# Answer to Question #12797 in Discrete Mathematics for okunade ademola

Question #12797

If A and B are two non-empty sets such that A x B = B x A, show that A=B

Expert's answer

Proof.

Notice that A x B consists of all pairs (a,b), where a belongs to A,

and b belongs to B, while

B x A consists of all pairs (b,a), where a

belongs to A, and b belongs to B.

Let a belongs to A.

We should prove

that a belongs to B as well.

Take any b from B.

Then

(a,b) belongs

to A x B = B x A,

whence

(a,b) = (b',a')

for some b' from B and a'

from A.

This means that

a=b' and b=a'.

Hence a belong to

B.

This proves that A is contained in B.

By similar argument B is contained

in A, and so A=B.

Notice that A x B consists of all pairs (a,b), where a belongs to A,

and b belongs to B, while

B x A consists of all pairs (b,a), where a

belongs to A, and b belongs to B.

Let a belongs to A.

We should prove

that a belongs to B as well.

Take any b from B.

Then

(a,b) belongs

to A x B = B x A,

whence

(a,b) = (b',a')

for some b' from B and a'

from A.

This means that

a=b' and b=a'.

Hence a belong to

B.

This proves that A is contained in B.

By similar argument B is contained

in A, and so A=B.

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