In set theory, the complement of a set A, often denoted by ${\displaystyle A^{c}}$ (or A′), is the set of elements not in A.

When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set U, the absolute complement of A is the set of elements in U that are not in A.

The relative complement of A with respect to a set B, also termed the set difference of B and A, written ${\displaystyle B\setminus A,}$ is the set of elements in B that are not in A.

If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the relative complement of A in U: $${\displaystyle A^{c}=U\setminus A=\{x\in U:x\notin A\}.}$$

The absolute complement of A is usually denoted by ${\displaystyle A^{c}}$. Other notations include ${\displaystyle {\overline {A}},A',}$ ${\displaystyle \complement _{U}A,{\text{ and }}\complement A.}$

Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements:

De Morgan's laws:

Complement laws: