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Let N be the set of natural numbers. Consider the following sets. P: Set of Rational numbers (positive and negative) Q: Set of functions from {0,1} to N R: Set of functions from N to {0,1} S: Set of finite subsets of N. Which of the sets above are countable?

1. Q and S only
2. P and S only
3. P and R only
4. P, Q and S only

Correct answer: P, Q and S only

Solution

The set of rational numbers (P) is countable because it can be listed in a sequence, while the set of functions from a finite set to natural numbers (Q) is also countable since it has a finite domain. The set of finite subsets of natural numbers (S) is countable as well, as each finite subset can be represented by a finite sequence of natural numbers.

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