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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?

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2 Answers 2

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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.

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  • $\begingroup$ 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
    Commented Mar 20, 2017 at 10:26
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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.

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