Question

19.
Let A and B be sets, if AX=BX= and AX=BX for some sets X,
Prove that A=B​

Answer 1

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∩(2)

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

A=B