hcju/proofwikips huggingface dataset:
Definition:Conditional/Language of Conditional
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''' : :$\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({z, x}\right) \in \mathcal R$
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''' : :$(1): \quad \left({R, +, \circ, \tau}\right)$ 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'''  $f$ 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'''  $\tuple {x_1, y_1} = \tuple {x_2, y_2}$.
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'''  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 $G$''' : === 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''' $f$ : :$\forall t \in T: \map f t \in T$
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)'''  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'''  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 $\mathcal S$ : :$\forall \gamma \in \Gamma: T_\gamma \subseteq S_\gamma$
Definition:Fixed Set of Sets
Let $\mathcal S$ be a set of sets. Then $\mathcal S$ is a '''fixed set of sets'''  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'''  :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'''  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: :$\map {\operatorname{par} } q := \map N q \bmod 2$.