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.


closed as off-topic by JMP, GNUSupporter 8964民主女神 地下教會, Arnaud D., Paramanand Singh, Servaes Mar 14 '18 at 15:41

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  • $\begingroup$ Even if you don't know how to prove that rigorously, at least tell us what you think about this question. Do you think that this set is countable or uncountable? And can you explain (informally, in your own words) why you think so? $\endgroup$ – zipirovich Mar 14 '18 at 15:32
  • $\begingroup$ @zipirovich for future reference, is my edited answer acceptable? $\endgroup$ – Cian Mc Sweeney Mar 15 '18 at 9:22
  • $\begingroup$ Yes, this is much better! Now you did demonstrate that you actually have some understanding and intuition about these concepts. Seeing some understanding and some effort from you makes this community much more willing to help you understand these things better. Back to your work, your intuition seems to be in the right place, but not your wording. Being a subset or $\mathbb{R}$ is not good enough -- sets like $\{1\}$ or $\mathbb{N}$ are also subsets of $\mathbb{R}$, but they are not uncountable. What you meant to say was that this is an interval in $\mathbb{R}$, and intervals are uncountable. $\endgroup$ – zipirovich Mar 15 '18 at 19:07

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.


Correct me if wrong:

Consider: $f:(-5,5) \rightarrow \mathbb{R},$

$f(x)= \dfrac{x}{5-|x|} $, continuos, bijection.


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