If A and B are two non-empty sets such that AxB = BxA, show that A=B?

Question

Mathematics

Guest

21 March, 11:50

If A and B are two non-empty sets such that AxB = BxA, show that A=B?

+4

Answers (1)

Know the Answer?

Matthew Glover · Answer 1

To prove that A=B, you need to show that A is a subset of B and that B is a subset of A. There is no need to assume that A≠B. We won't be proving this by contradiction.

First, let a1∈A. Then for any b1∈B, since A X B ⊆ B X A, there exists a2∈A and b2∈B such that a1=b2 and b1=a2. Since b2∈B and a1=b2, we have that a1∈B. Since a1 was arbitrarily chosen, we have that A ⊆B.

Second, let b1∈B. Then for any a1∈A, since A X B ⊆ B X A, there exists a2∈A and b2∈B such that b1=a2 and a1=b2. Since a2∈A and b1=a2, we have that b1∈A. Since b1 was arbitrarily chosen, we have that B ⊆A.

Thus we have shown that A=B.