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Absolute complement


In set theory, the complement of a set A refers to elements not in A. The relative complement of A with respect to a set B, also termed the difference of sets A and B, written BA, is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A.

If A and B are sets, then the relative complement of A in B, also termed the set-theoretic difference of B and A, is the set of elements in B but not in A.

The relative complement of A in B is denoted BA according to the ISO 31-11 standard. It is sometimes written BA, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all elements ba, where b is taken from B and a from A.

Formally:

Let A, B, and C be three sets. The following identities capture notable properties of relative complements:

If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A. In other words, if U is the universe that contains all the elements under study, and there is no need to mention it because it is obvious and unique, then the absolute complement of A is the relative complement of A in U:


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