Question

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

Answer 1

Given as R = {(a, b): a ≤ b³}

It is observed that (1/2, 1/2) in R as 1/2 > (1/2)3 = 1/8

∴ R is not reflexive.

Now,

(1, 2) ∈ R (as 1 < 23 = 8)

But, (2, 1) ∉ R (as 2 > 13 = 1)

Therefore, R is not symmetric.

We have (3, 3/2), (3/2, 6/5) in “R as” 3 < (3/2)³ and 3/2 < (6/5)³

But (3, 6/5) ∉ R as 3 > (6/5)³

∴ R is not transitive.

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