Binary set
Encyclopedia
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. In contrast, an ordered pair
(a, b) has a as its first element and b as its second element.
While there is general agreement that the two elements of an ordered pair (a, b) need not be distinct, a few authors only call {a, b} an unordered pair if a ≠ b.
But for most authors a singleton is also an unordered pair.
A set with precisely 2 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}.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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. In contrast, an ordered pair
Ordered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...
(a, b) has a as its first element and b as its second element.
While there is general agreement that the two elements of an ordered pair (a, b) need not be distinct, a few authors only call {a, b} an unordered pair if a ≠ b.
But for most authors a singleton is also an unordered pair.
A set with precisely 2 elements is also called a 2-set
N-set
In mathematics, an n-set is a set containing exactly n elements, where n is a natural number. Thus, every finite set is an n-set for some specific natural number n. If S is any set, then a subset of S containing k elements is called a k-subset, or a k-combination...
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
Axiom of pairing
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory.- Formal statement :...
.
More generally, an unordered n-tuple is a set of the form {a1, ,a2,... an}.