, , , , } Hence (A ′ ) ′ = { x : x ∈ U and x ∉ A ′ } = { , , , , } = A It is clear from the definition of the complement that for any subset of the universal set U, we have ( A ′ ) ′ = A Now, we want to find the results for ( A ∪ B ) ′ and A ′ ∩ B ′ in the followng example. Example Let U = { , , , , , }, A = { , } and B = { , , }. Find A ′ , B ′ , A ′ ∩ B ′ , A ∪ B and hence show that ( A ∪ B ) ′ = A ′ ∩ B ′ . Solution Clearly A ′ = { , , , }, B ′ = { , , }.
Hence A ′ ∩ B ′ = { , } Also A ∪ B = { , , , }, so that (A ∪ B ) ′ = { , } ( A ∪ B ) ′ = { , } = A ′ ∩ B ′ It can be shown that the above result is true in general. If A and B are any two subsets of the universal set U, then ( A ∪ B ) ′ = A ′ ∩ B ′ . Similarly, ( A ∩ B ) ′ = A ′ ∪ B ′ . These two results are stated in words as follows : MATHEMATICS The complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements.
These are called De Morgan’s laws . These are named after the mathematician De Morgan. The complement A ′ of a set A can be represented by a Venn diagram as shown in Fig . .
The shaded portion represents the complement of the set A. Some Properties of Complement Sets .