empty set

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