GATE | GATE CS 2018 | Question 58

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 above sets are countable?

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


Answer: (D)

Explanation: Set of Rational numbers (+ve or -ve) are countable. Refer this – https://math.stackexchange.com/questions/659302/how-to-prove-that-mathbbq-the-rationals-is-a-countable-set

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.

Sets P, Q and S are countable, therefore option (D) is Correct.


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