Let `U` be the universal set and `A` a subset of `U`. Then the complement of `A` is the set of all elements of `U` which are not the elements of `A`. Symbolically, we write `A′` to denote the complement of `A` with respect to `U.` Thus, `A′ = {x : x ∈ U and x ∉ A }.` Obviously `A′ = U - A` We note that the complement of a set `A` can be looked upon, alternatively, as the difference between a universal set `U` and the set `A`.

`text(Note:)` If `A` is a subset of the universal set `U,` then its complement `A′` is also a subset of `U.` It is clear from the definition of the complement that for any subset of the universal set `U,` we have `( A′ )′ = A` The complement `A′` of a set `A` can be represented by a Venn diagram as shown in Fig.

`text(Some Properties of Complement Sets :)`

`1.` Complement laws: `(i) A ∪ A′ = U \ \ \ \ \ \ \ \ \ \ (ii) A ∩ A′ = phi`

`2.` De Morgan's law: `(i) (A ∪ B)' = A′ ∩ B′ \ \ \ \ \ \ \ \ \ \ \ \ (ii) (A ∩ B )′ = A′ ∪ B′`

`3.` Law of double complementation :` (A′ )′ = A`

`4.` Laws of empty set and universal set `phi′ = U` and `U′ = phi.`

`5.` `U' = phi` and `phi' = U .`