Class 11 Maths Sets Complement of a Set

Complement of a Set

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.

E.g. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Then A′ = {2,4,6,8,10}

Properties of Complement of Set

  1. Complement laws: (i) A ∪ A′ = U (ii) A ∩ A′ = φ
  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 φ′ = U and U′ = φ.

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