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# Complement

From Mathematics Is A Science

Jump to navigationJump to searchGiven 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}.

**Example.**

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.