Question

Using set theory laws prove that (A ∩ B) ∪ (A ∩ B ' ) = A

Answer 1

For simplicity of the expression, I am representing ∩ as (.) and U as (+)

using various laws of set theory we can proceed as follows :

LHS

(A ∩ B) ∪ (A ∩ B ' )

= A.B + A.B'

taking double complement the eqn becomes

= [A.B + A.B']''

= [(A.B)' . (A.B')']'                      (using de-morgans law)

= [( A' + B') . (A' + B)]'                (using de-morgans law)

= [A'.(A' + B) + B'. (A' + B)]'             (using distributive law)

= [A'.A' + A'.B + B'.A' + B'.B]'

= [ A' + A'.( B + B') + 0]'

= [ A' + A'.(1)]'

= [ A' + A']'

= A''

= A

= RHS

Hence

(A ∩ B) ∪ (A ∩ B ' ) = A       (proved)