Question

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

Answer 1

R = {(a, b): a ≤ b³}
It is observed that
∴ R is not reflexive.

Now,
(1, 2) ∈ R (as 1 < 2³ = 8)
But,
(2, 1) ∉ R (as 2³ > 1)
∴ R is not symmetric. 

We have (3, \(\frac {3} {2}\)), (\(\frac {3} {2}\), \(\frac {6} {5}\)) ∈ R as 3<(\(\frac {3} {2}\))³ and \(\frac {3} {2}\)<(\(\frac {6} {5}\))³
But (3, \(\frac {6} {5}\)) ∉ R as 3>(\(\frac {6} {5}\))³
∴ R is not transitive.

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