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

X∖A={x∈X:x∉A}.

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

A¯=𝒰∖A.

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

A∩B‾=A¯∪B¯;
A∪B‾=A¯∩B¯.