Empty set

The empty set, denoted $\emptyset$, is the set with no elements:

In ZFC , existence of $\emptyset$ is ensured by an axiom; uniqueness follows from extensionality (a set is determined by its elements). The empty set is a subset of every set.

Examples:

• The solution set to $x^2 + 1 = 0$ in $\mathbb{R}$ is $\emptyset$.
• For any set $A$, $A \cap \emptyset = \emptyset$.
• For any set $A$, $A \cup \emptyset = A$.