Question

The relation R defined on the set of all positive integers by "mRn if m divides n"

Answer 1

condition for reflexive : R is said to be reflexive if a is related to a for all a $\in S$.

here given, The relation R defined on the set of all positive integers by "mRn if m divides n".

we know, every element is divided by itself.

so, m divides m for all m $\in R$ and n divides n for all n $\in R$.

hence, it is reflexive.

condition for symmetric : R is said to be symmetrical , if a is related to b $\implies$ b is related to a.

here given, m divides n, but we can't say n divides m. so, it is not symmetric.

condition for transitive : R is said to be transitive if a is related to b , b is related to c $\implies$ a is related to c.

here m divides n => n = mk ...(1)

Let n divides p => p = nl ....(2)

where k and l are constants.

from equations (1) and (2),

p = mkl = m(T) , where T is another constant i.e., T = kl

here it is clear that m divides p.

so, it is transitive .

hence, given relation is reflexive, transitive but not symmetric.

Answer 2

Answer:

it is a transitive relation