Complement (set theory)

In set theory, the complement of a subset of a given set is the "remainder" of the larger set.

Formally, if A is a subset of X then the (relative) complement of A in X is

$X\setminus A=\{x\in X:x\not \in A\}.\,$

In some version of set theory it is common to postulate a "universal set" ${\mathcal {U}}$ and restrict attention only to sets which are contained in this universe. We may then define the (absolute) complement

${\bar {A}}={\mathcal {U}}\setminus A.\,$

The relation of complementation to the other set-theoretic functions is given by De Morgan's laws:

${\overline {A\cap B}}={\bar {A}}\cup {\bar {B}};\,$

${\overline {A\cup B}}={\bar {A}}\cap {\bar {B}}.\,$