Since $\mathbb{Q}$ is countable and $\mathbb{R} \backslash \mathbb {Q}$ is not, what does this tell us about the density of rational and irrational numbers along the real number line? Saying that there exists more irrational numbers than rational numbers seems rather vague becuase we're comparing infinites. How do we even define density here?
[Math] Density of rational and irrational numbers
elementary-set-theoryreal numbers
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This is a tough concept for most people learning about the sizes of infinite sets for the first time. Number density, as you call it, is an intuitive concept, and it makes more sense up front. But there are a couple of problems with it. It seems like you are wanting to define size of a set $A$ by the limit as $n$ goes to infinity of the number of numbers in $A$ less than $n$ divided by $n$. This, again seems natural. But what if my set were not made of numbers? What if it were a set of polygons? Or lines? A set of functions? What if it was a set of sets? There is an even bigger problem. The digits we write down when enumerating a set are really just symbols$^\ast$. The set $\{1,2,3,4,...\}$ is just a collection of symbols. If I change the symbol "1" to "2" and change "2" to "4", "3" to "6", and so on, I get the set $\{2,4,6,...\}$. I changed the way that each symbol looks. Have I really changed the size of the set? Is there a universal way to define the size of a set? There is. There is no confusion about the size of finite sets. It is also easy to see that if a function from a finite set $A$ to another finite set $B$ is one-to-one, and hits everything in $B$, then $A$ and $B$ have the same size. We simply extend the idea to infinite sets. This avoids the problem of having to have a number system pre-defined on the sets. It avoids the problem of relabeling the elements of the sets. And most importantly, it acknowledges that the size of a set is whatever we define it to be. So we choose a definition that is useful. This is a useful definition. In other words, your question "what constitutes a proof" is ill-posed. We do not prove that two sets are the same size if there is a bijective function between them, we define it that way.
As for the set $[0,1]$, it is not hard to find a bijection from $(0,1)$ to $\mathbb R$, so the definition says they are the same size. There is another theorem that says if we add a finite number of elements to an infinite set, then we do not change its cardinality. Thus, $\mathbb R$ and $[0,1]$ have the same cardinality.
As for books, I would suggest Mathematical Proofs, by Gary Chartrand.
$^\ast$Thanks to Todd Wilcox for the revised wording here.
It is definitely not true that $\Bbb Q\subset\Bbb Z$. You will have to show that $\Bbb Q$ is countable to prove this result. One way you can do that is to define the sets $$ A_n = \{\:n/k : \text{$k$ is a nonzero integer}\:\},\qquad\text{where $n\in\Bbb Z_{\ge0}$,} $$ each of which is countable. Then $\bigcup_n A_n$ is the countable union of countable sets and is equal to $\Bbb Q$.
Best Answer
One can have any combination of (countably infinite, uncountable) and (dense in the line, not dense in the line). (Unless, as was suggested in the comments, you have some notion of density other than the one defined by Captain Falcon.)
Countably infinite and dense in the reals: Rationals
Countably infinite and not dense in the reals: Integers
Uncountable and dense in the reals: Irrationals
Uncountable and not dense in the reals: Unit interval