Let N be the set of natural numbers. Consider the following sets

Q. 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?

(A) Q and S only        (B) P and S only         (C) P and R only        (D) P, Q and S only

Ans: P, Q and S only

Sol:

Set of functions from {0, 1} to N are countable because it has one to one correspondence to N.

Set of functions from N to {0, 1} is uncountable, because it has one to one correspondence to set of real numbers between (0 and 1).

Set of finite subsets of N is countable.