Question

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

Answer 1

R = {(a, b): a ≤ b}

(i) R is reflexive. Replacing b by a, a ≤ a ⇒ a = a is true.

(ii) R is not symmetric : a < b and b < a which is not true.

e.g., 2 < 3, but 3 is not less than 2.

(iii) R is transitive : If a ≤ b and b ≤ c, then a ≤ c.

e.g. 2 < 3, 3 < 4 ⇒ 2 < 4.