Question

Show that the relation R = {(a, b): a > b} on N is transitive but neither reflexive nor symmetric.

Answer 1

It is given that

R = {(a, b): a > b}

Reflexive-

If a ∈ R we get (a > b)

So (a, b) ∉ R for a ∉ R

Hence, R is not reflexive.

Symmetric-

We know that

(3, 4) ∈ R but (4, 3) ∉ R

Hence, R is not symmetric.

Transitive-

Consider (a, b) ∈ R and (b, c) ∈ R

It can be written as

a > b and b > c

So we get

a > c where (a, c) ∈ R

Hence, R is transitive.

Therefore, R is transitive but neither reflexive nor symmetric.