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## Homework Statement

Suppose [itex]\mathbb{Q},\mathbb{R}[/itex] are the set of all rational numbers and the set of all real numbers, respectively. Then what is [itex]|\mathbb{R} \backslash \mathbb{Q}|[/itex]?

## Homework Equations

[itex]|\mathbb{Q}| = |\mathbb{Z^{+}}| < |P(\mathbb{Z^{+}})| = |\mathbb{R}|[/itex]

## The Attempt at a Solution

I can prove the relevant equations by the method of cut and general concepts of functions and logic. But actually my understanding of cut construction for R is kinda poor. So should I revise this concept to know the cardinality of the set of irrational numbers? Or do I need to know extra stuff?