# Thread: How do I know if it is countable?

1. ## How do I know if it is countable?

Hi there, I have to make some assignments(7), but I don't really get them. I have to 1) tell if a set is countable 2) tell if a set has the same cardinality as R (= all the real numbers).
But I really don't get how to do this!
For example:
The intersection of Q and [3,4). How do I proof that this is countable (or not), and what about the cardinality?

Can someone help me? Explain how to do this? I have to make 7 assignments exactly like this.
Tnx,
Mary

2. 1) A set is countable if it is either: 1) finite or 2) infinite and there exists a bijection from the set to the natural numbers.

2) A set has the same cardinality as $\mathbb{R}$ if there exists a bijection from the set to $\mathbb{R}$.

In your example, you should know that $\mathbb{Q}$ is countable. and also $\mathbb{Q} \cap [3,4) \subseteq \mathbb{Q}$, thus it is countable and does not have the same cardinality as $\mathbb{R}$.

Basically, you have to 'see' if it is possible to make a mapping from the set to $\mathbb{R}$ or $\mathbb{N}$..

3. Originally Posted by Defunkt
Basically, you have to 'see' if it is possible to make a mapping from the set to $\mathbb{R}$ or $\mathbb{N}$..
Okay I got this.
I also have to proof both 1) and 2).
I still have troubles with it...
If I take a look at the second: [2,3] x [2,3] I have no idea what to do!
1) is it countable?
2) is this true: | [2,3] x [2,3] | = |R|?

By the way, I have a question: Can I also delete a reply or question? I'm sorry thought I was editing my old reply!

5. Originally Posted by MaryB
Okay I got this.
I also have to proof both 1) and 2).
I still have troubles with it...
If I take a look at the second: [2,3] x [2,3] I have no idea what to do!
1) is it countable?
2) is this true: | [2,3] x [2,3] | = |R|?
1) It is not countable. A way to prove this would be to assume by contradiction that it is -- that is, there exists a bijection $f: \mathbb{N} \rightarrow [2,3]\times [2,3]$. Then look at f(n) and f(n+1) for some $n \in \mathbb{N}$ and conclude that it cannot happen.