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- Properties of Complement Sets

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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.

Example- Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′.

Solution- A′ = {2, 4, 6, 8,10}

**De Morgan's Law: **

1)The complement of the union of two sets is the intersection of their complements.

(A ∪ B)′= A′ ∩ B′

2) The complement of the intersection of two sets is the union of their complements.

(A ∩ B)′= A′ ∪ B′

**Some Properties of Complement Sets**

1) Complement laws:

i) A ∪ A′ = U

ii) A ∩ A′ = Ø

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 Ø′= U and U′ = Ø.