Question

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. 

Answer 1

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 × B 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.