Let A and B be two sets.


Let A and B be two sets. Show that the sets A × B and B × A have elements in common iff the sets A and B have an elements in common.


Case (i): Let:

A = (a, b, c)

B = (e, f)

Now, we have:

A × B = {(a, e}), (a, f), (b,e), (b, f), (c, e), (c, f)}

B × A  = {(e, a), (e, b), (e, c), (f, a), (f, b), (f, c)}

Thus, they have no elements in common.

Case (ii): Let:

A = (a, b, c)

B = (a, f)

Thus, we have:

A × B = {(a, a}), (a,f), (b, a), (b, f), (c,a), (c, f)}

B × A = {(a, a), (a, b), (a, c), (f, a), (f, b), (f, c)}

Here, A × and B × A have two elements in common.

Thus, A × B and B × A will have elements in common iff  sets A and B have elements in common.

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