Question

Given an example of a relation. Which is reflexive and transitive but not symmetric.

Answer 1

Define a relation R in R as:

R = {a, b): a³ ≥ b³}

Clearly (a, a) ∈ R as a³ = a³.

∴ R is reflexive.

Now,

(2, 1) ∈ R (as 2³ ≥ 1³)

But,

(1, 2) ∉ R (as 1³ < 2³)

∴ R is not symmetric.

Now,

Let (a, b), (b, c) ∈ R.

⇒ a³ ≥ b³ and b³ ≥ c³

⇒ a³ ≥ c³

⇒ (a, c) ∈ R

∴ R is transitive.

Hence, relation R is reflexive and transitive but not symmetric.