Set of irrationals between two reals is uncountable

Question

I know that between any two reals, there is an irrational number.

See: Proving that there exists an irrational number in between any given real numbers

Now let a, b $\in$ $R$ such that a < b. And let M be the set of irrationals between a and b.

I want to show that M is uncountable. To do this, I think I need to show that there does not exist a bijection from M to the natural numbers.

1. Can someone give me a hint about how to start to show this?

2. What's the best way to approach "non-existence" proofs in general?

Thanks

Answer 1

I assume that you know that the interval $(a, b)$ is uncountable and that the set of rational numbers in that interval is countable. So the problem reduces to the following:

If $Y \subseteq X$ are sets such that $X$ is uncountable and $Y$ is countable, then $Z = X \mathrel{\backslash} Y$ is uncountable.

To see this, note that $Z$ is either (a) finite, (b) countably infinite or (c) uncountable. But the union of a finite set (or a countably infinite set) and a countable set is countable so (a) and (b) can't hold. So (c) holds, which was what we wanted.

Answer 2

Instead you could show that there is a bijection between this set and interval (0,1)

Answer 3

First, show that there is a rational number between any two reals.

This can be done using the Archimedean axiom (or theorem depending on your axioms) that, for any two positive reals $x$ and $y$, there is an integer $n$ such that $nx > y$.

Then, using the same reasoning, show that there is another rational between the two reals.

Let these rationals be $r$ and $s$.

Finally, construct a linear mapping from $[0, 1]$ to $[r, s]$. This mapping maps rationals to rationals and irrationals to irrationals.

Since there an uncountable number of irrationals in $[0, 1]$, there are an uncountable number of irrationals in $[r, s]$.

Another proof, this time by contradiction, can be modeled on Cantor's original proof that the reals are uncountable, but I'll leave that as an exercise.