A subset of a set S is a set A whose elements all belong to S. Written $A \subseteq S$: for every x, if $x \in A$ then $x \in S$. Every set is a subset of itself. The empty set is a subset of every set.

Subset is not the same as element. The number 3 is an element of {1, 2, 3}. The set {3} is a subset of {1, 2, 3}. An element belongs to a set; a subset is contained within a set. These are different relations — $3 \in \{1, 2, 3\}$ but $3 \not\subseteq \{1, 2, 3\}$, and $\{3\} \subseteq \{1, 2, 3\}$ but $\{3\} \notin \{1, 2, 3\}$ (unless 3 itself happens to be the set {3}).

The collection of all subsets of S is the power set $\mathcal{P}(S)$. If S has n elements, $\mathcal{P}(S)$ has $2^n$ elements. The power set is ordered by inclusion: $A \subseteq B$ defines a partial order on $\mathcal{P}(S)$. Under this ordering, intersection is the meet (greatest lower bound) and union is the join (least upper bound). Every subset has a complement $S \setminus A$, making $\mathcal{P}(S)$ a Boolean algebra.

A binary relation from A to B is defined as a subset of $A \times B$. So subset is the concept that connects sets to relations: a relation IS a subset of a product. This makes subset one of the load-bearing concepts in the chain from sets to categories — without it, you cannot define relations, and without relations, you cannot define functions.

In a category, subsets generalize to subobjects: a subobject of an object A is an equivalence class of monomorphisms into A. Each subset of S corresponds to a characteristic function $S \to \{0, 1\}$ that says yes or no for each element. In categories beyond Set, this characteristic function targets a richer object — the subobject classifier — which may have more values than just true and false.