# 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 $R$: a<x<b} is uncountable.
b) Prove that the set {x e $Q$: 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 $\{x \in \mathbb{R}: a and $\{x \in \mathbb{R}: 0 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 $R$: a<x<b} is uncountable.
b) Prove that the set {x e $Q$: a<x<b} is countably infinite

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

For part b. The important part there is “countably infinite”.
Can you prove that $\bigcup\limits_{n = 2}^\infty {\left( {\frac{1}{{n + 1}},\frac{1}{n}} \right]} \cup \left( {\frac{1}{2},1} \right) = \left( {0,1} \right)$?
By the density of the rationals we have $\left( {\forall n} \right)\left( {\exists q_n \in \mathbb{Q}} \right)\left[ {\frac{1}{{n + 1}} < q_n < \frac{1}{n}} \right]$.
That proves that there are infinitely many rationals in $(0,1)$.
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???