GATE CS 2018 Set 1 — Question 37
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Question
Let be the set of natural numbers. Consider the following sets.
: Set of Rational numbers (positive and negative)
: Set of functions from to
: Set of functions from to
: Set of finite subsets of .Which of the sets above are countable?
: Set of Rational numbers (positive and negative)
: Set of functions from to
: Set of functions from to
: Set of finite subsets of .Which of the sets above are countable?
Correct answer
(D) P, Q and S only
Solution
To determine which sets are countable:
1.Set (Rational numbers): The set of all rational numbers is a well-known example of a countable set. Thus, is countable.
2.Set (Functions from to ): A function from to is uniquely determined by its values at 0 and 1. Thus, the set is equivalent to . Since the Cartesian product of two countable sets is countable, is countable.
3.Set (Functions from to ): This set is equivalent to the power set of , denoted by or . According to Cantor's theorem, the power set of a countably infinite set is uncountable (its cardinality is ). Thus, is uncountable.
4.Set (Finite subsets of ): The set of all finite subsets of a countable set is countable. This can be shown by expressing as the union of sets , where is the set of subsets of size . Each is countable, and a countable union of countable sets is countable. Thus, is countable.
Therefore, and are countable. The correct option is (D).Continue learning with Success Tracker
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