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

${\displaystyle X\setminus A=\{x\in X:x\in ̸A\}.}$

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

${\displaystyle \overline{A}={𝒰}\setminus A.}$

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

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

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