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 \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 . \,
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