In set theory, a complement of a set A refers to things not in (that is, things outside of), A. The relative complement of A with respect to a set B, 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 all elements in U but not in A.
If A and B are sets, then the relative complement of A in B, also known as 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 B ∖ A according to the ISO 31-11 standard (sometimes written B − A, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all b − a, where b is taken from B and a from A).
The following proposition lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection.
PROPOSITION 2: If A, B, and C are sets, then the following identities hold:
- C ∖ (A ∩ B) = (C ∖ A)∪(C ∖ B)
- C ∖ (A ∪ B) = (C ∖ A)∩(C ∖ B)
- C ∖ (B ∖ A) = (A ∩ C)∪(C ∖ B)
- (B ∖ A) ∩ C = (B ∩ C) ∖ A = B∩(C ∖ A)
- (B ∖ A) ∪ C = (B ∪ C) ∖ (A ∖ C)
- A ∖ A = Ø
- Ø ∖ A = Ø
- A ∖ Ø = A
If a universe U is defined, then the relative complement of A in U is called the absolute complement (or simply complement) of A, and is denoted by Ac or sometimes A′, also the same set often is denoted by or if U is fixed, that is:
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