Question

Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b²} is neither reflexive nor symmetric nor transitive.

Answer 1

(i) Reflexive:

R = {(a, b) : a ≤ b²}

Let a ∈ R

a ≤ a² which is false.

(a, a) ∉ R

∴ R is not reflexive.

(ii) Symmetric:

Let a, b ∈ R and (a, b) ∈ R

a ≤ b² and b ≤ a², which is false.

(a, b) ∈ R, but (b, a) ∉ R

∴ R is not symmetric.

(iii) Transitive:

Let a, b, c ∈ R

Consider (a, b) ∈ R, (b, c) ∈ R

a ≤ b² and b ≤ c²

a ≤ c² is false.

(a, c) ∉ R

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.