The cardinality of the set of irrational numbers

[julypraise]

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?

 

[mighty2000]

Thank you for your question. To answer your question, the cardinality of the set of irrational numbers is the same as the cardinality of the set of real numbers, which is denoted by |\mathbb{R}|. This means that the cardinality of the set of irrational numbers is also equal to |\mathbb{R}|. This can be proven using the concept of cuts, as you mentioned, or using other methods such as Cantor's diagonal argument.

Therefore, the cardinality of |\mathbb{R} \backslash \mathbb{Q}| is equal to the cardinality of the set of irrational numbers, which is |\mathbb{R}|.

I hope this helps. Let me know if you have any further questions.