Empty Set
The empty set is the unique set that contains no elements, written as {} or ∅.
Formula
\emptyset = \{\}
Definition
The empty set is a set with nothing in it, zero elements, like an empty bag: the bag exists, but it holds nothing. Written $\emptyset = \{\}$, it has cardinality $0$, is a subset of every set ($\emptyset \subseteq A$ for all $A$, vacuously true since there are no elements to violate the condition), and is the identity element for union ($A \cup \emptyset = A$) while $A \cap \emptyset = \emptyset$. In ZFC, the empty set is guaranteed by the Axiom of the Empty Set (or derived from other axioms) and is unique by Extensionality; in the von Neumann ordinal construction, $0 = \emptyset$, $1 = \{\emptyset\}$, $2 = \{\emptyset, \{\emptyset\}\}$, and so on, meaning all natural numbers are ultimately built from the empty set.
Example
The set of all even prime numbers greater than $2$ is $\emptyset$ (there are none), as is the set of all months with $32$ days. $A \cap B = \emptyset$ means $A$ and $B$ are disjoint: for example, the odd numbers and even numbers share no elements. The power set $P(\emptyset) = \{\emptyset\}$ has exactly one element, and $P(\{\emptyset\}) = \{\emptyset, \{\emptyset\}\}$ has two, showing how iterated power sets build the Von Neumann universe.
Key Insight
The empty set is still a set: having no elements is a perfectly valid description, the mathematical equivalent of zero. Vacuous truth is why $\emptyset \subseteq A$ always holds: the statement "every element of $\emptyset$ is in $A$" is true because there are no elements of $\emptyset$ to check. The entire natural number system can be encoded in set theory using only the empty set and the power set operation, illustrating the expressive power of axiomatic set theory.