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Given a set X and a subset A, the complement X \ A of A in X is "the rest of X":

X \ A = {x in X but not in A}.


R \ (0,1] = (-∞, 0] U (1, ∞).

In other words, a subset and its complement form a partition of the ambient set:

A U (X \ A) = X, and
A ∩ (X \ A) = ∅.

In topology, complements of open sets are closed (see Open and closed sets) and vice versa.

Compare to orthogonal complement.