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

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

Question
Irrational numbers
asked 2021-02-25
Prove that the set of irrational numbers in [0,1] is not countable

Answers (1)

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.
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