Question

Prove that the relation R in R defined by R = {(a, b): a ≤ b³} is neither reflexive nor symmetric nor transitive.

Answer 1

Given that R = {(a, b): a ≤ b³}

It is observed that (1/2, 1/2) ∈ R as 1/2 < (1/2)³ = 1/8

So, R is not reflexive. 

Now, (1, 2) (as 1< 2³ =8) 

But (2, 1) ∉ R (as 2³ > 1) 

So, R is not symmetric.

we have (3, 3/2), (3/2, 6/5) ∈ R as 3> (3/2)³ and < (6/5)³

But (3, 6/5) ∉ R as 3 > (6/5)³

Therefore, R is not transitive.

Hence, R is neither reflexive nor symmetric nor transitive.