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  • Let A and B be sets. If A intersection  X = B intersection X = phi and A union  X = B union X for some set X, show that A = B.

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

Answers (1)

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

To prove:   A = B

A = A \cap(A\cupX)              (A \cap X =B \cap X)

    = A \cap(B\cupX) 

    = (A\capB) \cup (A\capX)

    =  (A\capB) \cup \phi            (A \cap X = \phi)

    =  (A\capB) 

B = B \cap(B\cupX)              (A \cap X =B \cap X)

    = B \cap(A\cupX) 

    = (B\capA) \cup (B\capX)

    =  (B\capA) \cup \phi            (B \cap X = \phi)

    =  (B\capA)

 We know that    (A\capB) =  (B\capA) = A = B

Hence, A = B

 

 

 

 

Posted by

Divya Prakash Singh

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