Let A and B be sets. If A ∩ X = B ∩ X = f and A ∪ X = B ∪ X for some set X, show that A = B.

(Hints A = A ∩ (A ∪ X) , B = B ∩ (B ∪ X) and use Distributive law)

It is given in the question that,

A and B are sets such that

And, for some set X

Now, we have to show that A = B

As it can be seen that,

= (By using distributive law)

=

= (i)

Now, we have:

=

= (By using distributive law)

=

=

(ii)

∴ From the equation (i) and (ii), we have

A = B