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.