No. Irrational Numbers form a proper subset of real numbers. That means that all irrationals are real so non-reals cannot be irrational.
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All rational and irrational numbers are real numbers.
All irrational numbers are not rational.
All irrational numbers are real, but not all real numbers are irrational.
No. All irrational numbers are real, not all real numbers are irrational.
There are infiitelt many subsets of irrational numbers. One possible subset is the set of all positive irrational numbers.