# Prove that the set of irrational numbers in [0,1] is not countable

Question
Irrational numbers
Prove that the set of irrational numbers in [0,1] is not countable

2021-02-26
It is known that the set of rational numbers are countable.
Union of rational numbers and irrational numbers gives the real numbers.
So $$\displaystyle{\left[{0},{1}\right]}={Q}{\left[{0},{1}\right]}\cup{R}{Q}{\left[{0},{1}\right]}$$
Since Q[0,1] is countable and [0,1] is not countable.
So the only possibility is that $$\displaystyle{R}{Q}{\left[{0},{1}\right]}$$ is uncountable.
Hence, the set of irrational numbers in [0,1] is not countable.

### Relevant Questions

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