Complement (set theory)

related topics
{math, number, function}
{math, energy, light}
{law, state, case}

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.

Contents

Relative complement

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).

Formally

Examples:

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

Absolute complement

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 \complement_U A or \complement A if U is fixed, that is:

Full article ▸

related documents
Logical disjunction
Divisor
Binary function
Lagrange's theorem (group theory)
Simple LR parser
Lipschitz continuity
Greedy algorithm
Chomsky normal form
Ordered field
Metrization theorem
Graded algebra
Uncountable set
Amicable number
Nash embedding theorem
Kernel (category theory)
Goldbach's weak conjecture
Topological ring
De Moivre's formula
Regular language
Twin prime conjecture
Parity (mathematics)
Identity element
Separated sets
Logarithmic integral function
Magma (algebra)
Euphoria (programming language)
CLU (programming language)
Box-Muller transform
Heaviside step function
PILOT