empty set

An empty set is a set $\mathrm{\varnothing }$ that contains no elements. The Zermelo-Fraenkel Axioms of set theory imply that there exists an empty set. One constructs an empty set by starting with any set $X$ and then applying the axiom of separation to form the empty set $\mathrm{\varnothing }:=\{x\in X\mid x\ne x\}$.

An empty set is a subset of every other set, and any two empty sets are equal. Alternative notations for the empty set include $\{\}$ and $\mathrm{\varnothing }$.