hcju/proofwikips huggingface dataset:
Definition:Connected Relation
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. Then $\mathcal R$ is '''connected''' : :$\forall a, b \in S: a \ne b \implies \tuple {a, b} \in \mathcal R \lor \tuple {b, a} \in \mathcal R$ That is,  every pair of distinct elements is related (either or both ways round).
Definition:Well-Founded Ordered Set
Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is '''well-founded'''  it satisfies the '''minimal condition''': :Every non-empty subset of $S$ has a minimal element. The term '''well-founded''' can equivalently be said to apply to the ordering $\preceq$ itself rather than to the ordered set $\struct {S, \preceq}$ as a whole.
Definition:Partial Ordering
Let $\struct {S, \preceq}$ be an ordered set. Then the ordering $\preceq$ is a '''partial ordering''' on $S$  $\preceq$ is not connected. That is,  $\struct {S, \preceq}$ has at least one pair which is non-comparable: :$\exists x, y \in S: x \npreceq y \land y \npreceq x$
Definition:Well-Ordering
{{:Definition:Well-Ordering/Definition 1}}
Definition:Ordinal
{{:Definition:Ordinal/Definition 2}}
Definition:Mathematical Theory
A '''mathematical theory''', or just '''theory''', is a concept in mathematical logic. Let $U$ be a set of logical formulas. Let $\map \TT U$ be the set of all logical formulas $P$ such that $P$ is a semantic consequence of $U$. That is, let $\map \TT U = \set {P: U \models P}$. Then $\TT$ is called '''the (mathematical) theory of $U$'''. The elements of $\map \TT U$ are called theorems of $U$. The elements of $U$ are called the axioms of $\map \TT U$.
Definition:Sign (Mathematical Theory)
A '''sign''' of a mathematical theory is a symbol which has a specific meaning in the theory, as follows: # A logical sign; # A letter; # A specific sign. A logical sign is a sign which has a specific meaning and definition in a particular class of mathematical theory. A letter is a more or less arbitrary symbol whose definition depends on the specific context. For a given mathematical theory, there are signs which are specific to the use to which the theory is being put. These are the specific signs. {{wtd|This may need to go, as it covers the same ground as Sign (Formal Systems).}}
Definition:Logical Sign (Mathematical Theory)
A '''logical sign''' of a mathematical theory is a symbol which is specifically pre-defined for the class of the theory.
Definition:Letter (Mathematical Theory)
A '''letter''' of a mathematical theory is a more or less arbitrary symbol whose definition depends on the specific context. Depending on the nature of the mathematical theory, and the particular location in whatever assembly in which it occurs, the interpretation of the letter will vary.
Definition:Specific Sign (Mathematical Theory)
A '''specific sign''' of a mathematical theory is a symbol which is defined according to the use to which the theory is being put.
Definition:Assembly (Mathematical Theory)
An '''assembly''' in a mathematical theory is a succession of signs written one after another, along with other delineating marks according to the specific nature of the theory under consideration.
Definition:Link (Bourbaki Theory)
An assembly in {{AuthorRef|Nicolas Bourbaki|Bourbaki}}'s exposition of a mathematical theory is specified as being: * A succession of signs written one after another; * Certain signs which are not letters can be joined in pairs by '''links''', as follows: :$\overbrace {\tau A \Box}^{} A'$<ref>The specific symbology used by {{AuthorRef|Nicolas Bourbaki|Bourbaki}} in {{BookLink|Theory of Sets|Nicolas Bourbaki}} has not been rendered accurately here, as the author of this page has not been able to establish a method by which to do it. However, the intent has been expressed as accurately as possible.</ref>
Definition:Word (Formal Systems)
Let $\mathcal A$ be an alphabet. Then a '''word in $\mathcal A$''' is a juxtaposition of finitely many (primitive) symbols of $\mathcal A$. '''Words''' are the most ubiquitous of collations used for formal languages.
Definition:Concatenation (Formal Systems)
Let $\AA$ be an alphabet of symbols. '''Concatenation''' is the process of placing elements of $\AA$ and words in $\AA$ next to each other to form a longer word.
Definition:String
Let $\mathcal A$ be an alphabet of symbols. A '''string (in $\mathcal A$)''' is a sequence of symbols from $\mathcal A$. There is no limit to the number of times a particular symbol may appear in a given '''string'''. === Finite String === {{:Definition:String/Finite}} === Infinite String === {{:Definition:String/Infinite}}
Definition:Formal Grammar
Let $\LL$ be a formal language whose alphabet is $\AA$. The '''formal grammar''' of $\LL$ comprises of rules of formation, which determine whether collations in $\AA$ belong to $\LL$ or not. Roughly speaking, there are two types of '''formal grammar''': top-down grammar and bottom-up grammar.
Definition:Well-Formed Formula
Let $\FF$ be a formal language whose alphabet is $\AA$. A '''well-formed formula''' is a collation in $\AA$ which can be built by using the rules of formation of the formal grammar of $\FF$. That is, a collation in $\AA$ is a '''well-formed formula''' in $\FF$  it has a parsing sequence in $\FF$.
Definition:Formal System
A '''formal system''' is a formal language $\mathcal L$ together with a deductive apparatus for $\mathcal L$.
Definition:Deductive Apparatus
Let $\mathcal L$ be a formal language. A '''deductive apparatus for $\mathcal L$''' is a formally specified system for deriving conclusions about the well-formed formulas of $\mathcal L$. In mathematics and logic, '''deductive apparatuses''' can by and large be divided into proof systems and formal semantics. === Proof System === {{:Definition:Proof System}} === Formal Semantics === {{:Definition:Formal Semantics}}
Definition:Axiom/Formal Systems/Axiom Schema
An '''axiom schema''' is a well-formed formula $\phi$ of $\LL$, except for it containing one or more variables which are ''outside'' $\LL$ itself. This formula can then be used to represent an infinite number of individual axioms in one statement. Namely, each of these variables is allowed to take a specified range of values, most commonly WFFs. Each WFF $\psi$ that results from $\phi$ by a valid choice of values for all the variables is then an axiom of $\mathscr P$.
Definition:Proof System/Rule of Inference
A '''rule of inference''' is a specification of a valid means to conclude new theorems in $\mathscr P$ from given theorems and axioms of $\mathscr P$. Often, the formulation of '''rules of inference''' also appeals to the notion of provable consequence.
Definition:LAST
'''LAST''' stands for '''LAnguage of Set Theory'''. It is a formal system designed for the description of sets.
Definition:Zenzizenzizenzic
The '''zenzizenzizenzic''' of a number is its eighth power.
Definition:Predicate Symbol
Let $\mathcal L$ be a formal language (for example, the language of predicate logic $\mathcal L_1$). A '''predicate symbol''' is a letter of $\mathcal L$ used to describe a predicate or a relation. The name '''predicate symbol''' is a gesture to the reader to make clear what such a symbol should (intuitively) represent in the formal language $\mathcal L$.
Definition:Singular Statement
A '''singular statement''' is a statement whose subject is identified by means of a proper name. More generally, it is a statement which contains no variables, either bound or free. === Individuating Description === {{:Definition:Singular Statement/Individuating Description}} === Designatory Function === {{:Definition:Singular Statement/Designatory Function}}
Definition:Universal Statement
A '''universal statement''' is one which expresses the fact that all objects (in a particular universe of discourse) have a particular property. That is, a statement of the form: :$\forall x: P \paren x$ where: : $\forall$ is the universal quantifier : $P$ is a predicate symbol. It means: :All $x$ (in some given universe of discourse) have the property $P$. Note that if there exist no $x$ in this particular universe, $\forall x: P \paren x$ is always true: see vacuous truth. === Bound Variable === {{:Definition:Bound Variable/Examples/Universal Statement}}
Definition:Existential Statement
An '''existential statement''' is one which expresses the existence of at least one object (in a particular universe of discourse) which has a particular property. That is, a statement of the form: :$\exists x: P \paren x$ where: : $\exists$ is the existential quantifier : $P$ is a predicate symbol. It means: :There exists at least one $x$ (in some given universe of discourse) which has the property $P$. === Bound Variable === {{:Definition:Bound Variable/Examples/Existential Statement}}
Definition:Universe of Discourse
The '''universe of discourse''', or just '''universe''', is the term used to mean '''everything we are talking about'''. When introducing the symbols $\forall$ (the universal quantifier) or $\exists$ (the existential quantifier), it is understood that the objects referred to are those in the specified '''universe'''. It is usual to define that '''universe'''.
Definition:Logical Complement
The '''(logical) complement''' of a propositional formula $\mathbf A$ is the negation of $\mathbf A$, that is, $\neg \mathbf A$. Conversely, the '''complement''' of $\neg \mathbf A$ is defined to be $\mathbf A$. === Complementary Pair === {{:Definition:Logical Complement/Complementary Pair}}
Definition:Logical NAND
'''NAND''' (that is, '''not and'''), is a binary connective, written symbolically as $p \uparrow q$, whose behaviour is as follows: :$p \uparrow q$ is defined as: :'''it is not the case that $p$ and $q$ are both true.''' $p \uparrow q$ is voiced: :'''$p$ nand $q$'''
Definition:Logical NOR
'''NOR''' (that is, '''not or'''), is a binary connective, written symbolically as $p \downarrow q$, whose behaviour is as follows: :$p \downarrow q$ is defined as:  :'''neither $p$ nor $q$ is true.''' $p \downarrow q$ is voiced: : '''$p$ nor $q$'''
Definition:Backus-Naur Form
'''Backus-Naur Form''' (abbrevated '''BNF''') is a (formal) metalanguage for defining the syntax of a formal language $\LL$. As such, it is a formal grammar for $\LL$. '''BNF''' is only applicable to formal languages that use the collation system of words and concatenation. {{transclude:Definition:Backus-Naur Form/Alphabet   | section  = definition   | increase = 1   | title    = Alphabet   | header   = 3   | link     = true }} {{transclude:Definition:Backus-Naur Form/Rules of Formation   | section  = definition   | increase = 1   | title    = Rules of Formation   | header   = 3   | link     = true }} {{transclude:Definition:Backus-Naur Form/Specification   | section  = definition   | increase = 1   | title    = Specification   | header   = 3   | link     = true }} === Terminals and Non-Terminals === ==== Non-Terminal ==== {{:Definition:Backus-Naur Form/Non-Terminal}} ==== Terminal ==== {{:Definition:Backus-Naur Form/Terminal}} That is, non-terminals are analogous to grammatical clauses of a natural language, while terminals are analogous to its words.
Definition:Metalanguage
A '''metalanguage''' is a language (either formal or natural) which is used to make statements about another language (again, either formal or natural). === Formal Systems === {{:Definition:Metalanguage/Formal Systems}} === Object Language === {{:Definition:Metalanguage/Object Language}} === Metasyntax === {{:Definition:Metalanguage/Metasyntax}} === Metasymbol === {{:Definition:Metalanguage/Metasymbol}}
Definition:Syntax
The '''syntax''' of a language (either natural or formal) is its ''structure''.
Definition:Parsing Sequence
Let $\FF$ be a formal language with alphabet $\AA$. Let $S$ be a collation in $\AA$. A '''parsing sequence''' for $S$ in $\FF$ is a sequence of collations in $\FF$ formed by application of rules of formation of $\FF$ from previous collations in this sequence, and ending in the collation $S$. If $S$ has no '''parsing sequence''' in $\FF$, then it is not a well-formed formula in $\FF$. A parsing sequence for a given well-formed formula in any formal language is usually not unique. Thus, we can determine whether $S$ is a well-formed formula in any formal language by using a sequence of rules of formation of that language. To '''parse''' a collation in a formal language is to find a parsing sequence for that collation, and thereby to determine whether or not it is a well-formed formula.
Definition:Metatheorem
A '''metatheorem''' of a formal language $\mathcal F$ is a theorem about $\mathcal F$ which is not part of $\mathcal F$. {{mistake|It would seem that a formal system is intended instead of formal language.}}
Definition:Multigraph
A '''multigraph''' is a graph that can have more than one edge between a pair of vertices. That is, $G = \left({V, E}\right)$ is a '''multigraph''' if $V$ is a set and $E$ is a multiset of 2-element subsets of $V$.
Definition:Block Incidence Matrix of a Design
The '''block incidence matrix''' of a $(v,b,r,k,\lambda)$-BIBD, is a $v\times b$ incidence matrix $A=(a_{ij})$ such that: $a_{ij}=\begin{cases} 1 & : \text{ if the } i^{th} \text { point is in the } j^{th} \text { block} \\ 0 & : \text{ otherwise} \end{cases}$  gdc8tmjisbj7dr1loz33eonpe8ox3kz
Definition:Contrapositive Statement
The '''contrapositive''' of the conditional: :$p \implies q$ is the statement: :$\neg q \implies \neg p$
Definition:Walk (Graph Theory)
A '''walk''' on a graph is: : an alternating series of vertices and edges : beginning and ending with a vertex : in which each edge is incident with the vertex immediately preceding it and the vertex immediately following it. A '''walk''' between two vertices $u$ and $v$ is called a '''$u$-$v$ walk'''.
Definition:Connected (Graph Theory)
=== Vertices === {{:Definition:Connected (Graph Theory)/Vertices}} === Graph === {{:Definition:Connected (Graph Theory)/Graph}}
Definition:Isomorphism (Graph Theory)
Let $G = \struct {\map V G, \map E G}$ and $H = \struct {\map V H, \map E H}$ be graphs. Let there exist a bijection $F: \map V G \to \map V H$ such that for each edge $\set {u, v} \in \map E G$, there is an edge $\set {\map F u, \map F v} \in \map E H$. That is, that: :$F: \map V G \to \map V H$ is a homomorphism, and :$F^{-1}: \map V H \to \map V G$ is a homomorphism. Then $G$ and $H$ are '''isomorphic''', and this is denoted $G \cong H$. The function $F$ is called an '''isomorphism''' from $G$ to $H$.
Definition:Eulerian Graph
A loop-multigraph or loop-multidigraph is called '''Eulerian'''  it contains an Eulerian circuit.
Definition:Symmetric Design
A block design with parameters $v, b, r, k, \lambda$ is said to be '''symmetric''' if $b = v$, and (consequently) $r = k$. === Note === The '''symmetry''' does not mean the incidence matrix is symmetric (it often is not). {{explain|Why $b = v$ has as a consequence $r = k$.}}
Definition:Isomorphism (Abstract Algebra)/Group Isomorphism
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: G \to H$ be a (group) homomorphism. Then $\phi$ is a group isomorphism  $\phi$ is a bijection.
Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism
Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings. Let $\phi: R \to S$ be a (ring) homomorphism. Then $\phi$ is a ring isomorphism  $\phi$ is a bijection.
Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism
Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ and $\struct {T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}_R$ be $R$-algebraic structures. Let $\phi: S \to T$ be an $R$-algebraic structure homomorphism. Then $\phi$ is an $R$-algebraic structure isomorphism  $\phi$ is a bijection.
Definition:Isomorphism (Abstract Algebra)/Isomorphic Copy
Let $\phi: S \to T$ be an isomorphism. Let $x \in S$. Then $\map \phi x \in T$ is known as '''the isomorphic copy of $x$ (under $\phi$)'''.
Definition:Hamiltonian Graph
A '''Hamiltonian graph''' is an undirected graph that contains a Hamilton cycle.
Definition:Subgraph
A graph $H = \struct {\map V H, \map E H}$ is called a '''subgraph''' of a graph $G = \struct {\map V G, \map E G}$  $\map V H \subseteq \map V G$ and $\map E H \subseteq \map E G$.  That is to say, it contains no vertices or edges that are not in the original graph. If $H$ is a '''subgraph''' of $G$, then: :$G$ '''contains''' $H$; :$H$ '''is contained in''' $G$
Definition:Hamilton Cycle
A '''Hamilton cycle''' in a graph is a cycle that contains every vertex of the graph (but not necessarily every edge).
Definition:Eulerian Circuit
An '''Eulerian circuit''' (or '''eulerian circuit''') is a circuit that passes through every vertex of a graph and uses every edge exactly once. It follows that every Eulerian circuit is also an Eulerian trail.
Definition:Decision Procedure
Let $S$ be a set of propositional formulas. A '''decision procedure for $S$''' is an algorithm which, given a propositional formula $\mathbf A$, always terminates, returning the answer: * '''Yes''' if $\mathbf A \in S$; * '''No''' if $\mathbf A \notin S$. === Decision Procedure for Satisfiability === {{:Definition:Decision Procedure/Satisfiability}} === Decision Procedure for Tautologies === {{:Definition:Decision Procedure/Tautologies}} === Refutation Procedure === {{:Definition:Decision Procedure/Refutation Procedure}}
Definition:Semantic Consequence
Let $\mathscr M$ be a formal semantics for a formal language $\mathcal L$. Let $\mathcal F$ be a collection of WFFs of $\mathcal L$. Let $\map {\mathscr M} {\mathcal F}$ be the formal semantics obtained from $\mathscr M$ by retaining only the structures of $\mathscr M$ that are models of $\mathcal F$. Let $\phi$ be a tautology for $\map {\mathscr M} {\mathcal F}$. Then $\phi$ is called a '''semantic consequence of $\mathcal F$''', and this is denoted as: :$\mathcal F \models_{\mathscr M} \phi$ That is to say, $\phi$ is a '''semantic consequence of $\mathcal F$''' , for each $\mathscr M$-structure $\mathcal M$: :$\mathcal M \models_{\mathscr M} \mathcal F$ implies $\mathcal M \models_{\mathscr M} \phi$ where $\models_{\mathscr M}$ is the models relation. Note in particular that for $\mathcal F = \O$, the notation agrees with the notation for a $\mathscr M$-tautology: :$\models_{\mathscr M} \phi$ The concept naturally generalises to sets of formulas $\mathcal G$ on the : :$\mathcal F \models_{\mathscr M} \mathcal G$  $\mathcal F \models_{\mathscr M} \phi$ for every $\phi \in \mathcal G$.
Definition:Basis Element
Let $S^n$ be a cartesian space. An element $s \in S$ is known as a '''basis element of $S^n$'''. Alternatively, let the ordered tuple $X = \left({x_1, x_2, \ldots, x_n}\right)$ be any element of $S^n$. Let $x_j \in \left({x_1, x_2, \ldots, x_n}\right)$. Then $x_j$ is a '''basis element of $S^n$'''.
Definition:Logical Complement/Complementary Pair
For any propositional formula $\mathbf A$, the set $\left\{{\mathbf A, \neg \mathbf A}\right\}$ is called a '''complementary pair of formulas'''.
Definition:Disjunctive Normal Form
A propositional formula $P$ is in '''disjunctive normal form'''  it consists of a disjunction of: :$(1):\quad$ conjunctions of literals and/or: :$(2):\quad$ literals.
Definition:Multiset
A '''multiset''' is a pair $\struct {S, \mu}$ where: :$S$ is a set :$\mu: S \to \N_{>0}$ is a mapping to the strictly positive natural numbers For $s \in S$ the natural number $\map \mu s$ is called the '''multiplicity''' of $s$. Note that the '''multiplicities''' of elements is finite: we do not allow infinitely many occurrences of the same element, though the set $S$ itself may be finite, countably infinite or uncountably infinite.
Definition:Semantics
The '''semantics''' of a language (either natural or formal) is its '''meaning''' in a linguistic sense.
Definition:Trail
A '''trail''' is a walk in which all edges are distinct. A '''trail''' between two vertices $u$ and $v$ is called a '''$u$-$v$ trail'''. The set of vertices and edges which go to make up a '''trail''' form a subgraph. This subgraph itself is also referred to as a '''trail'''.
Definition:Circuit
A '''circuit''' is a closed trail with at least one edge. The set of vertices and edges which go to make up a circuit form a subgraph. This subgraph itself is also referred to as a '''circuit'''.
Definition:Walk (Graph Theory)/Closed
A '''closed walk''' is a walk whose first vertex is the same as the last. That is, it is a walk which ends where it starts.
Definition:Tree (Graph Theory)
=== Definition 1=== {{:Definition:Tree (Graph Theory)/Definition 1}} === Definition 2=== {{:Definition:Tree (Graph Theory)/Definition 2}}
Definition:Path (Graph Theory)
A '''path''' is a trail in which all vertices (except perhaps the first and last ones) are distinct. A path between two vertices $u$ and $v$ is called a '''$u$-$v$ path'''. The set of vertices and edges which go to make up a '''path''' form a subgraph. This subgraph itself is also referred to as a '''path'''. === Open Path === {{:Definition:Path (Graph Theory)/Open}}
Definition:Rooted Tree
A '''rooted tree''' is a tree with a countable number of nodes, in which a particular node is distinguished from the others and called the '''root node''': :300px
Definition:Cycle (Graph Theory)
A '''cycle''' is a circuit in which no vertex except the first (which is also the last) appears more than once. An '''$n$-cycle''' is a cycle with $n$ vertices. The set of vertices and edges which go to make up a cycle form a subgraph. This subgraph itself is also referred to as a '''cycle'''. === Odd Cycle === {{:Definition:Cycle (Graph Theory)/Odd}} === Even Cycle === {{:Definition:Cycle (Graph Theory)/Even}}
Definition:Walk (Graph Theory)/Length
The '''length''' of a walk is the number of edges it has, counting repeated edges as many times as they appear. A walk is said to be of '''infinite length'''  it has infinitely many edges.
Definition:Finite Graph
A '''finite graph''' is a graph with a finite number of edges and a finite number of vertices.
Definition:Simple Graph
A '''simple graph''' is a graph which is: :An undirected graph, that is, the edges are defined as doubleton sets of vertices and not ordered pairs :Not a multigraph, that is, there is no more than one edge between each pair of vertices :Not a loop-graph, that is, there are no loops, that is, edges which start and end at the same vertex :Not a weighted graph, that is, the edges are not mapped to a number.
Definition:Root of Polynomial
Let $R$ be a commutative ring with unity. Let $f \in R \sqbrk x$ be a polynomial over $R$. A '''root''' in $R$ of $f$ is an element $x \in R$ for which $\map f x = 0$, where $\map f x$ denotes the image of $f$ under the evaluation homomorphism at $x$.
Definition:Root of Mapping
Let $f: R \to R$ be a mapping on a ring $R$. Let $x \in R$. Then the values of $x$ for which $f \left({x}\right) = 0_R$ are known as the '''roots of the mapping $f$'''.
Definition:Tree (Graph Theory)/Leaf Node
Let $v$ be a node of a tree $T$. Then $v$ is a '''leaf node''' of a $T$  $v$ is of degree $1$. If $T$ is a rooted tree, this is equivalent to saying that $v$ has no child nodes.
Definition:Rooted Tree/Ancestor Node
Let $T$ be a rooted tree with root $r_T$. Let $t$ be a node of $T$. An '''ancestor node''' of $t$ is a node in the path from $t$ to $r_T$. This path is indeed unique, by Path in Tree is Unique. === Proper Ancestor === {{:Definition:Rooted Tree/Ancestor Node/Proper}}
Definition:Propositional Tableau
'''Propositional tableaus''' are a specific subset of the labeled trees for propositional logic. === Identification === {{:Definition:Propositional Tableau/Identification}} {{transclude:Definition:Propositional Tableau/Construction  |section = def  |increase = 1  |link = true  |title = Construction  |header = 3 }}
Definition:Rooted Tree/Parent Node
Let $T$ be a rooted tree whose root is $r_T$. Let $t$ be a node of $T$. From Path in Tree is Unique, there is only one path from $t$ to $r_T$. Let $\pi: T \setminus \left\{{r_T}\right\} \to T$ be the mapping defined by: :$\pi \left({t}\right) := \text{the node adjacent to $t$ on the path to $r_T$}$ Then $\pi \left({t}\right)$ is known as the '''parent node''' of $t$. The mapping $\pi$ is called the '''parent mapping'''.