Question

Let $R$ be the relation over the set of integers such that $mRn$ if and only if $m$ is a multiple of $n$.Then $R$ is

• A. reflexive
• B. symmetric
• C. transitive
• D. an equivalence relation

Answer 1

Answer: (A), (C)

reflexive
transitive

We know that a number is a factor as well as multiple of itself.

Since 'n' will be a factor of 'n' for any natural number, hence $(n,n)$ will be a subset of the relation. In other words $nRn$ for a natural number 'n' holds true. This makes the relation reflexive.

If 'n' is a factor of 'm' then simultaneously 'm' cannot be a factor of 'n' since $n\le m$.

Thus $mRn\to nRm$ is not true. Thus the relation is not symmetric.

Now, if 'm' is multiple of 'n' and 'p' be a multiple of 'm', (all being distinct natural numbers), then 'n' will be a common factor of 'm' and 'p'. Thus

$pRm\to mRn\to pRn$ is true.

Hence the relation is transitive.