# Complement of a Set

Let U be the universal set and A be a subset of U. We define the complement of A with respect to U to be the set {x : x âˆˆ U and x âˆ‰ A}. The complement of A is denoted by A'.

When the universal set is clearly understood, we usually omit the phrase "with respect to" and simply call A' as complement of A.

Illustration 1
Let U be the universal set of all the triangles in a plane. Let E denote the set of all the equilateral triangles in the plane, then E' = {x : x is not a equilateral triangle}.

Illustration 2
Let U be the universal set of the English alphabet and let V = {x : x is a vowel of the English alphabet}, then V' = {x : x is a consonant of the English alphabet}.
We now note some facts about the complement of a set.
1. De Morgan's Laws
(A âˆ© B)' = A' âˆª B'; (A âˆª B)' = A' âˆ© B';
2. Complement Laws
U' = Ï† and Ï† ' = U
That is, the complement of the universal set is the null set Ï† and vice versa.
3. Involution Law
(A')' = A
That is, the complement of the complement of a set is the set itself.