In mathematics, an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them , where {a, b} = {b, a}. In contrast, an ordered pair (a, b) has a as its first element and b as its second element, which means (a, b) ≠ (b, a).
While the two elements of an ordered pair (a, b) need not be distinct, modern authors only call {a, b} an unordered pair if a ≠ b.[1][2][3][4]
But for a few authors a singleton is also considered an unordered pair, although today, most would say that {a, a} is a multiset. It is typical to use the term unordered pair even in the situation where the elements a and b could be equal, as long as this equality has not yet been established.
A set with precisely two elements is also called a 2-set or (rarely) a binary set.
An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1.
In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing.
More generally, an unordered n-tuple is a set of the form {a1, a2,... an}.[5][6][7]
In mathematics, an unorderedpair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them...
from the ordered pair (b, a) unless a = b. (In contrast, the unorderedpair {a, b} equals the unorderedpair {b, a}.) Ordered pairs are also called 2-tuples...
of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced...
stratified comprehension; in ZFC, the existence of the unorderedpair is given by the Axiom of Pairing, the existence of the empty set follows by Separation...
{and}}\;x\neq y\}} , a set of edges (also called links or lines), which are unorderedpairs of vertices (that is, an edge is associated with two distinct vertices)...
there are 21 unorderedpairs of points, each of which may be mapped by a symmetry onto any other unorderedpair. For any unorderedpair there are 8 symmetries...
loops. A multigraph G is an ordered pair G := (V, E) with V a set of vertices or nodes, E a multiset of unorderedpairs of vertices, called edges or lines...
triangular numbers arise naturally as numbers of unorderedpairs of unorderedpairs of objects, including pairs where both objects are the same: An example...
something, a pairUnorderedpair, or pair set, in mathematics and set theory Ordered pair, or 2-tuple, in mathematics and set theory Pairing, in mathematics...
ISBN 978-0-321-39053-0. Dijkstra, Edsger W. (1997). "WLOG, or the misery of the unorderedpair (EWD1223)". In Broy, Manfred; Schieder, Birgit (eds.). Mathematical...
usually written ∀ unorderedpair A set of two elements where the order of the elements does not matter, distinguishing it from an ordered pair where the sequence...
empty or contains an unorderedpair from S, Each symbol occurs exactly once in each row and column of the array, and Every unorderedpair of symbols occurs...
contains an unorderedpair from the set of symbols Each symbol occurs exactly once in each row and column of the array Every unorderedpair of symbols...
ordinary or undirected graph, in that the latter is defined in terms of unorderedpairs of vertices, which are usually called edges, links or lines. The aforementioned...
types, featuring unordered elements accessed by label. A few programming languages combine ordered tuple product types and unordered record types into...
a similarity, turns one Brocard point into the other. However, the unorderedpair formed by both points is invariant under similarities. The midpoint...
{\displaystyle f} maps all unorderedpairs of elements drawn from that subset to zero, or it maps all such unorderedpairs to one. An equivalent formulation...
has a rank 3 action on fifty-five points from an induced action on unorderedpairs, as well as two five-dimensional faithful complex irreducible representations...
science, an adjacency list is a collection of unordered lists used to represent a finite graph. Each unordered list within an adjacency list describes the...
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