# Complement of a Set

The complement of a set A (also called “absolute complement” of A) is the set of all those elements of the universal set S which are not elements of A. It is denoted by Ac or A′. Clearly Ac or A′ = S - A.

Symbolically, Ac or A′ = {x : x S and x A}. Thus x Ac x A.

Complement of a set can be represented by Venn diagram as shown in Figure. The shaded region represents A′.

# Important properties of complement

1. A A′ = φ
2. A A′ = U
3. U′ = φ
4. (A′)′ = A
5. A B B A
6. A - B = B′ - A
7. A - B = A B
8. B - A = A B

# De Morgan’s laws

1. (A B)′ = A B
2. (A B)′ = A B
3. A - (B C) = (A - B) (A - C)
4. A - (B C) = (A - B) (A - C)