Complement set theory: Encyclopedia II - Complement set theory - Relative complement

Complement set theory - 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:

Examples:

• {1,2,3} − {2,3,4}   =   {1}
• {2,3,4} − {1,2,3}   =   {4}
• If is the set of real numbers and is the set of rational numbers, then 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

Adapted from the Wikipedia article "Relative complement", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki