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Question

Let A and B be two sets show that the set A×B and B×A an element in common IFF the sets A and B have an element in common.

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Solution

Proof by a simple examples:

Case 1:

Let A = {1,2} & B = {3,4,5}

A X B = {(1,3), (1,4),(1,5),(2,3),(2,4),(2,5)}

B X A ={(3,1),(3,2),(4,1),(4,2),(5,1),(5,2)}

Here, there is no common element in neither set A and set B nor in set A×B and B×A.

Case 2:

Let A = {1,2} & B = {2,3}

A X B = {(1,2), (2,3),(1,3),(2,2)}

B X A = {(2,1), (2,2),(3,1),(3,2)}

As we can see that (2,2) is common in both.

Hence, the set A×B and B×A an element in common If and only if, the sets A and B have an element in common


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