Definitions

# 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 $emptyset = \left\{\right\}$ exists. From the axiom of pairing it is then known that the set $\left\{emptyset,emptyset\right\} = \left\{emptyset\right\}$ exists, and thus the set $\left\{\left\{emptyset\right\},emptyset\right\}$ exists. This latter set has two elements.