Unordered pair

A binary set is a set with (exactly) two distinct elements, or, equivalently, a set whose cardinality is two.

Examples:

• The set {a,b} is binary.
• The set {a,a} is not binary, since it is equivalent to the set {a}, which is a singleton.

In axiomatic set theory, the existence of binary sets is a consequence of the axiom of empty set and the axiom of pairing. From these two axioms can be derived the existence of the singleton {{}}. From the axiom of empty set it is known that the set {} exists. From the axiom of pairing it is then known that the set {{},{{}}} exists, which contains both {} and {{}}. But the set {{},{{}}} is a binary set.

See also: ordered pair, binary relation.