What is the explanation for the elements of this set? thank you, that definitely helps a lot. I still have to ask: in order to "note that $B$ is not in the range of the function" we then need to consider the range of the function to be $\{ \{ a\}, \{ a,c\} \{ a,b,c\}\}$ as I wrote in the question? And that's why $B = \{b\} \notin f(A)$? Is that correct? In other words $B$ is a set, but the important observation is that it is not an element of the range of the function?
What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? @kahen: sorry, but I was having a little difficulty understanding the right to left direction of the statement you wrote. The way I was interpreting it, I would have thouhgt that if $p = 4$, $a=4$, $b=2$ so $ab = 8$, then $p$ divides $ab$ as well as $a$... But is part of the requirement that this is true for every $ab$? So if $p=4$ it divides $ab=12$ yet $4$ divides neither $a=6$ nor $b=2$ and therefore $p=4$ is not prime? Is that correct?
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