GATE CS 2015 Set 2 — Question 23
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Question
Let be the relation on the set of positive integers such that if and only if and are distinct and have a common divisor other than 1. Which one of the following statements about is true?
Correct answer
(D) R is symmetric but not reflexive and not transitive
Solution
The relation is defined on the set of positive integers as:
and .
and .
1.Reflexivity: For a relation to be reflexive, must hold for all . However, the definition requires and to be distinct (). Since is always true, is always false. Thus, is not reflexive.
2.Symmetry: For a relation to be symmetric, . If and , then and . This satisfies the condition for . Thus, is symmetric.
3.Transitivity: For a relation to be transitive, and . Let's test with a counter-example:
- Let .
- : and (True).
- : and (True).
- : and . Since the common divisor is not greater than 1, is False.
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