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# Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2. [1]

110 relations: Abstract rewriting system, Affine space, Algebraic structure, Antisymmetric relation, Arithmetic, Ascending chain condition, Asymmetric relation, Axiom of choice, Bell number, Bijection, Binary relation, Boolean algebra (structure), Cambridge University Press, Cartesian product, Category of relations, Category theory, Charles Sanders Peirce, Class (set theory), Closure (mathematics), Codomain, Complement (set theory), Completeness (order theory), Composition of relations, Computer science, Confluence (abstract rewriting), Congruence (geometry), Connex relation, Converse relation, Correspondence (mathematics), Dependency relation, Directed graph, Divisibility rule, Divisor, Domain of a function, Duality (order theory), Empty set, Equality (mathematics), Equivalence relation, Euclidean relation, Finitary relation, Finite set, Function (mathematics), Function composition, Geometry, Google Books, Graph (discrete mathematics), Graph theory, Greatest and least elements, Gunther Schmidt, Hasse diagram, ... Expand index (60 more) »

## Abstract rewriting system

In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviation ARS) is a formalism that captures the quintessential notion and properties of rewriting systems.

## Affine space

In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

## Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.

## Antisymmetric relation

In mathematics, a binary relation R on a set X is anti-symmetric if there is no pair of distinct elements of X each of which is related by R to the other.

## Arithmetic

Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.

## Ascending chain condition

In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.

## Asymmetric relation

In mathematics, an asymmetric relation is a binary relation on a set X where.

## Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

## Bell number

In combinatorial mathematics, the Bell numbers count the possible partitions of a set.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

## Boolean algebra (structure)

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Cartesian product

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

## Category of relations

In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms.

## Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

## Charles Sanders Peirce

Charles Sanders Peirce ("purse"; 10 September 1839 – 19 April 1914) was an American philosopher, logician, mathematician, and scientist who is sometimes known as "the father of pragmatism".

## Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

## Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.

## Codomain

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall.

## Complement (set theory)

In set theory, the complement of a set refers to elements not in.

## Completeness (order theory)

In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset).

## Composition of relations

In the mathematics of binary relations, the composition relations is a concept of forming a new relation from two given relations R and S. The composition of relations is called relative multiplication in the calculus of relations.

## Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

## Confluence (abstract rewriting)

In computer science, confluence is a property of rewriting systems, describing which terms in such a system can be rewritten in more than one way, to yield the same result.

## Congruence (geometry)

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

## Connex relation

In mathematics, a binary relation R on a set X is called a connex relation if it relates all pairs of elements from X in some way.

## Converse relation

In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation.

## Correspondence (mathematics)

In mathematics and mathematical economics, correspondence is a term with several related but distinct meanings.

## Dependency relation

In mathematics and computer science, a dependency relation is a binary relation that is finite, symmetric, and reflexive; i.e. a finite tolerance relation.

## Directed graph

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is a set of vertices connected by edges, where the edges have a direction associated with them.

## Divisibility rule

A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits.

## Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Duality (order theory)

In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd.

## Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

## Euclidean relation

In mathematics, Euclidean relations are a class of binary relations that formalizes "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other.".

## Finitary relation

In mathematics, a finitary relation has a finite number of "places".

## Finite set

In mathematics, a finite set is a set that has a finite number of elements.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

Google Books (previously known as Google Book Search and Google Print and by its codename Project Ocean) is a service from Google Inc. that searches the full text of books and magazines that Google has scanned, converted to text using optical character recognition (OCR), and stored in its digital database.

## Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

## Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

## Greatest and least elements

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.

## Gunther Schmidt

Gunther Schmidt (born 1939, Rüdersdorf) is a German mathematician who works also in informatics.

## Hasse diagram

In order theory, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction.

## Heterogeneous relation

In mathematics, a heterogeneous relation is a subset of a Cartesian product A × B, where A and B are distinct sets.

## Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

## Incidence structure

In mathematics, an abstract system consisting of two types of objects and a single relationship between these types of objects is called an incidence structure.

## Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

## Inequality (mathematics)

In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality).

## Infimum and supremum

In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Intersection (set theory)

In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

## Involution (mathematics)

In mathematics, an involution, or an involutory function, is a function that is its own inverse, for all in the domain of.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Lattice (order)

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## Loop (graph theory)

In graph theory, a loop (also called a self-loop or a "buckle") is an edge that connects a vertex to itself.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Maximal and minimal elements

In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.

## Morse–Kelley set theory

In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG).

## Multivalued function

In mathematics, a multivalued function from a domain to a codomain is a heterogeneous relation.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Order theory

Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.

## Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects.

## Ordinal number

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

## Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

## Parallel (geometry)

In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.

## Partial equivalence relation

In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) R on a set X is a relation that is symmetric and transitive.

## Partial function

In mathematics, a partial function from X to Y (written as or) is a function, for some subset X ′ of X.

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## Partition of a set

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

## Pecking order

Pecking order or peck order is the colloquial term for the hierarchical system of social organization.

## Power set

In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.

## Preference

A preference is a technical term in psychology, economics and philosophy usually used in relation to choosing between alternatives; someone has a preference for A over B if they would choose A rather than B.

## Preorder

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Range (mathematics)

In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Reflexive relation

In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.

## Relation algebra

In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation.

## Relational algebra

Relational algebra, first created by Edgar F. Codd while at IBM, is a family of algebras with a well-founded semantics used for modelling the data stored in relational databases, and defining queries on it.

## Restriction (mathematics)

In mathematics, the restriction of a function f is a new function f\vert_A obtained by choosing a smaller domain A for the original function f. The notation f is also used.

In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction.

## Serial relation

In set theory, a serial relation is a binary relation R for which every element of the domain has a corresponding range element (∀ x ∃ y x R y).

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

## Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a. In mathematical notation, this is: Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

## Ternary relation

In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three.

## Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

## Tournament (graph theory)

A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph.

## Transitive closure

In mathematics, the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive.

## Transitive relation

In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.

## Trichotomy (mathematics)

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.

## Tuple

In mathematics, a tuple is a finite ordered list (sequence) of elements.

## Two-dimensional graph

A two-dimensional graph is a set of points in two-dimensional space.

## Upper and lower bounds

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (K, ≤) is an element of K which is greater than or equal to every element of S. The term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

## Vacuous truth

In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property.

## Von Neumann–Bernays–Gödel set theory

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel set theory (ZFC).

## Weak ordering

In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other.

## Well-founded relation

In mathematics, a binary relation, R, is called well-founded (or wellfounded) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is an element m not related by sRm (for instance, "s is not smaller than m") for any s ∈ S. In other words, a relation is well founded if Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.

## Well-order

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

## References

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