Binary set

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 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