# Q. 11 Let A and B be sets. If A $\cap$ X $=$B $\cap$ X $=$ $\phi$and A $\cup$ X $=$ B $\cup$ X for some set X, show that A $=$B.

Given,  A $\cap$ X $=$B $\cap$ X $=$ $\phi$   and  A $\cup$ X $=$ B $\cup$ X

To prove:   A = B

A = A $\cap$(A$\cup$X)              (A $\cap$ X $=$B $\cap$ X)

= A $\cap$(B$\cup$X)

= (A$\cap$B) $\cup$ (A$\cap$X)

=  (A$\cap$B) $\cup$ $\phi$            (A $\cap$ X $=$ $\phi$)

=  (A$\cap$B)

B = B $\cap$(B$\cup$X)              (A $\cap$ X $=$B $\cap$ X)

= B $\cap$(A$\cup$X)

= (B$\cap$A) $\cup$ (B$\cap$X)

=  (B$\cap$A) $\cup$ $\phi$            (B $\cap$ X $=$ $\phi$)

=  (B$\cap$A)

We know that    (A$\cap$B) =  (B$\cap$A) = A = B

Hence, A = B

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