My question reads:
Which of the following sets are dense? (Take $p\in\mathbb{Z}$ and $q\in\mathbb{N}$)
($a$) the set of all rational numbers $\frac{p}{q}$ with $q\leq\ 10$
($b$) the set of all rational numbers $\frac{p}{q}$ with $q$ a power of 2.
I am not too sure how to go about determining if the set if dense or not. I do know that for a set to be dense there must be an element in this set that can be found between any two real numbers $a<b$.
For the first one I was looking at the fact that fractions would be a part of this set, but I am not sure if noticing this helps me. Then, for the second one, I think once I see a bit of how the first once works I will get it.
(a) is not dense, because for example in $(0,1)$ you have only a finite number of rational numbers with denominator less than $10$. So you can find an interval in the gaps, which does not contain any of the aforementioned rational numbers
(b) is dense, because you can make $1/q$ as small as you want. So if you take an interval $(a,b)$ you can find a $q=2^n$ such that $\frac{1}{q}<b-a$. So if you start in the origin and make steps of size $\frac{1}{q}$ towards the interval, you will eventually fall in the interval $(a,b)$. See also Proving that $\Bbb Q$ is dense in $\Bbb R$. The idea is the same.