Binary set

Binary set

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

* The set {"a","b"} is binary.
* The set {"a","a"} is not binary, since it is the same set as {"a"}, and is thus 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 the axiom of empty set it is known that the set emptyset = {} exists. From the axiom of pairing it is then known that the set {emptyset,emptyset} = {emptyset} exists, and thus the set {{emptyset},emptyset} exists. This latter set has two elements.

ee also

* ordered pair
* binary relation

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