I found one problem which asks following:
Show that the set of odd positive integers greater then 3 is countable.
At the begining I was thinking that such numbers could be represented by $2k+1$,where $k>1$; but in the answers paper there was written as $2n+3$ or general function is $f(n)=2n+3$ and when I was thinking how to prove that such answer is countable, the answer paper said this function is a one-to-one correspondence from the set of positive numbers set to the set of positive odd integers greater to 3.
My question is: is it enough to prove a one-to-one correspondence between two sets, that one of them is countable. If yes, then once my lecturer ask me to proof that rational numbers are countable, so in this case if I represent rational numbers by following function from set of positive numbers: $f(n)=\frac{n+1}{n}$ or maybe $f(n)=\frac{n}{n+1}$. They both are one-to-one correspondences from the set of positive numbers to the set of rational numbers (positives sure). Please help me, is my logic correct or not?