# Countables and Uncountable sets.

• Sep 8th 2008, 01:15 PM
ynn6871
Countables and Uncountable sets.
I have 2 problems that are due tomorrow that I can't figure out. I couldn't find any example of such problems anywhere in my book or the internet.
Anyway here they are:

1. Suppose that a and b are disctinct real numbers such that a<b.
a) Prove that the set {x e $\displaystyle R$: a<x<b} is uncountable.
b) Prove that the set {x e $\displaystyle Q$: a<x<b} is countably infinite

2. Use the fact that every real number has a unique decimal expansion that does not end in all 9's to prove that the interval (0,1) is an uncountable set.

Any help on these would be greatly appreciated.
Thanks.
• Sep 8th 2008, 01:25 PM
Matt Westwood
Cantor's diagonal argument - Wikipedia, the free encyclopedia

Then you can prove that there is a one-to-one correspondence between the set $\displaystyle \{x \in \mathbb{R}: a<x<b\}$ and $\displaystyle \{x \in \mathbb{R}: 0<x<1\}$ easily enough.

Proof that rational numbers are countable - from Homeschool Math

gives you the countability of the rationals.
• Sep 8th 2008, 01:59 PM
Plato
Quote:

Originally Posted by ynn6871
1. Suppose that a and b are disctinct real numbers such that a<b.
a) Prove that the set {x e $\displaystyle R$: a<x<b} is uncountable.
b) Prove that the set {x e $\displaystyle Q$: a<x<b} is countably infinite

For part a. Consider the mapping, $\displaystyle \alpha :(0,1) \to (a,b),\quad \alpha (x) = (b - a)x + a$.
Show that is a bijection. Then note that $\displaystyle (0,1)$ is uncountable.

For part b. The important part there is “countably infinite”.
Can you prove that $\displaystyle \bigcup\limits_{n = 2}^\infty {\left( {\frac{1}{{n + 1}},\frac{1}{n}} \right]} \cup \left( {\frac{1}{2},1} \right) = \left( {0,1} \right)$?
By the density of the rationals we have $\displaystyle \left( {\forall n} \right)\left( {\exists q_n \in \mathbb{Q}} \right)\left[ {\frac{1}{{n + 1}} < q_n < \frac{1}{n}} \right]$.
That proves that there are infinitely many rationals in $\displaystyle (0,1)$.
Now use part (a) to complete part (b).
• Sep 8th 2008, 04:29 PM
ynn6871
Thanks for your help but I think I am still somewhat confused! Don't we have to prove that (0,1) is uncountable, do we just accept it as a fact?? Can you show be an example of how to prove that something is a bijection???

For part b you got me lost!! How did you find these function 1/n etc... I am really confused.

What about the second problem any advise on how I could solve it???

Thanks again for your help!!
• Sep 8th 2008, 04:45 PM
ynn6871
ok, i think I was able to prove that it is a bijection but then what do we conclude from it??? Can we conclude that (0,1) is a subset of (a,b)???
• Sep 8th 2008, 10:40 PM
Matt Westwood
The link I posted originally was a proof that I found on the web (didn't have to look very far, it was on wikipedia) for the uncountability of the reals. It's one of the most famous proofs ever.

Please study this hard, as it's fundamental.