Empty Set
Calculus & Advanced MathThe 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. It is like an empty bag: the bag exists, but it holds nothing.
Example
The set of all even prime numbers greater than $2$ = $\emptyset$ (there are none). The set of all months with $32$ days = $\emptyset$.
Key Insight
The empty set is still a set! Having no elements is a perfectly valid description. It is the mathematical equivalent of zero.
Definition
The empty set $\emptyset = \{\}$ has cardinality $0$. It is a subset of every set: $\emptyset \subseteq A$ for all $A$ (vacuously true, since there are no elements to violate the condition). It is the identity element for union: $A \cup \emptyset = A$. $A \cap \emptyset = \emptyset$.
Example
$A \cap B = \emptyset$ means $A$ and $B$ are disjoint (share no elements). Example: $A = \{\text{odd numbers}\}$, $B = \{\text{even numbers}\}$ are disjoint: $A \cap B = \emptyset$.
Key Insight
Vacuous truth is why $\emptyset \subseteq A$: the statement "every element of $\emptyset$ is in $A$" is true because there are no elements of $\emptyset$ to check.
Definition
In ZFC, the empty set is guaranteed by the Axiom of the Empty Set (or derived from other axioms). It is unique by Extensionality. In the von Neumann ordinal construction: $0 = \emptyset$, $1 = \{\emptyset\}$, $2 = \{\emptyset, \{\emptyset\}\}$, etc. Thus all natural numbers are ultimately built from the empty set.
Example
The power set $P(\emptyset) = \{\emptyset\}$ has exactly one element. $P(\{\emptyset\}) = \{\emptyset, \{\emptyset\}\}$ has two elements. These iterated power sets build the Von Neumann universe.
Key Insight
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.