# 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*A*or^{c}*A*′. Clearly*A*or^{c}*A*′ =*S*-*A*.Symbolically,

*A*or^{c}*A*′ = {*x*:*x*∈*S*and*x*∉*A*}. Thus*x*∉*A*^{c}^{ }⇔*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

*A*∪*A*′ =*φ**A*∪*A*′ =*U**U*′ =*φ*- (
*A*′)′ =*A* *A*⊆*B*⇔*B*′ ⊆*A*′*A*-*B*=*B*′ -*A*′*A*-*B*=*A*∪*B*′*B*-*A*=*A*′ ∪*B*

# De Morgan’s laws

- (
*A*∪*B*)′ =*A*′ ∪*B*′ - (
*A*∪*B*)′ =*A*′ ∪*B*′ *A*- (*B*∪*C*) = (*A*-*B*) ∪ (*A*-*C*)*A*- (*B*∪*C*) = (*A*-*B*) ∪ (*A*-*C*)