Question

Let A and B be sets.if A intersection X=B intersection X=fi and A union X=B union X for some set Z,show that A=B (Hints A=A intersection (A union X), B=B intersection (B union X) and use distributive law)​

Answer 1

Answer:

ANSWER

Let A and B be two sets such that A∩X=B∩X=ϕ and A∪X=B∪X for some set X.

To show: A=B

It can be seen that

A=A∩(A∪X)

=A∩(B∪X) (A∪X=B∪X)

=(A∩B)∪(A∩X) (Distributive law)

=(A∩B)∪ϕ (∵A∩X=ϕ)

=A∩B ......(1)

Now, B=B∩(B∪X)

=B∩(A∪X) (∵A∪X=B∪X)

=(B∩A)∪(B∩X) (Distributive law)

=(B∩A)∪ϕ (∵B∩X=ϕ)

=B∩A

=A∩B ......(2)

Hence, from (1) and (2), we get

A=B.