Set

:This article is about sets in mathematics. For other senses, see set (disambiguation).

Complements

Two sets can also be "subtracted". The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B − A, (or B A) is the set of all elements which are members of B, but not members of A. Note that it is valid to "subtract" members of a set that are not in the set, such as removing green from {1,2,3}; doing so has no effect.

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In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U − A, is called the absolute complement or simply complement of A, and is denoted by A′.

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The relative complementof A in B

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The complement of A in U

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Examples:

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:*{1, 2} − {red, white} = {1, 2}

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:*{1, 2, green} − {red, white, green} = {1, 2}

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:*{1, 2} − {1, 2} = ø

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:*If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then the complement of E in U is O, or equivalently, E′ = O.

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Some basic properties of complements:

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:*A U A′ = U

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:*A ∩ A′ = ø

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:*(A′ )′ = A

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:*A − A = ø

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:*A − B = A ∩ B′

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For more information about complements of sets, see Complement (set theory).

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