GATE CSE 2015 Set 2 | Question: 16

Question

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

• A. $R$ is symmetric and reflexive but not transitive
• B. $R$ is reflexive but not symmetric not transitive
• C. $R$ is transitive but not reflexive and not symmetric
• D. $R$ is symmetric but not reflexive and not transitive

Comments

Reflexive → NO.

Irreflexive → YES.

Symmetric → YES.

Anti-Symmetric → NO.

Asymmetric → NO.

Transitive → NO.

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Be careful while checking for transitivity when the relation is not reflexive.

Answer 1

Answer: $D$

Take $(3, 6)$ and $(6, 2)$ elements of $R$. For transitivity $(3, 2)$ must be element of $R$, but $3$ and $2$ don't have a common divisor and hence not in $R$.

For any positive integer $n$, $(n, n)$ is not element of $R$ as only distinct $m$ and $n$ are allowed for $(m, n)$ in $R$. So, not reflexive also.

Comments

@Raj, I think Opt (C) is not correct for all possible cases, Even one case is not satisfied we can say that it will not satisfy, like an example given in the question.

or another example: (3, 6), (6, 4) if transitive relation (3,4) belongs to R, then 3 and 4 doesn’t have any common divisor other than one.

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(3,2) must be element of R, but 3 and 2 don't have a common divisor and hence not in R.

It should be  “not have a common divisor other than 1”.

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Then symmatric property also not satisfied bcz (1,2) and (2,1) is also not in the relation . 

so for symmetric property , is it necessary to all the element on the parent set must satisfied (x,y) ↔ (y,x) ?? 

Answer 2

The correct ans is (D) R is symmetric but not reflexive and not transitive

Comments

Always give an explanation for your answer , people tend to see the answers with explanation to get more clarity. Hope u understand :)

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@Warrior, please stop giving answer on all the question for earning some points. Please write answer when you see that another written answer are not good, OR you have different and easy way to explain the solution.

This is suggestion for all those gateoverflow users which answer without any explanation or copy content of other written answers.