbf C \to \mathbf D$ is full.


Definition:Subcategory

=== Definition 1 === Let $\mathbf C$ be a metacategory. A '''subcategory''' of $\mathbf C$ is a metacategory with: {{begin-axiom}} {{axiom|lc= Objects:        |t = Any collection of objects from $\mathbf C$ }} {{axiom|lc= Morphisms:        |t = Any collection of morphisms from $\mathbf C$ }} {{axiom|lc= Composition:        |t = Inherited from $\mathbf C$ }} {{axiom|lc= Identity morphisms:        |t = Inherited from $\mathbf C$ }} {{end-axiom}} Colloquially, a '''subcategory''' of $\mathbf C$ is a "part of $\mathbf C$ that is a category in its own right". === Definition 2 === Let $\mathbf C$ be a metacategory. A '''subcategory''' of $\mathbf C$ is a monic functor $F: \mathbf D \to \mathbf C$ to $\mathbf C$


Definition:Jordan Arc

Let $\tuple {x_1, y_1}, \tuple {x_2, y_2} \in \R^2$. Let $f: \closedint 0 1 \to \R^2$ be a path from $\tuple {x_1, y_1}$ to $\tuple {x_2, y_2}$. Then $f$ is a '''Jordan arc''' {{iff}} $f$ is an injection, except that we allow the possibility $\tuple {x_1, y_1} = \tuple {x_2, y_2}$.


Definition:Jordan Curve

Let $f$ be a Jordan arc from $\tuple {x_1, y_1}$ to $\tuple {x_2, y_2}$. Then $f$ is a '''Jordan curve''' {{iff}} $\tuple {x_1, y_1} = \tuple {x_2, y_2}$.


Definition:Orthogonal (Linear Algebra)/Real Vector Space

Let $\mathbf u$, $\mathbf v$ be vectors in $\R^n$. Then $\mathbf u$ and $\mathbf v$ are said to be '''orthogonal''' {{iff}} their dot product is zero: :$\mathbf u \cdot \mathbf v = 0$ As Dot Product is Inner Product, this is a special case of the definition of orthogonal vectors.


Definition:Normal Subgroup/Definition 1

:$\forall g \in G: g \circ N = N \circ g$


Definition:Normal Subset

Let $\left({G, \circ}\right)$ be a group. Let $S \subseteq G$ be a general subset of $G$. Then $S$ is a '''normal subset of $G$''' {{iff}}: === Definition 1=== {{:Definition:Normal Subset/Definition 1}} === Definition 2=== {{:Definition:Normal Subset/Definition 2}} === Definition 3=== {{:Definition:Normal Subset/Definition 3}} === Definition 4=== {{:Definition:Normal Subset/Definition 4}} === Definition 5 === {{:Definition:Normal Subset/Definition 5}} === Definition 6 === {{:Definition:Normal Subset/Definition 6}} === Definition 7 === {{:Definition:Normal Subset/Definition 7}}


Definition:Normal Subset/Definition 1

:$\forall g \in G: g \circ S = S \circ g$


Definition:Normal Subset/Definition 2

:$\forall g \in G: g \circ S \circ g^{-1} = S$ or, equivalently: :$\forall g \in G: g^{-1} \circ S \circ g = S$


Definition:Normal Subset/Definition 3

: $\forall g \in G: g \circ S \circ g^{-1} \subseteq S$ or, equivalently: : $\forall g \in G: g^{-1} \circ S \circ g \subseteq S$


Definition:Normal Subset/Definition 4

: $\forall g \in G: S \subseteq g \circ S \circ g^{-1}$ or, equivalently: : $\forall g \in G: S \subseteq g^{-1} \circ S \circ g$


Definition:Normal Subgroup/Definition 2

: Every right coset of $N$ in $G$ is a left coset that is: : The right coset space of $N$ in $G$ equals its left coset space.


Definition:Normal Subset/Definition 5

:$\forall x, y \in G: x \circ y \in S \implies y \circ x \in S$


Definition:Normal Subgroup/Definition 3

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


Definition:Normal Subgroup/Definition 4

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


Definition:Normal Subgroup/Definition 5

:$\forall g \in G: g \circ N \circ g^{-1} = N$ :$\forall g \in G: g^{-1} \circ N \circ g = N$


Definition:Normal Subgroup/Definition 6

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


Definition:Normal Subgroup/Definition 7

:$N$ is a normal subset of $G$.


Definition:Subset Product/Singleton

Let $A \subseteq S$ be a subset of $S$. Then: :$(1): \quad a \circ S := \set a \circ S$ :$(2): \quad S \circ a := S \circ \set a$


Definition:Convex Angle

A '''convex angle''' is an angle that is larger than a zero angle and smaller than a straight angle. That is, the measurement of a '''convex angle''' is larger than $0 \degrees$ and smaller than $180 \degrees$. In radians, the measurement of a '''convex angle''' is larger than $0$ and smaller than $\pi$.


Definition:Inverse of Subset/Monoid

Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$. Let $C \subseteq S$ be the set of cancellable elements of $S$. Let $X \subseteq C$. Then the '''inverse''' of the subset $X$ is defined as: :$X^{-1} = \set {y \in S: \exists x \in X: x \circ y = e_S}$ That is, it is the set of all the inverses of all the elements of $X$.


Definition:Inverse of Subset/Group

Let $\struct {G, \circ}$ be a group. Let $X \subseteq G$. Then the '''inverse''' of the subset $X$ is defined as: :$X^{-1} = \set {x \in G: x^{-1} \in X}$ or equivalently: :$X^{-1} = \set {x^{-1}: x \in X}$


Definition:Natural Deduction/Elementary Valid Argument Forms

In most treatments of PropLog various subsets of the following rules are treated as the axioms. Some of them are obvious. Others are more subtle. These rules are not all independent, in that it is possible to prove some of them using sequents constructed from combinations of others. However, when a set of proof rules is selected as the axioms for any particular treatment of this subject, those rules are usually selected carefully so that they ''are'' independent. === {{ProofRuleLink|Rule of Assumption}} === {{:Rule of Assumption/Proof Rule}} === {{ProofRuleLink|Rule of Conjunction}} === {{:Rule of Conjunction/Proof Rule}} === {{ProofRuleLink|Rule of Simplification}} === {{:Rule of Simplification/Proof Rule}} === {{ProofRuleLink|Rule of Addition}} === {{:Rule of Addition/Proof Rule}} === {{ProofRuleLink|Proof by Cases}} === {{:Proof by Cases/Proof Rule}} === {{ProofRuleLink|Modus Ponendo Ponens}} === {{:Modus Ponendo Ponens/Proof Rule}} === {{ProofRuleLink|Modus Tollendo Tollens}} === {{:Modus Tollendo Tollens/Proof Rule}} === {{ProofRuleLink|Modus Tollendo Ponens}} === {{:Modus Tollendo Ponens/Proof Rule}} === {{ProofRuleLink|Modus Ponendo Tollens}} === {{:Modus Ponendo Tollens/Proof Rule}} === {{ProofRuleLink|Rule of Implication}} === {{:Rule of Implication/Proof Rule}} === {{ProofRuleLink|Double Negation Introduction}} === {{:Double Negation Introduction/Proof Rule}} === {{ProofRuleLink|Double Negation Elimination}} === {{:Double Negation Elimination/Proof Rule}} === {{ProofRuleLink|Biconditional Introduction}} === {{:Biconditional Introduction/Proof Rule}} === {{ProofRuleLink|Biconditional Elimination}} === {{:Biconditional Elimination/Proof Rule}} === {{ProofRuleLink|Principle of Non-Contradiction}} === {{:Principle of Non-Contradiction/Proof Rule}} === {{ProofRuleLink|Proof by Contradiction}} === {{:Proof by Contradiction/Proof Rule}} === {{ProofRuleLink|Rule of Explosion}} === {{:Rule of Explosion/Proof Rule}} === {{ProofRuleLink|Law of Excluded Middle}} === {{:Law of Excluded Middle/Proof Rule}} === {{ProofRuleLink|Reductio ad Absurdum}} === {{:Reductio ad Absurdum/Proof Rule}}


Definition:Invariant Subset

Let $S$ be a set. Let $T \subseteq S$ be a subset of $S$. Let $f: S \to S$ be a self-map on $S$. Then $T$ is '''invariant under''' $f$ {{iff}}: :$\forall t \in T: \map f t \in T$


Definition:Inversion Mapping

Let $\struct {G, \circ}$ be a group. The '''inversion mapping''' on $G$ is the mapping $\iota: G \to G$ defined by: :$\forall g \in G: \map \iota g = g^{-1}$ That is, $\iota$ assigns to an element of $G$ its inverse.


Definition:Isomorphism of Categories

Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor. Then $F$ is an '''isomorphism (of categories)''' {{iff}} there exists a functor $G: \mathbf C \to \mathbf D$ such that: :$G F: \mathbf C \to \mathbf C$ is the identity functor $I_{\mathbf C}$ :$F G: \mathbf D \to \mathbf D$ is the identity functor $I_{\mathbf D}$


Definition:Isomorphism of Categories/Isomorphic Categories

Let $F: \mathbf C \to \mathbf D$ be an isomorphism of categories. Then $\mathbf C$ and $\mathbf D$ are said to be '''isomorphic''', and we write $\mathbf C \cong \mathbf D$.


Definition:Equivalence of Categories

Let $\mathbf C$ and $\mathbf D$ be metacategories. An '''equivalence of $\mathbf C$ and $\mathbf D$''' comprises: :Functors $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf C$ :Natural isomorphisms $\alpha: G F \overset{\sim}{\longrightarrow} \operatorname{id}_{\mathbf C}$ and $\beta: F G \overset{\sim}{\longrightarrow} \operatorname{id}_{\mathbf D}$ === Equivalent Categories === {{:Definition:Equivalence of Categories/Equivalent Categories}}


Definition:Equivalence of Categories/Equivalent Categories

Let there exist an equivalence of categories between $\mathbf C$ and $\mathbf D$. Then $\mathbf C$ and $\mathbf D$ are said to be '''equivalent''', denoted $\mathbf C \simeq \mathbf D$.


Definition:Conjunction/Semantics of Conjunction

The '''conjunction''' is used to symbolise ''any'' statement in natural language such that two substatements are held to be true simultaneously. Thus it is also used to symbolise the concept of '''but''' as well as '''and'''. Thus $p \land q$ can be also interpreted as: * '''$p$ ''and'' $q$''' * '''$p$ ''but'' $q$''' * '''$p$; ''however'', $q$''' * '''$p$; ''on the other hand'' $q$''' * '''''Not only'' $p$ ''but also'' $q$''' * '''''Despite'' $p$, $q$'''


Definition:Biconditional/Semantics of Biconditional

The concept of the biconditional has been defined such that $p \iff q$ means: :'''If $p$ is true then $q$ is true, and if $q$ is true then $p$ is true.''' $p \iff q$ can be considered as a ''shorthand'' to replace the use of the longer and more unwieldy expression involving two conditionals and a conjunction. If we refer to ways of expressing the conditional, we see that: * $q \implies p$ can be interpreted as '''$p$ is true if $q$ is true''', and * $p \implies q$ can be interpreted as '''$p$ is true only if $q$ is true'''. Thus we arrive at the usual way of reading '''$p \iff q$''' which is: '''$p$ is true ''if and only if'' $q$ is true.''' This can also be said as: * '''The truth value of $p$ is ''equivalent'' to the truth value of $q$.''' * '''$p$ is ''equivalent'' to $q$.''' * '''$p$ and $q$ are ''equivalent''.''' * '''$p$ and $q$ are ''coimplicant''.''' * '''$p$ and $q$ are ''logically equivalent''.''' * '''$p$ and $q$ are ''materially equivalent''.''' * '''$p$ is true ''exactly when'' $q$ is true.''' * '''$p$ is true ''iff'' $q$ is true.''' This is another convenient and useful (if informal) shorthand which is catching on in the mathematical community.


Definition:Free Set of Sets

Let $\mathcal S$ be a set of sets. Then $\mathcal S$ is a '''free set of sets''' {{iff}} the intersection of $\mathcal S$ is empty. That is: :$\displaystyle \bigcap \mathcal S = \varnothing$


Definition:Gaussian Integral/One Variable

The '''Gaussian Integral''' (of one variable) is the following improper integral, considered as a real function: :$\phi_1: \R \to \R$: :$\map {\phi_1} x = \displaystyle \int_{\mathop \to -\infty}^x \frac 1 {\sqrt {2 \pi} } \map \exp {-\frac {t^2} 2 } \rd t$ where $\exp$ is the real exponential function.


Definition:Precise Refinement of Cover

Let: :$\mathcal S = \left\{ {S_\gamma: \gamma \in \Gamma}\right\}$ be a cover of a set $X$. {{explain|What is $\Gamma$? Is this an indexing set? If so, this needs to be noted.}} Also let: :$\mathcal T = \left\{ {T_\gamma: \gamma \in \Gamma}\right\}$ be a cover of $X$. Then $\mathcal T$ is a '''precise refinement''' of $\mathcal S$ {{iff}}: :$\forall \gamma \in \Gamma: T_\gamma \subseteq S_\gamma$ Category:Definitions/Covers oim53wijy4z95p55k1lpgihgipfd2lw


Definition:Fixed Set of Sets

Let $\mathcal S$ be a set of sets. Then $\mathcal S$ is a '''fixed set of sets''' {{iff}} the intersection of $\mathcal S$ is non-empty. That is: :$\displaystyle \bigcap \mathcal S \ne \varnothing$


Definition:Gaussian Integral/Two Variables

The '''Gaussian Integral''' (of two variables) is the following definite integral, considered as a real-valued function: :$\phi_2: \left\{{\left({a, b}\right) \in \R^2: a \le b}\right\} \to \R$:  :$\phi_2 \left({a, b}\right) = \displaystyle \int_a^b \frac 1 {\sqrt{2 \pi}} \exp \left({- \frac {t^2} 2}\right) \ \mathrm d t$ where $\exp$ is the real exponential function.


Definition:Preorder Category/Definition 1

Let $\left({S, \precsim}\right)$ be a preordered set. One can interpret $\left({S, \precsim}\right)$ as being a category, with: {{DefineCategory |ob = The elements of $S$ |mor = Precisely one morphism $a \to b$ for every $a, b \in S$ with $a \precsim b$ }} More formally, we let the morphisms be the elements of the relation ${\precsim} \subseteq S \times S$. Thus, $a \to b$ in fact denotes the ordered pair $\left({a, b}\right)$. The category that so arises is called a '''preorder category'''.


Definition:Preorder Category/Definition 2

Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is a '''preorder category''' {{iff}} :For all objects $C, C'$ of $\mathbf C$, there is at most one morphism $f: C \to C'$


Definition:Order Category/Definition 1

Let $\left({S, \preceq}\right)$ be an ordered set. One can interpret $\left({S, \preceq}\right)$ as being a category, with: {{DefineCategory |ob = The elements of $S$ |mor = Precisely one morphism $a \to b$ for every $a, b \in S$ with $a \preceq b$ }} More formally, we let the morphisms be the elements of the relation ${\preceq} \subseteq S \times S$. Thus, $a \to b$ in fact denotes the ordered pair $\left({a, b}\right)$. The category that so arises is called an '''order category'''.


Definition:Order Category/Definition 2

Let $\mathbf C$ be a metacategory. Then $\mathbf C$ is an '''order category''' iff: :For all objects $C, C'$ of $\mathbf C$, there is at most one morphism $f: C \to C'$ :Whenever $f: C \to C'$ is an isomorphism, $C = C'$ Thus, an '''order category''' is a skeletal preorder category.


Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient

Let: :$p \iff q$ where $\iff$ denotes the biconditional operator. Then it can be said that '''$p$ is ''necessary and sufficient'' for $q$.''' This is a consequence of the definitions of necessary and sufficient conditions.


Definition:Star Convex Set

Let $V$ be a vector space over $\R$ or $\C$. A subset $A \subseteq V$ is said to be '''star convex''' iff there exists $a \in A$ such that: :$\forall x \in A: \forall t \in \left[{0 \,.\,.\, 1}\right]: t x + \left({1 - t}\right) a \in A$ === Star Center === {{:Definition:Star Convex Set/Star Center}} A '''star convex set''' can thus be described as a set containing all line segments between the '''star center''' and an element of the set.


Definition:Jordan Curve/Interior

It follows from the Jordan Curve Theorem that $\R^2 \setminus \Img f$ is a union of two disjoint connected components, one of which is bounded. This bounded component is called the '''interior of $f$''', and is denoted as $\Int f$.


Definition:Jordan Curve/Exterior

It follows from the Jordan Curve Theorem that $\R^2 \setminus \operatorname{Im} \left({f}\right)$ is a union of two disjoint connected components, one of which is unbounded. This unbounded component is called the '''exterior of $f$''', and is denoted as $\operatorname{Ext} \left({f}\right)$.


Definition:Crossing (Jordan Curve)

Let $P$ be a polygon embedded in $\R^2$. Let $q \in \R^2 \setminus \partial P$. Let $\mathbf v \in R^2 \setminus \set {\mathbf 0}$ be a non-zero vector. Let $\LL = \set {q + s \mathbf v: s \in \R_{\ge 0} }$ be a ray with start point $q$. Then $\LL \cap \partial P$ consists of a finite number of line segments, where $\partial P$ denotes the boundary of $P$. As two adjacent sides in $P$ do not form a straight angle by the definition of polygon, each line segment is either a single point or an entire side of $P$. Each of these line segments is called a '''crossing''' {{iff}} the line segment is one of these: :a single point which is not a vertex of $P$ :a single vertex of $P$, and its adjacent sides lie on opposite sides of $\LL$ :a side $S$ of $P$, and the two sides adjacent to $S$ lie on opposite sides of $\LL$. === Parity === {{:Definition:Crossing (Jordan Curve)/Parity}}


Definition:Crossing (Jordan Curve)/Parity

Let $\map N q$ be the number of '''crossings''' between $\LL$ and the boundary $\partial P$ of $P$. Then the '''parity of $q$''' is defined as: :$\map {\operatorname{par} } q := \map N q \bmod 2$.


Definition:Local Sub-Basis

Let $\left({X, \tau}\right)$ be a topological space. Let $x \in X$. Then a '''local sub-basis''' of $x$ relative to $\tau$ is a neighborhood sub-basis of $x$ consisting of open sets.


Definition:Neighborhood Sub-Basis

Let $\struct {S, \tau}$ be a topological space. Let $x \in S$. Let $\BB$ be a set of neighborhoods of $x$. Then $\BB$ is a '''neighborhood sub-basis of $x$ relative to $\tau$''' {{iff}}: :for each neighborhood $N$ of $x$, there exists a finite subset $K$ of $\BB$ such that $\bigcap K \subseteq N$.


Definition:Star Convex Set/Star Center

The point $a \in A$ is called a '''star center of $A$'''.


Definition:Sublattice

Let $\left({L,\wedge_L,\vee_L,\preceq_L}\right)$ be a lattice. Let $S$ be a subset of $L$ and let $\wedge_S$, $\vee_S$, and $\preceq_S$ be the restrictions to $S$ of $\wedge_L$, $\vee_L$, and $\preceq_L$, respectively. Then $\left({S,\wedge_S,\vee_S,\preceq_S}\right)$ is a '''sublattice''' of $\left({L,\wedge_L,\vee_L,\preceq_L}\right)$ iff $S$ is closed under  $\wedge_S$ and $\vee_S$. If in addition $L$ is a bounded lattice and its top and bottom elements are in $S$, then $S$ is called a '''$0,1$-sublattice''' of $L$. If in addition $L$ is a complete lattice and for each subset $T$ of $S$, $\sup_L T,\inf_L T \in S$, then $S$ is called a '''complete sublattice''' of $L$.


Definition:Generalized Ordered Space

Let $\struct {S, \preceq}$ be a totally ordered set. Let $\tau$ be a topology for $S$. === Definition 1 === {{:Definition:Generalized Ordered Space/Definition 1}} === Definition 2 === {{:Definition:Generalized Ordered Space/Definition 2}} === Definition 3 === {{:Definition:Generalized Ordered Space/Definition 3}}


Definition:Generalized Ordered Space/Definition 2

$\left({S, \preceq, \tau}\right)$ is a '''generalized ordered space''' {{iff}}: :$(1): \quad$ there exists a linearly ordered space $\left({S', \preceq', \tau'}\right)$ :$(2): \quad$ there exists a mapping $\phi: S \to S'$ such that $\phi$ is both an order embedding and a topological embedding.


Definition:Generalized Ordered Space/Definition 1

$\left({S, \preceq, \tau}\right)$ is a '''generalized ordered space''' {{iff}}: :$(1): \quad \left({S, \tau}\right)$ is a Hausdorff space :$(2): \quad$ there exists a basis for $\left({S, \tau}\right)$ whose elements are convex in $S$.


Definition:Argumentum ad Hominem

An '''argumentum ad hominem''' is a logical argument that concludes the falsehood of a statement because it is made by a particular person or group of people. It can be argued that it is reasonable to be suspicious of an argument if most of the assertions made by said person or group regarding the topic under discussion are false. However, such reasoning is still fallacious.


Definition:Argumentum ad Baculum

An '''argumentum ad baculum''' is a logical argument that, rather than prove or present evidence for a claim, threatens any who dare argue with the person or group making the claim. The presence of such a threat is often a good reason to avoid ''stating'' an opposing view, but manifestly does not support the truth or falsity of the claim under consideration.


Definition:Conjunction/General Definition

Let $p_1, p_2, \ldots, p_n$ be statements. The '''conjunction''' of $p_1, p_2, \ldots, p_n$ is defined as: :$\displaystyle \bigwedge_{i \mathop = 1}^n \ p_i = \begin{cases} p_1 & : n = 1 \\ & \\ \displaystyle \left({\bigwedge_{i \mathop = 1}^{n-1} \ p_i}\right) \land p_n & : n > 1 \end{cases}$ That is: :$\displaystyle \bigwedge_{i \mathop = 1}^n \ p_i = p_1 \land p_2 \land \cdots \land p_{n-1} \land p_n$ In terms of the set $P = \left\{{p_1, \ldots, p_n}\right\}$, this can also be rendered: :$\displaystyle \bigwedge P$ and is referred to as the '''conjunction of $P$'''.


Definition:Disjunction/General Definition

Let $p_1, p_2, \ldots, p_n$ be statements. The '''disjunction''' of $p_1, p_2, \ldots, p_n$ is defined as: :$\displaystyle \bigvee_{i \mathop = 1}^n \ p_i = \begin{cases} p_1 & : n = 1 \\ & \\ \displaystyle \paren {\bigvee_{i \mathop = 1}^{n - 1} \ p_i} \lor p_n & : n > 1 \end{cases}$ That is: :$\displaystyle \bigvee_{i \mathop = 1}^n \ p_i = p_1 \lor p_2 \lor \cdots \lor p_{n - 1} \lor p_n$ In terms of the set $P = \set {p_1, \ldots, p_n}$ this can also be rendered: :$\displaystyle \bigvee P$ and is referred to as the '''disjunction of $P$'''.


{{:Definition:Countable Set/Countably Infinite/Definition 1}}


Definition:Logical Not/Boolean Interpretation

The truth value of $\neg \mathbf A$ under a boolean interpretation $v$ is given by: :$\map v {\neg \mathbf A} = \begin{cases} \T & : \map v {\mathbf A} = \F \\ \F & : \map v {\mathbf A} = \T \end{cases}$


Definition:Collation

A '''collation''' is a structured alignment with certain placeholders that underpins the construction of formal languages. These placeholders may be replaced by elements of an alphabet $\mathcal A$ under consideration. A '''collation in $\mathcal A$''' is one where all placeholders are replaced by symbols from $\mathcal A$. For example, if we take $\square$ to denote a placeholder, then $\square\square\square\square\square$ represents the collation "a word of length $5$". We can see that then the word "sheep" is an instance of the '''collation''' "a word of length $5$" '''in the English alphabet''', as is "axiom". Typical examples of '''collations''' encountered in mathematics are words or structured graphics like labeled trees. {{transclude:Definition:Collation/Collation System   |section = def   |title = Collation System   |header = 3   |link = true }} {{transclude:Definition:Collation/Unique Readability|   |section = def   |title = Unique Readability   |header = 3   |link = true }}


Definition:Conditional/Boolean Interpretation

The truth value of $\mathbf A \implies \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \implies \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \F \text{ or } \map v {\mathbf B} = \T \\ \F & : \text{otherwise} \end{cases}$ and the truth value of $\mathbf A \impliedby \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \impliedby \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \T \text{ or } \map v {\mathbf B} = \F \\ \F & : \text{otherwise} \end{cases}$


Definition:Conjunction/Boolean Interpretation

The truth value of $\mathbf A \land \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \land \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \map v {\mathbf B} = \T \\ \F & : \text{otherwise} \end{cases}$


Definition:Disjunction/Boolean Interpretation

The truth value of $\mathbf A \lor \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \lor \mathbf B} = \begin{cases} \F & : \map v {\mathbf A} = \map v {\mathbf B} = \F \\ \T & : \text{otherwise} \end{cases}$


Definition:Biconditional/Boolean Interpretation

The truth value of $\mathbf A \iff \mathbf B$ under a boolean interpretation $v$ is given by: :$\map v {\mathbf A \iff \mathbf B} = \begin{cases} \T & : \map v {\mathbf A} = \map v {\mathbf B} \\ \F & : \text{otherwise} \end{cases}$


Definition:Exclusive Or/Boolean Interpretation

The truth value of $\mathbf A \oplus \mathbf B$ under a boolean interpretation $v$ is given by: :$v \left({\mathbf A \oplus \mathbf B}\right) = \begin{cases} F & : v \left({\mathbf A}\right) = v \left({\mathbf B}\right) \\ T & : \text{otherwise} \end{cases}$


Definition:Order Automorphism

Let $\left({S, \preceq}\right)$ be an ordered set. Let $\phi: S \to S$ be an order isomorphism from $\left({S, \preceq}\right)$ to $\left({S, \preceq}\right)$. Then $\phi$ is an '''(order) automorphism'''.


Definition:Logical NAND/Boolean Interpretation

The truth value of $\mathbf A \uparrow \mathbf B$ under a boolean interpretation $v$ is given by: :$v \left({\mathbf A \uparrow \mathbf B}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = F \text{ or } v \left({\mathbf B}\right) = F\\ F & : \text{otherwise} \end{cases}$


Definition:Logical NOR/Boolean Interpretation

The truth value of $\mathbf A \downarrow \mathbf B$ under a boolean interpretation $v$ is given by: :$v \left({\mathbf A \downarrow \mathbf B}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = v \left({\mathbf B}\right) = F\\ F & : \text{otherwise} \end{cases}$


Definition:Dual Isomorphism (Order Theory)

Let $\left({S, \preceq_S}\right)$ and $\left({T, \preceq_T}\right)$ be ordered sets. Let $\phi:S \to T$ be a bijection. Then $\phi$ is a '''dual isomorphism''' between $\left({S, \preceq_S}\right)$ and $\left({T, \preceq_T}\right)$ iff $\phi$ and $\phi^{-1}$ are decreasing mappings. If there is a dual isomorphism between $\left({S, \preceq_S}\right)$ and $\left({T, \preceq_T}\right)$, then $\left({S, \preceq_S}\right)$ is '''dual''' to $\left({T, \preceq_T}\right)$. Equivalently, $\left({S, \preceq_S}\right)$ is '''dual''' to $\left({T, \preceq_T}\right)$ iff $S$ with the dual ordering is isomorphic to $T$.


Definition:Self-Dual (Order Theory)

Let $\left({S, \preceq}\right)$ be an ordered set. If $S$ is dual to $S$, then $S$ is '''self-dual'''.


Definition:Labeled Tree

A '''labeled tree''' is a finite rooted tree in which each node has attached to it a '''label'''. These '''labels''' may be any symbol or other object that one can think of, but usually consist of one letter or symbol. {{expand|A picture like [http://mathworld.wolfram.com/images/eps-gif/PrueferCode_700.gif this], exemplifying the below}} The parent function is usually depicted by placing the ancestor above the child and connecting them with a line. The various children of a node may also be considered to be in a particular order. This is usually depicted by placing them in a left-to-right order. Because of the intuitive clarity of these relations, '''labeled trees''' may be considered collations as basic as strings are. However, because of the space limitations that most printed media have to take into account, it is usual that one quickly resorts to representing trees by strings in one way or another. Category:Definitions/Formal Systems g30ri96v5v2842jfjrgnt4jzs1jw7iq


Definition:Upper Set

Let $\left({S, \preceq}\right)$ be an ordered set. Let $U \subseteq S$.


Definition:Lower Set

Let $\left({S, \preceq}\right)$ be an ordered set. Let $L \subseteq S$. Then $L$ is a '''lower set''' in $S$ {{iff}}: : For all $l \in L$ and $s \in S$: if $s \preceq l$ then $s \in L$. That is, $L$ is a '''lower set''' {{iff}} it contains its own lower closure.


Definition:Generalized Ordered Space/Definition 3

$\left({S, \preceq, \tau}\right)$ is a '''generalized ordered space''' {{iff}}: :$(1): \quad \left({S, \tau}\right)$ is a Hausdorff space :$(2): \quad$ there exists a sub-basis for $\left({S, \tau}\right)$ each of whose elements is an upper set or lower set in $S$.


Definition:Singular Statement/Individuating Description

An '''individuating description''' is a predicate whose purpose is to uniquely identify a particular object.


Let $T$ be a binary labeled tree. '''In-order traversal''' of $T$ is an algorithm designed to obtain a string representation of $T$. The steps are as follows: === Variant 1 === {{:Definition:In-Order Traversal of Labeled Tree/Variant 1}} === Variant 2 === {{:Definition:In-Order Traversal of Labeled Tree/Variant 2}}


Definition:In-Order Traversal of Labeled Tree/Variant 1

$\mathtt{Inorder} (T):$ * $n \gets t$, where $t$ is the root node of $T$. * If $n$ is a leaf node, output the label of $n$, and stop. * Let $T_1$ and $T_2$ be the left and right subtrees of $T$. * If $n$ has only one child, skip this step. Output $\mathtt{Inorder} (T_1)$. * Output the label of $n$. * Output $\mathtt{Inorder} (T_2)$. * Stop.


Definition:In-Order Traversal of Labeled Tree/Variant 2

$\mathtt{Inorder} (T):$ * $n \gets t$, where $t$ is the root node of $T$. * If $n$ is a leaf node, output the label of $n$, and stop. * Let $T_1$ and $T_2$ be the left and right subtrees of $T$. * Output a left bracket $($. * If $n$ has only one child, skip this step. Output $\mathtt{Inorder} (T_1)$. * Output the label of $n$. * Output $\mathtt{Inorder} (T_2)$. * Output a right bracket $)$. * Stop.


Definition:Pre-Order Traversal of Labeled Tree

Let $T$ be a binary labeled tree. '''Pre-order traversal''' of $T$ is an algorithm designed to obtain a string representation of $T$. The steps are as follows: $\mathtt{Preorder} (T):$ * $n \gets t$, where $t$ is the root node of $T$. * Output the label of $n$. * If $n$ is a leaf node, stop. * Let $T_1$ and $T_2$ be the left and right subtrees of $T$. * If $n$ has only one child, skip this step. Output $\mathtt{Preorder} (T_1)$. * Output $\mathtt{Preorder} (T_2)$. * Stop. The resulting string will be in Polish notation.


Definition:Polish Notation

'''Polish notation''' is a method of writing expressions without the need for parentheses. The main method in writing an expression in '''Polish notation''' is to output an operator $\Box$, directly followed by the output of all its arguments $p, q, \ldots$, yielding: :$\Box p q \ldots$ If we had another, operator $\diamond$ taking only one argument, then also: :$\Box {\diamond} p {\diamond} q \ldots$ is in '''Polish notation'''. Note that for the applicability of '''Polish notation''', it is vital to be able to efficiently distinguish an argument from an operator, or a sequence of arguments. === Formal Definition === {{:Definition:Polish Notation/Formal Definition}} === Reverse Polish Notation === {{:Definition:Polish Notation/Reverse Polish Notation}} {{expand|Could benefit from being defined in BNF or otherwise formally, as an exercise in formal languages}}


Definition:Many-to-One Relation/Defined/Element

Let $f \subseteq S \times T$ be a many-to-one relation. <section notitle="true" name="tc-defn"> Let $s \in S$. Then $f$ is '''defined at $s$''' {{iff}} $s \in \Dom f$, the domain of $f$. </section>


Definition:Many-to-One Relation/Defined/Set

Let $f \subseteq S \times T$ be a many-to-one relation. <section notitle="true" name="tc-defn"> Let $R \subseteq S$. Then $f$ is '''defined on $R$''' {{iff}} it is defined at all $r \in R$. Equivalently, {{iff}} $R \subseteq \Dom f$, the domain of $f$. </section>


Definition:Logical Not/Notational Variants

Various symbols are encountered that denote the concept of the logical not: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $\neg p$ |   |   |- | $\mathsf{NOT}\ p$ |   |   |- | $\sim p$ or $\tilde p$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''tilde''' or '''curl''' |- | $- p$ |  |   |- | $\bar p$ |   | '''Bar $p$''' |- | $p'$ |   | '''$p$ prime''' or '''$p$ complement''' |- | $! p$ |   | '''Bang $p$''' |- | $\operatorname{N} p$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |  |}


Definition:Deductive Method

'''The deductive method''' is a general term for the process of building a logical argument from a (usually) limited number of premises. Thus from a small number of statements which have been accepted as true, and a small number of rules of deduction, a theoretically unlimited number of further statements are determined to be true as a result.


Definition:Conjunction/Notational Variants

Various symbols are encountered that denote the concept of logical conjunction: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \land q$ |   | '''wedge''' |- | $p\ \mathsf{AND} \ q$ |   |   |- | $p \ . \ q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''dot''' |- | $p \ \And \ q$ |  | '''Ampersand''' |- | $\operatorname K p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |  |}


Definition:Conditional/Notational Variants

Various symbols are encountered that denote the concept of the conditional: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \implies q$ |  | '''Implies''' |- | $p \to q$ |   | often used when space is limited |- | $p \supset q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''hook''' or '''horseshoe''' |- | $p \, \mathop {-\!\!\!<} q$ | {{AuthorRef|Charles Sanders Peirce}} | '''sign of illation''' |- | $\operatorname C p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |  |} In mathematics, as opposed to works concerned purely with logic, it is usual to use "$\implies$", as then it can be ensured that it is understood to mean exactly the same thing when we use it in the "mathematical" context. There are other uses in mathematics for the other symbols.


Definition:Biconditional/Notational Variants

Various symbols are encountered that denote the concept of biconditionality: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  |- | $p \iff q$ |   |- | $p\ \mathsf{EQ} \ q$ |   |- | $p \equiv q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} |- | $p = q$ | |- | $p \leftrightarrow q$ | |- | $\operatorname E p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |} It is usual in mathematics to use $\iff$, as there are other uses for the other symbols.


Definition:Universal Quantifier/Notational Variants

Various symbols are encountered that denote the concept of universal quantifier: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  |- | $\forall x$ | {{AuthorRef|Gerhard Gentzen}}: ''Untersuchungen über das logische Schließen'' (1935) |- | $\paren x$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} |- | $\Pi x$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |- | $\wedge x$ or $\bigwedge x$ |  |- | $\ds \operatorname{\Large {\textsf A} } \limits_{x, y, \dotsc}$ | {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski}} |}


Definition:Disjunction/Notational Variants

Various symbols are encountered that denote the concept of disjunction: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \lor q$ | {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell}} | '''vee''' or '''vel''' |- | $p\ \mathsf{OR} \ q$ |   |   |- | $p + q$ |   |   |- | $\operatorname A p q$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |   |}


Definition:Existential Quantifier/Notational Variants

Various symbols are encountered that denote the concept of existential quantifier: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  |- | $\exists x$ | {{AuthorRef|Giuseppe Peano}}: ''Formulario Mathematico (2nd ed.)'' (1896) |- | $\Sigma x$ | {{AuthorRef|Jan Łukasiewicz|Łukasiewicz}}'s Polish notation |- | $\lor x$ or $\bigvee x$ |  |- | $\ds \operatorname {\Large {\textsf E} } \limits_{x, y \dotsc}$ | {{BookReference|Introduction to Logic and to the Methodology of Deductive Sciences|1946|Alfred Tarski}} |}


Definition:Exclusive Or/Notational Variants

Various symbols are encountered that denote the concept of exclusive or: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \oplus q$ | | sometimes called '''o-plus''' |- | $p\ \mathsf{XOR} \ q$ | | |- | $p + q$ | | |- | $p \not \Leftrightarrow q$ | | |- | $p \not \equiv q$ | | |- | $p \ne q$ | | |- | $p \ \dot \lor \ q$ | | |- | $p \ \_ \lor \ q$ | | |}


Definition:Logical NAND/Notational Variants

Various symbols are encountered that denote the concept of logical NAND: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \mid q$ | {{AuthorRef|Henry Maurice Sheffer|Henry Sheffer}} | Sheffer stroke |- | $p \uparrow q$ |  | Also sometimes referred to as the '''Sheffer stroke''' |- | $p \ \mathsf{NAND} \ q$ |  |   |- | $p / q$ |{{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor}} | who refer to it as the '''stroke function''' |- | $p \bar \curlywedge q$ | {{AuthorRef|Charles Sanders Peirce}} | Modified Ampheck |} The all-uppercase rendition NAND originates from the digital electronics industry, where, because NAND is Functionally Complete, this operator has a high importance.


Definition:Logical NOR/Notational Variants

Various symbols are encountered that denote the concept of logical NOR: {| border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; |- ! Symbol  ! Origin  ! Known as |- | $p \downarrow q$ | {{AuthorRef|Willard Van Orman Quine|Willard Quine}} | Quine arrow |- | $p \ \mathsf{NOR} \ q$ |   |   |- | $p \mathop \bot q$ |  | |- | $p \curlywedge q$ | {{AuthorRef|Charles Sanders Peirce}} | Ampheck |} The all-uppercase rendition NOR originates from the digital electronics industry, where, because NOR is Functionally Complete, this operator has a high importance.


Definition:Axiom/Logic

An '''axiom''' in logic is a statement which is taken as '''self-evident'''. Note, however, that there has been disagreement for as long as there have been logicians and philosophers as to whether particular statements are true or not. For example, the Law of Excluded Middle is accepted as axiomatic by philosophers and logicians of the Aristotelian school but is denied by the intuitionist school.


Definition:Axiom/Formal Systems

Let $\mathcal L$ be a formal language. Part of defining a proof system $\mathscr P$ for $\mathcal L$ is to specify its '''axioms'''. An '''axiom of $\mathscr P$''' is a well-formed formula of $\mathcal L$ that $\mathscr P$ approves of by definition.


Definition:Axiom/Mathematics

The term '''axiom''' is used throughout the whole of mathematics to mean a statement which is accepted as true for that particular branch. Different fields of mathematics usually have different sets of statements which are considered as being '''axiomatic'''. So statements which are taken as axioms in one branch of mathematics may be theorems, or irrelevant, in others.


Definition:Strictly Precede/Definition 2

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a \preceq b$ such that $a \ne b$. Then '''$a$ strictly precedes $b$'''.


Definition:Strictly Precede/Definition 1

Let $\left({S, \prec}\right)$ be a strictly ordered set. Let $a, b \in S$ and $a \prec b$. Then '''$a$ strictly precedes $b$'''.


Definition:Strictly Ordered Set

A '''strictly ordered set''' is a relational structure $\left({S, \prec}\right)$ such that the relation $\prec$ is an strict ordering.


Definition:Strictly Well-Ordered Set

{{expand|Add a definition to the effect that a '''strictly ordered set''' is a relational structure where the relation is a strict well-ordering.}} Let $\left({S, \prec}\right)$ be a strictly totally ordered set. Then $\left({S, \prec}\right)$ is a '''strictly well-ordered set''' {{iff}} $\prec$ is a foundational relation.


Definition:Minimal/Relation

Let $\left({S, \mathcal R}\right)$ be a relational structure. Let $T \subseteq S$ be a subset of $S$. An element $x \in T$ is an '''$\mathcal R$-minimal element of $T$''' {{iff}}: :$\forall y \in T: y \not \mathrel {\mathcal R} x$


Definition:Maximal/Relation

Let $\left({S, \mathcal R}\right)$ be a relational structure. Let $T \subseteq S$ be a subset of $S$. An element $x \in T$ is an '''$\mathcal R$-maximal element of $T$''' {{iff}}: :$\forall y \in T: x \not \mathrel {\mathcal R} y$


Definition:Dual Order Embedding

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. A '''dual order embedding''' is a mapping $\phi: S \to T$ such that: :$\forall x, y \in S: x \preceq_1 y \iff \phi \left({y}\right) \preceq_2 \phi \left({x}\right)$ That is: : if $\phi$ is an order embedding of $\left({S, \preceq_1}\right)$ into $\left({T, \succeq_2}\right)$ where $\succeq_2$ is the dual of $\preceq_2$. Category:Definitions/Order Embeddings fyhwwnrx9b6ea2fmrab1iyhqq1jglui


Definition:Backus-Naur Form/Non-Terminal

Symbols that appear on the {{LHS}} of a specification in Backus-Naur form are called '''non-terminals.'''


Definition:Backus-Naur Form/Terminal

Symbols that never appear on the {{LHS}} of a specification in Backus-Naur form are called '''terminals.'''


Definition:Polish Notation/Reverse Polish Notation

For stack-based programming languages, '''reverse Polish language''' is a useful variant of Polish notation, because it naturally coincides with how the input is to be structured for the language. As the name suggests, a string $\mathsf P$ is in '''reverse Polish language''' {{iff}} reversing it gives a string $\tilde {\mathsf P}$ in Polish notation. Thus the '''reverse Polish language''' equivalents of these examples of Polish notation: :$\Box p q \ldots$ :$\Box {\diamond} p {\diamond} q \ldots$ are: :$\ldots q p \Box$ :$\ldots q {\diamond} p {\diamond} \Box$


Definition:Top (Logic)

'''Top''' is a constant of propositional logic interpreted to mean the canonical, undoubted tautology whose truth nobody could possibly ever question. The symbol used is $\top$.


Definition:Bottom (Logic)

'''Bottom''' is a constant of propositional logic interpreted to mean the canonical, undoubted contradiction whose falsehood nobody could possibly ever question. The symbol used is $\bot$.


Definition:Top (Logic)/Boolean Interpretation

There is only one boolean interpretation for $\top$: :$\map v \top = \T$ where $\T$ symbolises true.


Definition:Bottom (Logic)/Boolean Interpretation

There is only one boolean interpretation for $\bot$: :$\map v \bot = \F$ where $\F$ symbolises false.


Definition:Order Topology/Definition 2

Let $\struct {S, \preceq}$ be a totally ordered set. Define: :$\map {\Uparrow} S = \set {s^\succ: s \in S}$ :$\map {\Downarrow} S = \set {s^\prec: s \in S}$ where $s^\succ$ and $s^\prec$ denote the strict upper closure and strict lower closure of $s$, respectively. The '''order topology''' $\tau$ on $S$ is the topology on $S$ generated by $\map {\Uparrow} S \cup \map {\Downarrow} S$.


Definition:Interval/Ordered Set/Open

The '''open interval between $a$ and $b$''' is the set: :$\openint a b := a^\succ \cap b^\prec = \set {s \in S: \paren {a \prec s} \land \paren {s \prec b} }$ where: :$a^\succ$ denotes the strict upper closure of $a$ :$b^\prec$ denotes the strict lower closure of $b$.


Definition:Interval/Ordered Set/Left Half-Open

The '''left half-open interval between $a$ and $b$''' is the set: :$\hointl a b := a^\succ \cap b^\preccurlyeq = \set {s \in S: \paren {a \prec s} \land \paren {s \preccurlyeq b} }$ where: : $a^\succ$ denotes the strict upper closure of $a$ : $b^\preccurlyeq$ denotes the lower closure of $b$.


Definition:Interval/Ordered Set/Right Half-Open

The '''right half-open interval between $a$ and $b$''' is the set: :$\hointr a b := a^\succcurlyeq \cap b^\prec = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }$ where: :$a^\succcurlyeq$ denotes the upper closure of $a$ :$b^\prec$ denotes the strict lower closure of $b$.


Definition:Interval/Ordered Set/Closed

The '''closed interval between $a$ and $b$''' is the set: :$\closedint a b := a^\succcurlyeq \cap b^\preccurlyeq = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }$ where: : $a^\succcurlyeq$ denotes the upper closure of $a$ : $b^\preccurlyeq$ denotes the lower closure of $b$.


Definition:Interval/Ordered Set

Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $a, b \in S$. The '''intervals between $a$ and $b$''' are defined as follows: === Open Interval === {{:Definition:Interval/Ordered Set/Open}} === Left Half-Open Interval === {{:Definition:Interval/Ordered Set/Left Half-Open}} === Right Half-Open Interval === {{:Definition:Interval/Ordered Set/Right Half-Open}} === Closed Interval === {{:Definition:Interval/Ordered Set/Closed}}


Definition:Upper Closure/Set

Let $\struct {S, \preceq}$ be an ordered set or preordered set. Let $T \subseteq S$. The '''upper closure of $T$ (in $S$)''' is defined as: :$T^\succeq := \bigcup \set {t^\succeq: t \in T}$ where $t^\succeq$ denotes the upper closure of $t$ in $S$. That is: :$T^\succeq := \set {u \in S: \exists t \in T: t \preceq u}$


Definition:Order Topology/Definition 1

Let $\struct {S, \preceq}$ be a totally ordered set. Let $\XX$ be the set of open rays in $S$. Let $\tau$ be the topology on $S$ generated by $\XX$. Then $\tau$ is called the '''order topology''' on $S$.


Definition:Lower Closure/Element

Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $a \in S$. The '''lower closure of $a$ (in $S$)''' is defined as: :$a^\preccurlyeq := \set {b \in S: b \preccurlyeq a}$ That is, $a^\preccurlyeq$ is the set of all elements of $S$ that precede $a$.


Definition:Lower Closure/Set

Let $\struct {S, \preccurlyeq}$ be an ordered set or preordered set. Let $T \subseteq S$. The '''lower closure of $T$ (in $S$)''' is defined as: :$T^\preccurlyeq := \bigcup \set {t^\preccurlyeq: t \in T}$ where $t^\preccurlyeq$ is the lower closure of $t$. That is: :$T^\preccurlyeq := \set {l \in S: \exists t \in T: l \preccurlyeq t}$


Definition:Upper Set/Definition 2

$U$ is an '''upper set''' in $S$ {{iff}}: :$U^\succeq \subseteq U$ where $U^\succeq$ is the upper closure of $U$.


Definition:Upper Set/Definition 3

$U$ is an '''upper set''' in $S$ {{iff}}: :$U^\succeq = U$ where $U^\succeq$ is the upper closure of $U$.


Definition:Closed Set/Closure Operator

Let $S$ be a set. Let $\cl: \powerset S \to \powerset S$ be a closure operator. Let $T \subseteq S$ be a subset. === Definition 1 === The subset $T$ is '''closed''' (with respect to $\cl$) {{iff}}: :$\map \cl T = T$ === Definition 2 === The subset $T$ is '''closed''' (with respect to $\cl$) {{iff}} $T$ is in the image of $\cl$: :$T \in \Img \cl$


Definition:Kuratowski Closure Operator

=== Definition 1 === {{:Definition:Kuratowski Closure Operator/Definition 1}} === Definition 2 === {{:Definition:Kuratowski Closure Operator/Definition 2}}


Definition:Kuratowski Closure Operator/Definition 1

Let $S$ be a set. Let $\cl: \powerset S \to \powerset S$ be a mapping from the power set of $S$ to itself. Then $\cl$ is a '''Kuratowski closure operator''' {{iff}} it satisfies the following '''Kuratowski closure axioms''' for all $A, B \subseteq S$: {{begin-axiom}} {{axiom | n =1         | m = A \subseteq \map \cl A         | rc= $\cl$ is inflationary }} {{axiom | n = 2         | m = \map \cl {\map \cl A} = \map \cl A         | rc= $\cl$ is idempotent }} {{axiom | n = 3         | m = \map \cl {A \cup B} = \map \cl A \cup \map \cl B         | rc= $\cl$ preserves binary unions }} {{axiom | n = 4         | m = \map \cl \O = \O }} {{end-axiom}}


Definition:Kuratowski Closure Operator/Definition 2

Let $S$ be a set. Let $\cl: \powerset S \to \powerset S$ be a mapping from the power set of $S$ to itself. Then $\cl$ is a '''Kuratowski closure operator''' {{iff}} it satisfies the following axioms for all $A, B \subseteq X$: {{begin-axiom}} {{axiom | n = 1         | t = $\cl$ is a closure operator }} {{axiom | n = 2         | t = $\map \cl {A \cup B} = \map \cl A \cup \map \cl B$         | rc= $\cl$ preserves binary unions }} {{axiom | n = 3         | t = $\map \cl \O = \O$ }} {{end-axiom}}


Definition:Supremum of Mapping

Let $S$ be a set. Let $\struct {T, \preceq}$ be an ordered set. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $f \sqbrk S$, the image of $f$, admit a supremum. Then the '''supremum''' of $f$ (on $S$) is defined by: :$\displaystyle \sup_{x \mathop \in S} \map f x = \sup f \sqbrk S$ === Real-Valued Function === {{:Definition:Supremum of Mapping/Real-Valued Function}}


Definition:Projective Space/Over a Field

Let $V$ be a vector space over a field $K$ of dimension $n + 1 \ge 1$.   Let $\sim$ be the equivalence relation defined on the set $V \setminus \left\{{0}\right\}$ by: :$x, y \in V \setminus \left\{{0}\right\}: x \sim y \iff \exists \lambda \in K: x = \lambda y$ The '''projective space''' associated to $V$ of dimension $n$ over $K$ is the quotient set $\left({V \setminus \left\{{0}\right\}}\right) / \sim$ and is denoted $\mathbb P \left({ V }\right)$. If $V = K^{n+1}$ for $n \ge 0$ a natural number, projective space is sometimes denoted $\mathbb P\left( K^{n+1} \right) = \mathbb P^n\left( K \right)$. This is because while $K^{n+1}$ is an $\left(n+1\right)$-dimensional vector space, the projective space $\mathbb P\left( K^{n+1} \right)$ has dimension $n$. The notation $\mathbb P\left( K^{n+1} \right) = K\mathbb P^n$ is also in use.


Definition:Universal Bounds

Let $\left({S, \preceq}\right)$ be an ordered set. Let $S$ have a smallest element, $\bot$. Let $S$ have a greatest element, $\top$. Then $\bot$ and $\top$ are the '''universal bounds''' of $S$.


Definition:Reflexive Closure/Union with Diagonal

Let $\mathcal R$ be a relation on a set $S$. The '''reflexive closure''' of $\mathcal R$ is denoted $\mathcal R^=$, and is defined as: :$\mathcal R^= := \mathcal R \cup \set {\tuple {x, x}: x \in S}$ That is: :$\mathcal R^= := \mathcal R \cup \Delta_S$ where $\Delta_S$ is the diagonal relation on $S$.


Definition:Reflexive Closure/Intersection of Reflexive Supersets

Let $\mathcal R$ be a relation on a set $S$. Let $\mathcal Q$ be the set of all reflexive relations on $S$ that contain $\mathcal R$. The '''reflexive closure''' of $\mathcal R$ is denoted $\mathcal R^=$, and is defined as: :$\mathcal R^= := \bigcap \mathcal Q$ That is: :$\mathcal R^=$ is the intersection of all reflexive relations on $S$ containing $\mathcal R$.


Definition:Reflexive Closure/Smallest Reflexive Superset

Let $\RR$ be a relation on a set $S$. The '''reflexive closure''' of $\RR$ is defined as the smallest reflexive relation on $S$ that contains $\RR$ as a subset. The '''reflexive closure''' of $\RR$ is denoted $\RR^=$.


Definition:Transitive Closure (Relation Theory)/Smallest Transitive Superset

Let $\RR$ be a relation on a set $S$. The '''transitive closure of $\RR$''' is defined as the smallest transitive relation on $S$ which contains $\RR$ as a subset.


Definition:Transitive Closure (Relation Theory)/Intersection of Transitive Supersets

Let $\mathcal R$ be a relation on a set $S$. The '''transitive closure''' of $\mathcal R$ is defined as the intersection of all transitive relations on $S$ which contain $\mathcal R$.


Definition:Transitive Closure (Relation Theory)/Finite Chain

Let $\mathcal R$ be a relation on a set or class $S$. The '''transitive closure''' of $\mathcal R$ is the relation $\mathcal R^+$ defined as follows: For $x, y \in S$, $x \mathrel {\mathcal R^+} y$ {{iff}} for some $n \in \N_{>0}$ there exist $s_0, s_1, \dots, s_n \in S$ such that $s_0 = x$, $s_n = y$, and: {{begin-eqn}} {{eqn | l = s_0       | o = \mathcal R       | r = s_1 }} {{eqn | l = s_1       | o = \mathcal R       | r = s_2 }} {{eqn | o = \vdots }} {{eqn | l = s_{n - 1}       | o = \mathcal R       | r = s_n }} {{end-eqn}}


Definition:Transitive Closure (Relation Theory)/Union of Compositions

Let $\mathcal R$ be a relation on a set $S$. Let: :$\mathcal R^n := \begin{cases} \mathcal R & : n = 0 \\ \mathcal R^{n-1} \circ \mathcal R & : n > 0 \end{cases}$ where $\circ$ denotes composition of relations. Finally, let: :$\displaystyle \mathcal R^+ = \bigcup_{i \mathop \in \N} \mathcal R^i$ Then $\mathcal R^+$ is called the '''transitive closure''' of $\mathcal R$.


Definition:Affine Frame

Let $\mathcal E$ be an affine space with difference space $V$. === Definition 1 === {{:Definition:Affine Frame/Definition 1}} === Definition 2 === {{:Definition:Affine Frame/Definition 2}} Category:Definitions/Affine Geometry pz5qivyva9bgvrp3nv82l8s4yxut4l6


Definition:Coordinate System/Coordinates on Affine Space

Let $\mathcal E$ be an affine space of dimension $n$ over a field $k$. Let $\mathcal R = \left({p_0, e_1, \ldots, e_n}\right)$ be an affine frame in $\mathcal E$. Let $p \in \mathcal E$ be a point. Since Affine Coordinates are Well-Defined, there exists a unique ordered tuple $\left({\lambda_1, \ldots, \lambda_n}\right) \in k^n$ such that: :$\displaystyle p = p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i$ The numbers $\lambda_1, \ldots, \lambda_n$ are the '''coordinates''' of $p$ in the frame $\mathcal R$.


Definition:Barycenter

Let $\mathcal E$ be an affine space over a field $k$. Let $p_1,\ldots,p_n \in \mathcal E$ be points. Let $\lambda_1,\ldots,\lambda_n \in k$ such that $\displaystyle \sum_{i \mathop = 1}^n \lambda_i = 1$. The '''barycenter''' of $p_1,\ldots,p_n$ with '''weights''' $\lambda_1,\ldots,\lambda_n$ is the unique point $q$ of $\mathcal E$ such that for every point $r \in \mathcal E$ :$\displaystyle q = r + \sum_{i \mathop = 1}^n\lambda_i \vec{r p_i}$


Definition:Negatively Transitive Relation

Let $\RR$ be a relation on a set $S$. Then $\RR$ is '''negatively transitive''' {{iff}}: :$\forall x, y, z \in S: \neg \paren {x \mathrel \RR y} \land \neg \paren {y \mathrel \RR z} \implies \neg \paren {x \mathrel \RR z}$ By De Morgan's Laws, this can be given the alternative form: :$\forall x, y, z \in S: \paren {x \mathrel \RR z} \implies \paren {x \mathrel \RR y} \lor \paren {y \mathrel \RR z}$


Definition:Reflexive Transitive Closure/Transitive Closure of Reflexive Closure

The '''reflexive transitive closure''' of $\mathcal R$ is denoted $\mathcal R^*$, and is defined as the transitive closure of the reflexive closure of $\mathcal R$: :$\mathcal R^* = \left({\mathcal R^=}\right)^+$


Definition:Reflexive Transitive Closure/Reflexive Closure of Transitive Closure

The '''reflexive transitive closure''' of $\mathcal R$ is denoted $\mathcal R^*$, and is defined as the reflexive closure of the transitive closure of $\mathcal R$: :$\mathcal R^* = \left({\mathcal R^+}\right)^=$


Definition:Reflexive Transitive Closure/Smallest Reflexive Transitive Superset

The '''reflexive transitive closure''' of $\mathcal R$ is denoted $\mathcal R^*$, and is defined as the smallest reflexive and transitive relation on $S$ which contains $\mathcal R$.


Definition:Closure Operator/Ordering

{{:Definition:Closure Operator/Ordering/Definition 1}}


Definition:Closed Element

Let $\left({S, \preceq}\right)$ be an ordered set. Let $\operatorname{cl}$ be a closure operator on $S$. Let $x \in S$. === Definition 1 === {{:Definition:Closed Element/Definition 1}} === Definition 2 === {{:Definition:Closed Element/Definition 2}}


Definition:Affine Subspace

Let $\mathcal E$ be an affine space with tangent space $E$. Let $\mathcal F \subseteq \mathcal E$ be a subset of $\mathcal E$. Then $\mathcal F$ is an '''affine subspace''' of $\mathcal E$ {{iff}} there exists a point $p \in \mathcal E$ such that: :$F_p := \set {q - p: q \in \mathcal F}$ is a vector subspace of the vector space $E$.


Definition:Product of Affine Spaces

Let $\mathcal E, \mathcal F$ be affine spaces with difference spaces $E, F$ respectively. Let $\mathcal G = \mathcal E \times \mathcal F$ be the cartesian product of the sets $\mathcal E,\mathcal F$. Let $G = E \times F$ be the direct product of the vector spaces $E, F$. Define sum and difference operations $+ : \mathcal G \times G \to \mathcal G$ and $- : \mathcal G \times \mathcal G \to G$ by, for all $p, p' \in \mathcal E$ and $q, q' \in \mathcal F$: :$\left({p, q}\right) + \left({p', q'}\right) := \left({p + p', q + q'}\right)$ :$\left({p, q}\right) - \left({p', q'}\right) := \left({p - p', q - q'}\right)$ Then the set $\mathcal G$ together with the vector space $G$ and the operations $+, -$ is called the '''product''' of the affine spaces $\mathcal E$ and $\mathcal F$.


Definition:Standard Affine Structure on Vector Space

Let $E$ be a vector space. Let $\mathcal E$ be the underlying set of $E$. Let $+$ denote the addition operation $E \times E \to E$, viewed as a mapping $\mathcal E \times E \to \mathcal E$. Let $-$ denote the subtraction operation $E \times E \to E$, viewed as a mapping $\mathcal E \times \mathcal E \to E$. Then the set $\mathcal E$, together with the vector space $E$ and the operations $+,-$, is called the '''standard affine structure''' on the vector space $E$.


Definition:Vectorialization of Affine Space

Let $\mathcal E$ be an affine space over a field $k$ with difference space $E$. Let $\mathcal R = \left({p_0, e_1, \ldots, e_n}\right)$ be an affine frame in $\mathcal E$. Define a mapping $\Theta_{\mathcal R} : k^n \to \mathcal E$ by: :$\displaystyle \Theta_{\mathcal R} \left({\lambda_1, \ldots, \lambda_n}\right) = p_0 + \sum_{i \mathop = 1}^n \lambda_i e_i$. By Affine Coordinates are Well-Defined, $\Theta_{\mathcal R}$ is a bijection. For any $\mu \in k$, $p, q \in \mathcal E$ let: :$\mu \cdot p = \Theta_{\mathcal R}\left({ \mu \cdot \Theta_{\mathcal R}^{-1} \left({p}\right) }\right)$ and: :$p + q = \Theta_{\mathcal R} \left({ \Theta_{\mathcal R}^{-1} \left({p}\right) + \Theta_{\mathcal R}^{-1} \left({q}\right) }\right)$ We call the set $\mathcal E$, together with the operations $\cdot,+$ the '''vectorialization of $\mathcal E$ with origin $p_0$'''.


Definition:Strict Ordering/Asymmetric and Transitive

Let $\RR$ be a relation on a set $S$. Then $\RR$ is a '''strict ordering (on $S$)''' {{iff}} the following two conditions hold: {{begin-axiom}} {{axiom | n = 1         | lc= Asymmetry         | q = \forall a, b \in S         | ml= a \mathrel \RR b         | mo= \implies         | mr= \neg \paren {b \mathrel \RR a} }} {{axiom | n = 2         | lc= Transitivity         | q = \forall a, b, c \in S         | ml= \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c}         | mo= \implies         | mr= a \mathrel \RR c }} {{end-axiom}}


Definition:Strict Ordering/Antireflexive and Transitive

Let $\mathcal R$ be a relation on a set $S$. Then $\mathcal R$ is a '''strict ordering (on $S$)''' {{iff}} the following two conditions hold: {{begin-axiom}} {{axiom | n = 1         | lc= Antireflexivity         | q = \forall a \in S         | m = \neg \paren {a \mathrel {\mathcal R} a} }} {{axiom | n = 2         | lc= Transitivity         | q = \forall a, b, c \in S         | m = \paren {a \mathrel {\mathcal R} b} \land \paren {b \mathrel {\mathcal R} c} \implies a \mathrel {\mathcal R} c }} {{end-axiom}}


Definition:Closure Operator

=== Ordering === {{:Definition:Closure Operator/Ordering}} === Power Set === When the ordering in question is the subset relation on a power set, the definition can be expressed as follows: {{:Definition:Closure Operator/Power Set}}


Definition:Subband

Let $\struct {S, \circ}$ be an algebraic structure. Let $T \subseteq S$ such that $\struct {T, \circ {\restriction_T} }$ is a band, where $\circ {\restriction_T}$ denotes the restriction of $\circ$ to $T$. Then $\struct {T, \circ {\restriction_T} }$ is a '''subband''' of $S$.


Definition:Closure Operator/Ordering/Definition 1

Let $\struct {S, \preceq}$ be an ordered set. A '''closure operator''' on $S$ is a mapping: :$\cl: S \to S$ which satisfies the following conditions for all elements $x, y \in S$: {{begin-axiom}} {{axiom | lc= $\cl$ is inflationary         | ml= x         | mo= \preceq          | mr= \map \cl x }} {{axiom | lc= $\cl$ is increasing         | ml= x \preceq y         | mo= \implies         | mr= \map \cl x \preceq \map \cl y }} {{axiom | lc= $\cl$ is idempotent         | ml= \map \cl {\map \cl x}          | mo= =         | mr= \map \cl x }} {{end-axiom}}


Definition:Closure Operator/Ordering/Definition 2

Let $\struct {S, \preceq}$ be an ordered set. A '''closure operator''' on $S$ is a mapping: :$\cl: S \to S$ which satisfies the following condition for all elements $x, y \in S$: :$x \preceq \map \cl y \iff \map \cl x \preceq \map \cl y$


Definition:Closed Element/Definition 1

The element $x$ is a '''closed element of $S$ (with respect to $\cl$)''' {{iff}} $x$ is a fixed point of $\cl$: :$\map \cl x = x$


Definition:Closed Element/Definition 2

The element $x$ is a '''closed element of $S$ (with respect to $\cl$)''' {{iff}} $x$ is in the image of $\cl$: :$x \in \Img \cl$


Definition:Univalent Relation

Let $\RR$ be a relation on a set $S$. Then $\RR$ is '''univalent''' {{iff}}: :$\RR \circ \RR^{-1} \subseteq \Delta_S$ That is, $\RR$ composed with its inverse $\RR^{-1}$ is a subset of the diagonal relation on $S$.


Definition:Dual Relation/Complement of Inverse

Let $\RR \subseteq S \times T$ be a binary relation. Then the '''dual''' of $\RR$ is denoted $\RR^d$ and is defined as: :$\RR^d := \overline {\paren {\RR^{-1} } }$ where: :$\RR^{-1}$ denotes the inverse of $\RR$ :$\overline {\paren {\RR^{-1} } }$ denotes the complement of the inverse of $\RR$.


Definition:Dual Relation/Inverse of Complement

Let $\RR \subseteq S \times T$ be a binary relation. Then the '''dual''' of $\RR$ is denoted $\RR^d$ and is defined as: :$\RR^d := \paren {\overline \RR}^{-1}$ where: :$\overline \RR$ denotes the complement of $\RR$ :$\paren {\overline \RR}^{-1}$ denotes the inverse of the complement of $\RR$.


Definition:Dual Relation

=== Inverse of Complement === {{:Definition:Dual Relation/Inverse of Complement}} === Complement of Inverse === {{:Definition:Dual Relation/Complement of Inverse}}


Definition:Trivial Gradation

Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({M, \cdot, e}\right)$ be a monoid. Define a gradation on $R$ as follows: :$\displaystyle R_e = R$ :$\displaystyle \forall m \in M \setminus \left\{{e}\right\} : R_m = \mathbf 0$ where $\mathbf 0$ is the zero ring. This is called the '''trivial $M$-grading''' on $R$.


=== Linear Representation === {{:Definition:Irreducible (Representation Theory)/Linear Representation}} === G-Module === {{:Definition:Irreducible (Representation Theory)/G-Module}}


Definition:Transcendental (Abstract Algebra)/Ring

Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $x \in R$. Then $x$ is '''transcendental over $D$''' {{iff}}: :$\displaystyle \forall n \in \Z_{\ge 0}: \sum_{k \mathop = 0}^n a_k \circ x^k = 0_R \implies \forall k: 0 \le k \le n: a_k = 0_R$ That is, $x$ is '''transcendental over $D$''' {{iff}} the only way to express $0_R$ as a polynomial in $x$ over $D$ is by the null polynomial.


Definition:Transcendental (Abstract Algebra)/Field Extension/Element

Let $E / F$ be a field extension. Let $\alpha \in E$. Then $\alpha$ is '''transcendental over $F$''' {{iff}}: : $\nexists f \left({x}\right) \in F \left[{x}\right] \setminus \left\{{0}\right\}: f \left({\alpha}\right) = 0$ where $f \left({x}\right)$ denotes a polynomial in $x$ over $F$.


Definition:Transcendental (Abstract Algebra)/Field Extension

A field extension $E / F$ is said to be '''transcendental''' {{iff}}: :$\exists \alpha \in E: \alpha$ is transcendental over $F$ That is, a field extension is '''transcendental''' {{iff}} it contains at least one transcendental element.


Definition:Monomial of Free Commutative Monoid/Multiplication

The set of monomials over $\family {X_j: j \in J}$ has '''multiplication''' $\circ$ defined by: :$\ds \paren {\prod_{j \mathop \in J} X_j^{k_j} } \circ \paren {\prod_{j \mathop \in J} X_j^{k_j'} } = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$ which using multiindex addition notation reads: :$\mathbf X^k \circ \mathbf X^{k'} = \mathbf X^{k + k'}$


Definition:Monomial of Free Commutative Monoid/Degree

The '''degree''' of a monomial is defined as: :$\ds \sum_{j \mathop \in J} k_j$ that is, the modulus of the corresponding multiindex.


Definition:Degree of Polynomial/Null Polynomial

The null polynomial $0_R \in S \left[{X}\right]$ does ''not'' have a degree.


Definition:Degree of Polynomial/Zero

A polynomial $f \in S \sqbrk x$ in $x$ over $S$ is of '''degree zero''' {{iff}} $x$ is a non-zero element of $S$, that is, a constant polynomial.


Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +, \circ}$ be a subring of $R$. Let $x \in R$. Let: :$\displaystyle f = \sum_{j \mathop = 0}^n a_j x^j$ :$\displaystyle g = \sum_{k \mathop = 0}^n b_k x^k$ be polynomials in $x$ over $S$ such that $a_n \ne 0$ and $b_m \ne 0$. {{questionable|this does not need to be defined; it's just the product of ring elements}} The '''product of $f$ and $g$''' is defined as: :$\displaystyle f g := \sum_{l \mathop = 0}^{m + n} c_l x^l$ where: :$\displaystyle \forall l \in \set {0, 1, \ldots, m + n}:  c_l = \sum_{\substack {j \mathop + k \mathop = l \\ j, k \mathop \in \Z}} a_j b_k$ === Polynomial Forms === {{:Definition:Multiplication of Polynomials/Polynomial Forms}} === Polynomials as Sequences === {{:Definition:Multiplication of Polynomials/Sequence}}


Definition:Ring of Polynomials in Ring Element

The subring of $R$ consisting of all the polynomials in $x$ over $D$ is called the '''ring of polynomials over $D$''' and is denoted $D \sqbrk x$.


Definition:Order Embedding/Definition 1

$\phi$ is an '''order embedding of $S$ into $T$''' {{iff}}: :$\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$


Definition:Order Embedding/Definition 4

Let $T' = \Img S$ be the image of $S$ under $\phi$. $\phi$ is an '''order embedding of $S$ into $T$''' {{iff}}: :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 $T$''' {{iff}} 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 {{ProofWiki}} 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 $P \preceq Q$ {{iff}}: :$\exists p \in P: \exists q \in Q: p \precsim q$ 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}} Category:Definitions/Polynomial Theory c5gvtv5plr8rzi4rsqy8dwzxpc63f2i


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 $\partial f$.


=== 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''' {{iff}}: :$x \in C \implies x \subseteq \map f x$ That is, {{iff}} 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 $A \subseteq B$, {{iff}}: :$\forall x: \paren {x \in A \implies x \in B}$ where $x \in A$ denotes that $x$ is an element of $A$.


Definition:Inductive Class

Let $A$ be a class. Then $A$ is '''inductive''' {{iff}}: {{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 {{ProofWiki}}, 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}} Category:Definitions/Number Theory Category:Definitions/Eisenstein Integers ivyums544kha0szun3kswtcm6p9gxw2


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''' {{iff}} 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. Category:Definitions/Affine Geometry r6vqbj6ch7xapphyz93jf4n3uj6uxj8


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) {{iff}}: :$(1): \quad M \in A$ :$(2): \quad \forall S: \paren {S \in A \implies S \subseteq M}$


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. {{Disambiguation}} * 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 $\mathbf A$''' {{iff}} $\mathbf B$ 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''' {{iff}}: : 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''' {{iff}} 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''' {{iff}} $A$ 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''' {{iff}}: :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}} Category:Definitions/Class Theory 3cuwtmdpfv67ljjzn1hhicw9nvskb8c


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. {{EuclidSaid}} :''{{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.