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 . Number of Elements in a set If A , B and C are finite sets , then o n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )
Explanation for n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B ): Since the common elements A ∩ B is counted twice with both n ( A ) & n ( B ) , we subtract it .