Question

Check whether the relation R, defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive.

Answer 1

(i) a ≤ a³ is not true. e.g. \(\frac{1}{2}\) is not less than (\(\frac{1}{2}\))³
∴ R is not reflexive.
(ii) If a ≤ b³, then b is not less than or equal a3, e.g. 1 ≤ 2³, but 2 is not less than 1³.
⇒ R is not symmetric.
(iii) If a < b³ and b < c³, then but a is not necessarily less than c³.
e.g. a = 7, b = 2, c = 1.5
a < b³, ∵ 7 < 2³ = 8; b < c³, ∵ 2 < (1.5)³ = 3.375
But a > c³, ∵ 7 > (1.5)³ = 3.375
∴ R is not Transitive