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 Sep7 comment What is the explanation for the elements of this set? by the way I hope that $f(A)$ is the correct notation for the range of the function. Sep7 comment 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? Sep7 comment What is the explanation for the elements of this set? Sorry, those weren't actually my solutions, I've updated the question to hopefully make that more clear... Sep7 revised What is the explanation for the elements of this set? added 103 characters in body Sep7 asked What is the explanation for the elements of this set? Aug21 awarded Electorate Aug20 comment 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? Aug20 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? thank you rbojohn Aug20 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? thank you for this Aug20 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? thank you very much Aug20 accepted What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? Aug19 revised What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? added 51 characters in body Aug19 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? @Bill: for context, this is me trying to go through the book Understanding Analysis by Stephen Abbott. Unfortunately I do not know any number theory or abstract algebra... Aug19 comment What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? @kahen and John, thank you for your comments! Aug19 asked What are the explanations for certain steps in these proofs for the irrationality/rationality of certain numbers? Jul6 comment For which $k \in \mathbb{R}$ will this be a euclidean vector space? thank you for the explanation! Jul6 accepted For which $k \in \mathbb{R}$ will this be a euclidean vector space? Jul5 asked For which $k \in \mathbb{R}$ will this be a euclidean vector space? Jul4 comment Help understanding this example of a linear operator which rotates each vector $v$ about the z-axis by an angle $\theta$ [at]Jason: thank you! Jul4 accepted Help understanding this example of a linear operator which rotates each vector $v$ about the z-axis by an angle $\theta$