Complement (set theory)

In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement.

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 usually written B − A (also B \ A).

Formally:

<math>B – A = \{ x\in B \, | \, x \not \in A \}. <math>

Examples:

• {1,2,3} − {2,3,4}   =   {1}
• {2,3,4} − {1,2,3}   =   {4}
• If <math>\mathbb{R}<math> is the set of real numbers and <math>\mathbb{Q}<math> is the set of rational numbers, then <math> \mathbb{R}-\mathbb{Q}<math> is the set of irrational numbers.

The following proposition lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection.

PROPOSITION 1: 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 universal set 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 A^C, that is:

A^C  = U − A

For example, if the universal set is the set of natural numbers, then the complement of the set of odd numbers is the set of even numbers.

The following proposition lists some important properties of absolute complements in relation to the set-theoretic operations of union and intersection.

PROPOSITION 2: If A and B are subsets of a universal set U, then the following identities hold:

De Morgan's laws:

• (A ∪ B)^C  = A^C ∩ B^C
• (A ∩ B)^C  = A^C ∪ B^C

Complement laws:

• A ∪ A^C  =  U
• A ∩ A^C  =  Ø
• Ø^C  =  U
• U^C  =  Ø

Involution or double complement law:

• A^CC  =  A.

Relationship between relative and absolute complements:

• A − B = A ∩ B^C

The above shows that if A is a non-empty subset of U, then {A, A^C} is a partition of U.

See also

• Algebra of sets
• Naive set theory
• Symmetric difference