# Interval Countability proof

• May 12th 2011, 03:50 PM
jstarks44444
Interval Countability proof
Every interval [x,y] is uncountable.

How to prove this? Thank you!
• May 12th 2011, 04:16 PM
Plato
Quote:

Originally Posted by jstarks44444
Every interval [x,y] is uncountable.
How to prove this?

Without knowing what theorems go before this question, there is no way to give you an answer. For example: if \$\displaystyle a<b\$ then the interval \$\displaystyle [a,b]\$ is uncountable.
The rational numbers are countable.
So what does mean about \$\displaystyle [a,b]\setminus \mathbb{Q}~?\$
• May 12th 2011, 04:29 PM
jstarks44444
Ah, you're right. The theorems that come before this include,

R (Reals) is not countable.
The set R - Q (Rationals) of irrational numbers is uncountable.
The set of transcendental numbers is uncountable.

The proof of the reals not being countable shows that the set of decimals

{0.d1d2d3...: each dj = 3 or 4} which is a subset of R

is uncountable. Consequently, the interval [0,1] = {x in R : 0 <= x <= 1} is uncountable. This can apparently be modified to prove the theorem in question. Thanks!