# Q. 11 Let A and B be sets. If A X B X  and A X B X for some set X, show that A B.

Given,  A X B X  and  A X B X

To prove:   A = B

A = A (A X)              (A X B X)

= A (B X)

= (A B) (A X)

=  (A B)  (A X  )

=  (A B)

B = B (B X)              (A X B X)

= B (A X)

= (B A) (B X)

=  (B A)  (B X  )

=  (B A)

We know that    (A B) =  (B A) = A = B

Hence, A = B

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