hcju/proofwikips huggingface dataset:
Definition:Algebraic Geometry
'''Algebraic geometry''' is the branch of geometry which studies objects in multi-dimensional space using the techniques of abstract algebra. In particular, techniques from commutative algebra are mainly used.
Definition:Brill-Noether Theory
'''Brill-Noether Theory''' is the branch of algebraic geometry which studies special divisors.
Definition:Tarski-Grothendieck Set Theory
'''Tarski-Grothendieck set theory''' is an implementation of axiomatic set theory which allows for the existence of a universe.
Definition:Kummer Theory
'''Kummer Theory''' is a branch of abstract algebra and number theory which investigates certain types of field extensions.
Definition:Tensor Theory
'''Tensor theory''' is the branch of linear algebra which studies tensors.
Definition:Vector Analysis
'''Vector analysis''' is the branch of linear algebra concerned with differentiation and integration of vector spaces, primarily in Euclidean space of $3$-dimensional space.
Definition:Fluid Mechanics
'''Fluid mechanics''' is the branch of mechanics concerned with the behaviour of fluids.
Definition:Electromagnetism
'''Electromagnetism''' is the branch of physics concerned with the study of the combined phenomena of electricity and magnetism, and their interconnected nature.
Definition:Partition Theory
'''Partition theory''' is the subfield of combinatorics which studies integer partitions.
Definition:Proof Theory
'''Proof theory''' is the subfield of mathematical logic which studies proofs as mathematical objects, thus allowing them to be analysed mathematically.
Definition:Fluid Dynamics
'''Fluid dynamics''' is the branch of fluid mechanics concerned with the behaviour of fluids in motion.
Definition:Class Theory
'''Class theory''' is an extension of set theory which allows the creation of collections that are not sets by classes.
Definition:Descriptive Set Theory
'''Descriptive set theory''' is a branch of set theory which studies certain classes subsets of the real number line and other Polish spaces.
Definition:Recursion Theory
'''Recursion theory''' is a branch of computability theory which concerns itself with recursive structures.
Definition:Inner Model Theory
'''Inner model theory''' is a branch of set theory and model theory which studies certain models of ZFC.
Definition:Divisor (Algebra)/Notation
The conventional notation for '''$x$ is a divisor of $y$''' is "$x \mid y$", but there is a growing trend to follow the notation "$x \divides y$", as espoused by {{AuthorRef|Donald Ervin Knuth|Knuth}} etc. From {{BookReference|Concrete Mathematics: A Foundation for Computer Science||Ronald L. Graham|author2 = Donald E. Knuth|author3 = Oren Patashnik|ed = 2nd|edpage = Second Edition}}: :''The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.'' An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$. Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \ $ or similar to denote non-divisibility.
Definition:Cantor-Bendixson Derivative
Let $\struct {X, \tau}$ be a topological space. Let $S \subseteq X$. Then for all ordinals $\beta$, the $\beta$th '''Cantor-Bendixson derivative''' of $S$ is defined by Transfinite Recursion thus: :$S^{\paren \beta} = \begin {cases} S & : \beta = 0 \\ \paren {S^{\paren \alpha} }' & : \beta = \alpha^+ \\ \displaystyle \bigcap_{\alpha \mathop < \lambda} S^{\paren \alpha} & : \beta = \lambda \end{cases}$ where: :$\paren {S^{\paren \alpha} }'$ is the derived set of $S^{\paren \alpha}$ :$\lambda$ is a limit ordinal. {{NamedforDef|Georg Cantor|name2 = Ivar Otto Bendixson|cat = Cantor|cat2 = Bendixson}}
Definition:Directed Hamilton Cycle Problem/Function Version
:Given a directed graph $G$ with $n$ vertices, to find a Hamilton cycle in $G$.
Definition:Directed Hamilton Cycle Problem
There are two versions of the '''Directed Hamilton Cycle Problem'''. === Function Version === {{:Definition:Directed Hamilton Cycle Problem/Function Version}} === Decision Version === {{:Definition:Directed Hamilton Cycle Problem/Decision Version}}
Definition:Finite Subset
Let $S$ be a set. Let $T$ be a subset of $S$ which is a finite set. Then $T$ is a '''finite subset''' of $S$.
Definition:Quotient Module
Let $M$ be an $R$-module and $M^\ast$ be the underlying group. Let $N$ be a submodule of $M$ and $N^\ast$ be the underlying subgroup. Let $a + N$ denote the coset of the quotient group $M^\ast / N^\ast$. Define the $R$-action on $M^\ast / N^\ast$ as: :$\forall r \in R, \forall a \in M / N: r \left({a + N^\ast}\right) := r a + N^\ast$ Then $(M^\ast / N^\ast,\circ)$ is a '''quotient $R$-module'''.
Definition:Right Module Axioms
{{begin-axiom}} {{axiom | n = \text {RM} 1         | lc= Scalar Multiplication Right Distributes over Module Addition         | q = \forall \lambda \in R: \forall x, y \in G         | ml= \paren {x +_G y} \circ \lambda         | mo= =         | mr= \paren {x \circ \lambda} +_G \paren {y \circ \lambda} }} {{axiom | n = \text {RM} 2         | lc= Scalar Multiplication Left Distributes over Scalar Addition         | q = \forall \lambda, \mu \in R: \forall x \in G         | ml= x \circ \paren {\lambda +_R \mu}         | mo= =         | mr= \paren {x \circ \lambda} +_G \paren {x\circ \mu} }} {{axiom | n = \text {RM} 3         | lc= Associativity of Scalar Multiplication         | q = \forall \lambda, \mu \in R: \forall x \in G         | ml= x \circ \paren {\lambda \times_R \mu}         | mo= =         | mr= \paren {x \circ \lambda} \circ \mu }} {{end-axiom}}
Definition:Tensor Algebra
'''Tensor algebra''' is the branch of module theory which studies tensor products.
Definition:Module Direct Product
=== Finite Case === {{:Definition:Module Direct Product/Finite Case}} === General Case === {{:Definition:Module Direct Product/General Case}}
Definition:Ramsey Theory
'''Ramsey theory''' is a branch of combinatorics that studies the conditions under which order must appear.
Definition:Angle Inscribed in Circle
:300px Let $AB$ and $BC$ be two chords of a circle which meet at $B$. The angle $\angle ABC$ is the '''angle inscribed at $B$ (with respect to $A$ and $C$)'''.
Definition:Path (Topology)/Endpoint
The initial point and final point of $\gamma$ can be referred to as the '''endpoints of $\gamma$'''
Definition:Infinite Path
An '''infinite path''' is a path of infinite length
Definition:Supremum/Also known as
Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the '''least upper bound of $T$''' and denoted $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$. Some sources refer to the '''supremum of a set''' as the '''supremum ''on'' a set'''. Some sources refer to the '''supremum of a set''' as the '''join of the set''' and use the notation $\bigvee S$. Some sources introduce the notation $\displaystyle \sup_{y \mathop \in S} y$, which may improve clarity in some circumstances. Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a '''supremum''' as an '''upper limit'''.
Definition:Supremum/Also defined as
Some sources refer to the supremum as being '''''the'' upper bound'''. Using this convention, any element greater than this is not considered to be an upper bound.
Definition:Infimum/Also known as
Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the '''greatest lower bound of $T$''' and denoted $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$. Some sources refer to the '''infimum of a set''' as the '''infimum ''on'' a set'''. Some sources introduce the notation $\displaystyle \inf_{y \mathop \in S} y$, which may improve clarity in some circumstances. Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to a '''infimum''' as a '''lower limit'''.
Definition:Infimum/Also defined as
Some sources refer to the infimum as being '''''the'' lower bound'''. Using this convention, any element less than this is not considered to be a lower bound.
Definition:Differential Equation/Solution
Let $\Phi$ be a differential equation defined on a domain $D$. Let $\phi$ be a function which satisfies $\Phi$ on the whole of $D$. Then $\phi$ is known as '''a solution''' of $\Phi$. Note that, in general, there may be more than one '''solution''' to a given differential equation. On the other hand, there may be none at all. === General Solution === {{:Definition:Differential Equation/Solution/General Solution}} === Particular Solution === {{:Definition:Differential Equation/Solution/Particular Solution}}
Definition:Differential Equation/Solution/General Solution
The '''general solution''' of $\Phi$ is the set of ''all'' functions $\phi$ that satisfy $\Phi$.
Definition:Linear Representation/Group
Let $\left({\mathbb k, +, \circ}\right)$ be a field. Let $V$ be a vector space over $\mathbb k$ of finite dimension. Let $\operatorname {GL} \left({V}\right)$ be the general linear group of $V$. Let $\left({G, \cdot}\right)$ be a finite group. A '''linear representation of $G$ on $V$''' is a group homomorphism $\rho: G \to \operatorname {GL} \left({V}\right)$.
Definition:Linear Representation/Algebra
Let $K$ be a field. Let $A$ be an associative unitary algebra over $K$. Then a '''(linear) representation of $A$''' is a vector space $V$ over $K$ equipped with a homomorphism of algebras: :$\rho: A \to \map {\operatorname {End} } V$ where $\map {\operatorname {End} } V$ is the endomorphism ring of $V$.
Definition:Scalar Triple Product
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf a       | r = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k }} {{eqn | l = \mathbf b       | r = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k }} {{eqn | l = \mathbf c       | r = c_i \mathbf i + c_j \mathbf j + c_k \mathbf k }} {{end-eqn}} where $\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis of $\mathbf V$. The '''scalar triple product''', denoted as $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$, is defined as: :$\mathbf a \cdot \paren {\mathbf b \times \mathbf c} = \begin {vmatrix} a_i & a_j & a_k \\ b_i & b_j & b_k \\ c_i & c_j & c_k \\ \end {vmatrix}$
Definition:Singleton Graph
The '''singleton graph''' $N_1$ is the simple graph with one vertex: :60px
Definition:Complete Distributive Lattice
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Then $S$ is a '''complete distributive lattice'''  :$(1): \quad S$ is a complete lattice :$(2): \quad S$ is a distributive lattice.
Definition:Cauchy Sequence/Complex Numbers
Let $\sequence {z_n}$ be a sequence in $\C$. Then $\sequence {z_n}$ is a '''Cauchy sequence''' : : $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {z_n - z_m} < \epsilon$ where $\size {z_n - z_m}$ denotes the complex modulus of $z_n - z_m$.
Definition:Commutative and Unitary Ring/Axioms
A commutative and unitary ring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = \text A 0         | q = \forall a, b \in R         | m = a * b \in R         | lc= Closure under addition }} {{axiom | n = \text A 1         | q = \forall a, b, c \in R         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of addition }} {{axiom | n = \text A 2         | q = \forall a, b \in R         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = \text A 3         | q = \exists 0_R \in R: \forall a \in R         | m = a * 0_R = a = 0_R * a         | lc= Identity element for addition: the zero }} {{axiom | n = \text A 4         | q = \forall a \in R: \exists a' \in R         | m = a * a' = 0_R = a' * a         | lc= Inverse elements for addition: negative elements }} {{axiom | n = \text M 0         | q = \forall a, b \in R         | m = a \circ b \in R         | lc= Closure under product }} {{axiom | n = \text M 1         | q = \forall a, b, c \in R         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of product }} {{axiom | n = \text M 2         | q = \forall a, b \in R         | m = a \circ b = b \circ a         | lc= Commutativity of product }} {{axiom | n = \text M 3         | q = \exists 1_R \in R: \forall a \in R         | m = a \circ 1_R = a = 1_R \circ a         | lc= Identity element for product: the unity }} {{axiom | n = \text D         | q = \forall a, b, c \in R         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}         | lc= Product is distributive over addition }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} }} {{end-axiom}} These criteria are called the '''commutative and unitary ring axioms'''.
Definition:Accumulation Point/Set
Let $x \in S$. Then $x$ is an '''accumulation point''' of $A$ : :$x \in \cl {A \setminus \set x}$ where $\operatorname{cl}$ denotes the (topological) closure of a set.
Definition:Set Derivative
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A \subseteq S$. The '''derivative''' of $A$ in $T$ is the set of all accumulation points of $A$. It is usually denoted on  as $A'$.
Definition:Pythagoras's Constant
'''Pythagoras's constant''' is sometimes used to mean $\sqrt 2$, the square root of $2$. === Decimal Expansion === {{:Square Root of 2}} {{NamedforDef|Pythagoras of Samos|cat = Pythagoras}}
Definition:Existential Quantifier/Unique
The symbol $\exists !$ denotes the existence of a unique object fulfilling a particular condition. :$\exists ! x: \map P x$ means: :'''There exists exactly one object $x$ such that $\map P x$ holds''' or: :'''There exists one and only one $x$ such that $\map P x$ holds'''. Formally: :$\exists ! x: \map P x \dashv \vdash \exists x: \map P x \land \forall y: \map P y \implies x = y$ In natural language, this means: ::''There exists exactly one $x$ with the property $P$'' : is logically equivalent to: ::''There exists an $x$ such that $x$ has the property $P$, and for every $y$, $y$ has the property $P$ only if $x$ and $y$ are the same object.''
Definition:Existential Quantifier/Exact
The symbol $\exists_n$ denotes the existence of an exact number of objects fulfilling a particular condition. :$\exists_n x: \map P x$ means: :'''There exist exactly $n$ objects $x$ such that $\map P x$ holds'''.
Definition:Set Union/General Definition
Let $S$ be a collection, which could be either a set or a class. The '''union of $S$''' is: :$\ds \bigcup S := \set {x: \exists X \in S: x \in X}$ That is, the set of all elements of all elements of $S$ which are themselves sets.
Definition:Ordered Pair/Informal Definition
An '''ordered pair''' is a two-element set together with an ordering. In other words, one of the elements ''is'' distinguished above the other - it comes first. Such a structure is written: :$\tuple {a, b}$ and it means: : '''first $a$, then $b$'''.
Definition:Ordered Pair/Kuratowski Formalization
The concept of an ordered pair can be formalized by the definition: :$\tuple {a, b} := \set {\set a, \set {a, b} }$ This formalization justifies the existence of ordered pairs in Zermelo-Fraenkel set theory.
Definition:Inverse Mapping/Also defined as
Let $S$ and $T$ be sets. Let $f: S \to T$ be an injection. Then its '''inverse mapping''' is the mapping $g$ such that: :its domain $\Dom g$ equals the image $\Img f$ of $f$ :$\forall y \in \Img f: \map f {\map g y} = y$ Thus $f$ is seen to be a surjection by tacit use of Restriction of Mapping to Image is Surjection.
Definition:Weight of Topological Space
Let $T$ be a topological space. Let $\mathbb B$ be the set of all bases of $T$. === Definition 1 === {{:Definition:Weight of Topological Space/Definition 1}} === Definition 2 === {{:Definition:Weight of Topological Space/Definition 2}}
Definition:Operation Compatible with Equivalence
Let $F$ be a (unary) operation which can be applied to sets. Then $F$ is '''compatible with set equivalence''' : :$F \left({A}\right) = F \left({B}\right) \iff A \sim B$ where: :$A$ and $B$ are arbitrary sets :$\sim$ denotes set equivalence.
Definition:Weight of Topological Space/Definition 1
The '''weight''' of $T$ is defined as: :$\displaystyle \map w T := \bigcap_{\BB \mathop \in \mathbb B} \card \BB$ where $\card \BB$ denotes the cardinality of $\BB$.
Definition:Weight of Topological Space/Definition 2
The '''weight''' of $T$ is the smallest cardinality of the elements of $\mathbb B$: :$\map w T := \min \set {\card \BB: \BB \in \mathbb B}$
Definition:Character of Point in Topological Space
Let $T$ be a topological space. Let $x$ be a point of $T$. Let $\mathbb B \left({x}\right)$ be the set of all local bases at $x$. The '''character of (the point) $x$ in $T$''' is the smallest cardinality of the elements of $\mathbb B \left({x}\right)$: :$\chi \left({x,T}\right) := \min \left\{ {\left\vert{\mathcal B}\right\vert: \mathcal B \in \mathbb B \left({x}\right)}\right\}$
Definition:Character of Topological Space
Let $T$ be a topological space. The '''character of (the topological space) $T$''' is the supremum of the set of characters of the points of $T$: :$\chi \left({T}\right) := \sup \left\{ {\chi \left({x, T}\right): x \in T}\right\}$
Definition:Bound Variable/Examples/Universal Statement
In the universal statement: :$\forall x: \map P x$ the symbol $x$ is a '''bound variable'''. Thus, the meaning of $\forall x: \map P x$ does not change if $x$ is replaced by another symbol. That is, $\forall x: \map P x$ means the same thing as $\forall y: \map P y$ or $\forall \alpha: \map P \alpha$. And so on.
Definition:Bound Variable/Examples/Existential Statement
In the existential statement: :$\exists x: \map P x$ the symbol $x$ is a '''bound variable'''. Thus, the meaning of $\exists x: \map P x$ does not change if $x$ is replaced by another symbol. That is, $\exists x: \map P x$ means the same thing as $\exists y: \map P y$ or $\exists \alpha: \map P \alpha$. And so on.
Definition:Existential Statement/Conditional
A '''conditionally existential statement''' is an existential statement which states the existence of an object fulfilling a certain propositional function dependent upon the existence of certain other objects.
Definition:Conditional/Formal Implication
'''Formal implication''' is a usage of an implication in which it is necessary for there to be a formal connection between the antecedent and the consequent in order for the implication to have any semantic meaning.
Definition:Stoic School
The '''stoic school''' was a school of Greek philosophy which emphasized the importance of goodness and peace of mind gained from living a life of Virtue in accordance with Nature. The writings of its philosophers reveal the first beginnings of the discipline of propositional logic.
Definition:Truth Table/Matrix Presentation
A two-element truth table can be presented in '''matrix form''': :{{:Definition:Conditional/Truth Table/Matrix Form}}$\qquad${{:Definition:Conjunction/Truth Table/Matrix Form}}$\qquad${{:Definition:Disjunction/Truth Table/Matrix Form}}$\qquad${{:Definition:Biconditional/Truth Table/Matrix Form}} that is, in the form of a Cayley table. This, however, can be used only when the operation being displayed has two elements.
Definition:Fundamental Truth Table
The '''fundamental truth tables''' are the characteristic truth tables for: : Logical Not : Conjunction : Disjunction : Conditional : Biconditional as follows: === Logical Not === {{:Definition:Logical Not/Truth Table}} === Conjunction === {{:Definition:Conjunction/Truth Table}} === Disjunction === {{:Definition:Disjunction/Truth Table}} === Conditional === {{:Definition:Conditional/Truth Table}} === Biconditional === {{:Definition:Biconditional/Truth Table}}
Definition:Derivative Truth Table
A '''derivative truth table''' is a truth table for a statement form which is not mapped by one of the fundamental truth tables: : Logical Not : Conjunction : Disjunction : Conditional : Biconditional
Definition:Truth Table/Row
A '''row''' of a truth table is one of the horizontal lines that consists of instances of the symbols $T$ and $F$. Each '''row''' contains the truth values of each of the boolean interpretations of the statement forms according to the propositional variables that comprise them. There are as many '''rows''' in a truth table as there are combinations of $T$ and $F$ for all the propositional variables that constitute the statement forms.
Definition:Truth Table/Column
A '''row''' of a truth table is one of the vertical lines headed by a statement form presenting all the truth values that the statement form takes. Each entry in the '''column''' corresponds to one specific combination of truth values taken by the propositional variables that the statement form comprises.
Definition:Conjugate Statements
Let $p$ and $q$ be statements. The conditional statement: :$p \implies q$ its inverse: :$\lnot p \implies \lnot q$ its converse: :$q \implies p$ and its contrapositive: :$\lnot q \implies \lnot p$ are all '''conjugate statements'''.
Definition:Rubik's Cube
'''Rubik's cube''' is a toy consisting of a plastic cube divided into $27$ smaller cubes, interlocking at the center cube and allowing each face to rotate independently of the others. Each of the $6$ faces of the main cube is colored with one of six contrasting colors. Rotating the faces of the cube causes these $6$ faces to become mixed up. The object of the exercise is to restore the $6$ faces to their original single colors.
Definition:Huntington Algebra
A '''Huntington algebra''' is an algebraic structure $\struct {S, \circ, *}$ such that: {{begin-axiom}} {{axiom | n = \text {HA} 0         | lc=         | t =$S$ is closed under both $\circ$ and $*$ }} {{axiom | n = \text {HA} 1         | lc=         | t = Both $\circ$ and $*$ are commutative }} {{axiom | n = \text {HA} 2         | lc=         | t = Both $\circ$ and $*$ distribute over the other }} {{axiom | n = \text {HA} 3         | lc=         | t = Both $\circ$ and $*$ have identities $e^\circ$ and $e^*$ respectively, where $e^\circ \ne e^*$ }} {{axiom | n = \text {HA} 4         | lc=         | t = $\forall a \in S: \exists a' \in S: a \circ a' = e^*, a * a' = e^\circ$ }} {{end-axiom}} The element $a'$ in $(\text {HA} 4)$ is often called the '''complement''' of $a$. A '''Huntington algebra''' can also be considered as a mathematical system $\set {S, O, A}$ where $O = \set {\circ, *}$ and $A$ consists of the set of axioms $(\text {HA}  0)$ to $(\text {HA}  4)$ as defined above.
Definition:Cardinality/Finite
Let $S$ be a finite set. The '''cardinality''' $\card S$ of $S$ is the '''number of elements in $S$'''. That is, if: :$S \sim \N_{< n}$ where: :$\sim$ denotes set equivalence :$\N_{<n}$ is the set of all natural numbers less than $n$ then we define: :$\card S = n$
Definition:Rubik's Cube/Basic Moves
The set of '''basic moves''' of the Rubik's cube can be denoted: :$\set {U, D, L, R, F, B}$ where $U$, $D$, $L$, $R$, $F$, $B$ are in Singmaster notation.
Definition:Singmaster Notation
Let the cube be facing the user with the $U$ and $D$ faces horizontal and the $F$ face towards the user. In the following, '''clockwise''' denotes rotation of a layer of the cube in the direction of the clock '''as though that face were towards the user'''. :$U$: turns the up face one quarter turn clockwise. :$D$: turns the down face one quarter turn clockwise. :$L$: turns the left face one quarter turn clockwise. :$R$: turns the right face one quarter turn clockwise. :$F$: turns the front face one quarter turn clockwise. :$B$: turns the back face one quarter turn clockwise. Appending a $'$ symbol to one of the letters denotes a turn of that layer one quarter turn anticlockwise. Appending a $2$ to one of the letters denotes a half turn of that layer.