Question

Is the set of all rational numbers \( \mathbb{Q} \) a countable or uncountable set?

• A. Finite set
• B. Countable set
• C. Uncountable set
• D. Empty set

Hint:

Even though rationals are infinite, they are still countable!

Answer 1

The Correct Option is B

Concept: Countability of sets
A set is called:

• Countable if its elements can be put in one-to-one correspondence with \( \mathbb{N} \)
• Uncountable if this is not possible

Step 1: Form of rational numbers
Every rational number can be written as: \[ \frac{p}{q}, \quad p \in \mathbb{Z},\; q \in \mathbb{N} \]
Step 2: Arrangement idea
We can arrange all such fractions in a grid and traverse them diagonally (Cantor’s method), ensuring every rational number is listed.
Step 3: Conclusion
Since all rational numbers can be listed in a sequence: \[ \mathbb{Q} \text{ is countable} \]