# Complement (set theory)

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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

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

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

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

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

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