0
$\begingroup$

Since there is a rational number between any two irrational numbers, so can we say that rational number and irrational number have the same cardinality?

$\endgroup$
1
$\begingroup$

No. $\mathbb Q$ is countable. Suppose that the irrational numbers have the same cardinality, then $ \mathbb R \setminus \mathbb Q$ is also countable. This gives

$ \mathbb R = \mathbb Q \cup (\mathbb R \setminus \mathbb Q)$ is countable, a contradiction.

$\endgroup$
  • $\begingroup$ I know your argument. But what I mean is intuitively if there is a rational number between any two irrational numbers, then there are as many rational numbers as irrational numbers. $\endgroup$ – Bella Mar 20 '17 at 10:26
-1
$\begingroup$

Of course NOT.

The argument is not correct. For example, I can say there always exists a rational number between any two distinct real numbers, but apparently $\mathbb{Q}$ and $\mathbb{R}$ don't have the same cardinality.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.