I know and understand the basic theory behind what makes a set finite/infinite and countable/uncountable but I have no clue how to apply this to an actual question, my maths lectures are very theory heavy yet lack good examples, hopefully someone could help me with the below question to point me in the right direction, thanks in advance!
What I understand(from my lecture notes) is, if a set A is finite or the cardinal value of A is = the cardinal value of N than A is countable, also if $ B\leqslant\ A$ then B is also countable. I also know that sets N,Z and Q are countable while R is not,for context I'm a first year computer science student, more used to traditional "solve this" approach to maths, not much experience of using math theory to solve problems.
Q1) Decide whether the following set is a)countable or b)uncountable.
$\{x\in\ R_{}:\mid x\mid<5\}$
based on the knowledge I have I'm assuming that the above set is uncountable as its a subset of R, not sure how to properly proove this or whether my reasoning for my answer is correct.
We know that $\mathbb{R}$ is uncountable, hence so is $\mathbb{R^{>0}}$ since you can construct a $2$-to-$1$ function if you exclude the $0$. Consider the interval $(0,1)$. You can construct a bijection $f:(0,1)\rightarrow \mathbb{R^{>0}}$ \ $(0,1)$ by taking $f(x)=\frac{1}{x}$. Hence $(0,1)$ is uncountable - otherwise $\mathbb{R^{>0}}$ \ $(0,1)$ would be countable and the $\mathbb{R^{>0}}$ would too be, being the union of $2$ contable sets.
Since $(0,1)$ is uncoutable, then so is your set since it contains it.