Let R be the relation on the set of positive integers such that aRb if and only if 'a 'and 'b' are distinct and have a common divisor other than 1. Which one of the following statements about 'R' is true?

Let R be the relation on the set of positive integers such that aRb if and only if 'a 'and 'b' are distinct and have a common divisor other than 1. Which one of the following statements about 'R' is true? Correct Answer 𝑅 is symmetric but not reflexive and not transitive

R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1.

Let A = {1, 2, 3, 4, 5, 6…}

aRb ≡ (a, b}

• R cannot be reflexive because aRa, a and a are not distinct.

Example: R = {(4,6)}

• R is symmetric because if aRb is there then bRa is also possible, in both a and b are distinct.

Example:  R = {(6, 4), (4,6)}  

• R is not transitive as aRb and bRc doesn’t mean aRc.

Example: R = {(6, 4), (4,6)} since (6,6) cannot be included and hence it cannot be transitive

So, given relation is symmetric but not transitive and not reflexive.