Definition:Empty Set

Definition
The empty set is a set which has no elements.

It is usually denoted by some variant of a zero with a line through it, for example $$\varnothing$$ or $$\empty$$, and can always be represented as $$\left\{{}\right\}$$.

It is justifiable to refer to it as the empty set because there is only one empty set. That is, any two empty sets are necessarily equal, and therefore the same set.

See Empty Set Unique for a proof of this.

Axiomatic Set Theory
The concept of the empty set is axiomatised in the Axiom of Existence in Zermelo-Fraenkel set theory.

Alternative Terms
This is sometimes called the null set, but this name is discouraged because there is another concept for null set which ought not to be confused with this.

Some sources, for example, call this the vacuous set.

Notes on Symbology
The symbols $$\varnothing$$ and $$\empty$$ are properly considered as stylings of $$0$$ (zero), and not variants of the Greek "Phi": $$\Phi, \phi, \varphi$$.

Some sources maintain that it is a variant on the Norwegian / Danish / Faeroese letter Ø.

Some sources use $$\Box$$ as the symbol for the empty set, but this is rare.