Question

Show that the relation R in R defined as R = {(a, b) : a <, b}, is reflexive and transitive but not symmetric.

Answer 1

(a, a): a ≤ a is true ∀ a ∈ R

hence reflexive

(a, b) ⇒ a ≤ b  b  (b,a) ∈ R

hence R is not symmetric

i.e. (1,3) ⇒ 1 ≤ 3 but (3, 1)  3 ≤ 1

(a, b) ⇒ a ≤ b, (b, c)

⇒ b ≤ c ⇒ a ≤ c = (a, c) ∈ R

hence R is transitive.