Question

Let R be a relation defined over N, where N is set of natural numbers, defined as “mRn if and only if m is a multiple of n, m, n ∈ N.” Find whether R is reflexive, symmetric and transitive or not.

Answer 1

Given relation R on natural numbers defined as mRn if m is a multiple of n, m, n ∈ N.

R is reflexive: A relation is reflexive if mRn for all m ∈ N. It means that every number must be a multiple of itself. 

R is not symmetric: mRn, m is a multiple of n and nRm, not always n is multiple of m.

For example, m = 8, n =2

8R2 = 8 is a multiple of 2 

But 2R8 = 2 is not a multiple of 8

∴ R is reflexive but not symmetric

R is transitive

Let m, n, p ∈ N 

Whenever, mRn

⇒ m = nk   ...(1)

nRp

⇒ n = pk₁   ...(2) 

Now substituting (2) in (1),

m = nk

m =  pk₁k

m = pt, where, t = k₁k

Therefore, mRp, m is a multiple of p

∴ R is transitive.

Hence, R is reflexive and transitive but not symmetric.