I have a few questions regarding this problem below:
Prove that if A and B are finite sets, then A ≈ B if and only if |A| = |B|.
Would I assume that |A| = |B|? Which would obviously make A ≈ B and B ≈ A for all sets A and B. Then would I say that A = {1, 2 ,3} and since |A| = |B| doesn't that mean they have the same cardinality which would make A = B instead of A ≈ B. I'm confused in this aspect. Any help is appeciated!
Assuming $\approx$ means "there is a bijection between" a (sketch of a) proof will be as follows:
$(\Leftarrow)$ Assume $|A|=|B|$. Then, since they are finite sets, by definition there is an $n\in\omega$, and there are bijections $f:A\to n$ and $g:B\to n$ (precisely, $n$ is the cardinality of $A$ and $B$). Therefore $g^{-1}\circ f$ is a bijection from $A$ to $B$.
$(\Rightarrow)$ Assume $A\approx B$. Then there is a bijection $f:A\to B$. Since $A$ and $B$ are finite there are $n,m\in\omega$ and bijections $g:A\to n$ and $h:B\to m$. Now one can easily show that there is a bijection between $n$ and $m$. Hence $n=m$