Question

Show that for any sets A and B,

A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)​

Answer 1

Answer:

(¡) Let A and B be two sets

Now,

A

=A∩U

=A∩(BUB')

=(A∩B) U(A∩B')

=(A∩B) U(A-B) [since P∩Q'=P-Q]

(¡¡)

AU(B-A)

=AU(B∩A') [ since P∩Q'=P-Q]

=(AUB)∩(AUA')

=(AUB)∩ U [since PUP'=U]

=(AUB)