Show that A x B does not equal B x A, when A and B are nonempty unless A=B.

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Mathematics

25 December, 07:22

Show that A x B does not equal B x A, when A and B are nonempty unless A=B.

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Walter Osborne · Answer 1

We will assume A*B = B*A and show that A and B are necessarily the same.

Assume A*B = B*A. Let a ∈ A and b ∈ B. Then (a, b) ∈ A*B. Since A*B = B*A, we have (a, b) ∈ B*A. That is, a ∈ B and b ∈ A. Therefore, a ∈ A implies a ∈ B and b ∈ B implies b ∈ A. So A = B, as we had set out to do.