# Prove equal powers (probably fairly primitive)

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• March 1st 2011, 06:06 AM
Pranas
Prove equal powers (probably fairly primitive)
Hello.

My discrete mathematics course has just started, so I'm not very good even at probably fairly easy things:

Prove if $\displaystyle \[G = A \times B\]$ is a bijection between finite sets $\displaystyle \[A\]$ and $\displaystyle \[B\]$ then $\displaystyle \[\left| A \right| = \left| B \right|\]$.

It looks pretty obvious because of the nature of this bijection, but I just can't seem to figure out notations to prove it in a mathematical way.

Thanks.
• March 1st 2011, 06:15 AM
emakarov
A bijection is a function, but if A is nonempty and |B| > 1, then A x B is not a function. If there is no mistake in the question, then the fact that A x B is a bijection implies that |A| = |B| = 1 or |A| = |B| = 0, so |A| = |B|.
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