# The cardinality of the set of irrational numbers

• julypraise
In summary: Your Name]In summary, the cardinality of |\mathbb{R} \backslash \mathbb{Q}| is equal to the cardinality of the set of irrational numbers, which is |\mathbb{R}|. This can be proven using the concept of cuts or other methods such as Cantor's diagonal argument.
julypraise

## Homework Statement

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

## Homework Equations

$|\mathbb{Q}| = |\mathbb{Z^{+}}| < |P(\mathbb{Z^{+}})| = |\mathbb{R}|$

## 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?

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.

## What is the definition of cardinality?

Cardinality is a mathematical concept that refers to the size or number of elements in a set.

## How is the cardinality of a set determined?

The cardinality of a set is determined by counting the number of elements in the set. This can be done by listing out all the elements or using mathematical formulas.

## What is the cardinality of the set of irrational numbers?

The cardinality of the set of irrational numbers is uncountable, meaning that it cannot be represented by a finite or even infinite list of numbers. This is because there are infinitely many irrational numbers between any two rational numbers.

## How does the cardinality of the set of irrational numbers compare to the set of natural numbers?

The cardinality of the set of irrational numbers is greater than the cardinality of the set of natural numbers. This means that there are more irrational numbers than natural numbers even though both sets are infinite.

## Why is the cardinality of the set of irrational numbers important?

The concept of cardinality is important in mathematics as it helps us understand the size and properties of different sets. The uncountable cardinality of the set of irrational numbers also has implications in the study of real numbers and the nature of infinity.

• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
11
Views
2K
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
45
Views
3K
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
17
Views
2K