In mathematics, the empty set is the unique set having no elements, or a cardinality of zero. It is unique because in set theory, two sets are equal if they have the same elements. As a result there can only be one set with no elements – *the* (not *an*) empty set.

The empty set is usually denoted {} or $\emptyset$.

**Properties**

- The empty set is a subset of all sets.

$\forall A : \emptyset \subset A$ - The union of a set with the empty set is the set itself.

$\forall A : A \cup \emptyset = A$ - The intersection of any set with the empty set is the empty set.

$\forall A : A \cap \emptyset = \emptyset$ - The Cartesian product of any set and the empty set is the empty set.

$\forall A : A \times \emptyset = \emptyset$