Prove that the set of all intervals with rational endpoints is countable.

To prove that the set of all intervals with rational endpoints is countable, we proceed step by step:

### 1. Define the set of interest:
Let \( I \) be the set of all intervals with rational endpoints. This means each interval can be expressed in the form \( (a, b) \) where \( a \) and \( b \) are rational numbers, and \( a < b \). We want to show that this set is countable.

### 2. The set of rational numbers is countable:
We know that the set of rational numbers, denoted by \( \mathbb{Q} \), is countable. This means there exists a bijection between \( \mathbb{Q} \) and the set of natural numbers \( \mathbb{N} \). In other words, we can list all rational numbers as \( q_1, q_2, q_3, \dots \), where each \( q_i \in \mathbb{Q} \).

### 3. Pairs of rational numbers:
Next, consider the set of all pairs \( (a, b) \) where \( a, b \in \mathbb{Q} \) and \( a < b \). This set consists of ordered pairs of rational numbers. 

Since \( \mathbb{Q} \) is countable, the Cartesian product \( \mathbb{Q} \times \mathbb{Q} \) is also countable. That is, the set of all ordered pairs \( (a, b) \) where \( a \) and \( b \) are rational is countable.

However, we are only interested in pairs where \( a < b \). The condition \( a < b \) simply excludes some pairs from the countable set \( \mathbb{Q} \times \mathbb{Q} \), but excluding a subset from a countable set does not make the resulting set uncountable. Therefore, the set of all pairs \( (a, b) \) with \( a < b \) is still countable.

### 4. Intervals with rational endpoints:
Each interval \( (a, b) \) corresponds uniquely to a pair \( (a, b) \) of rational numbers with \( a < b \). Since the set of such pairs is countable, the set of all intervals with rational endpoints is also countable.

### Conclusion:
The set of all intervals with rational endpoints is countable because it can be put in one-to-one correspondence with a subset of the countable set \( \mathbb{Q} \times \mathbb{Q} \).

Thus, the set of all intervals with rational endpoints is countable.

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Do you have any questions or want more details on a particular part of the proof?

### Related questions:
1. What is the definition of a countable set?
2. How can we prove that the rational numbers are countable?
3. Can a subset of an uncountable set be countable?
4. How does the Cartesian product of two countable sets remain countable?
5. Why does excluding a subset from a countable set not affect its countability?

### Tip:
When working with countable sets, remember that the union of a finite or countably infinite number of countable sets is also countable!

Explore the detailed proof that shows the set of all intervals with rational endpoints is countable. This proof involves key set theory concepts, such as the countability of rational numbers and the Cartesian product. Ideal for undergraduate mathematics students.

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