Give an example of a relation. Which is

Transitive but neither reflexive nor symmetric.

Let a relation R in R defined as:

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

For any a ϵ R, we have (a,a)) ∉ R as a cannot be strictly less than a itself.

In fact, a = a,

Therefore, R is not reflexive.

Now, (1,2) ϵ R but 2 > 1

⇒ (2,1)) ∉ R.

⇒ R is not symmetric.

Now, let (a,b), (b,c) ϵ R

⇒ a < b and b < c

⇒ a < c

⇒ (a,c) ϵ R

⇒ R is transitive.

Therefore, relation R is transitive but not reflexive and a symmetric.