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.