I am trying to understand whether the set is finite, countable, or uncountable.
$$\{x \in\Bbb Q \mid 1<x<2 \} \qquad\text{is countable. }$$
but i dont understand why though. is it countable because there is finite numbers between 1 and 2? how could you count all the numbers between 1 and 2? why cant it be uncountable??
An extra problem i was givien: $$\left\{ \frac mn \mid m,n \in \Bbb N, m<100, 5<n<105\right\}$$ is finite. i think it is finite because $\Bbb N$ is greater than or equal to 1. N > 1. i would like to know if im wrong or right. I want to understand everything i do.
Thank you for efforts and time
Sincerely
Best Answer
Countable means you can put it in bijection with $\Bbb N$ (some people include finite sets as countable, but you seem to mean countably infinite). Have you seen the proof that all of $\Bbb Q$ or $\Bbb {N \times N}$ is countable? Your first is a subset of this, so is clearly countable. There are an infinite number of rationals between $1$ and $2$.
For the second, there are only $100$ choices for $m$ (or $99$ if you do not include $0$) and $99$ choices for $n$, and $100 \times 99$ is clearly finite.