# 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 (AX)              (A  X B  X)

= A (BX)

= (AB)  (AX)

=  (AB)              (A  X  )

=  (AB)

B = B (BX)              (A  X B  X)

= B (AX)

= (BA)  (BX)

=  (BA)              (B  X  )

=  (BA)

We know that    (AB) =  (BA) = A = B

Hence, A = B

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