Question

show that the relation R on the set R of real numbers defined as R = { (a,b) =a b^2 } is neither reflexive,nor symmetric and transitive.

Answer 1

R = {(a, b): a ≤ b²}

It can be observed that

∴R is not reflexive.

Now, (1, 2) ∈ R as 1 < 2²

But, 2 is not less than 1².

∴ (2, 1) ∉ R

∴R is not symmetric.

Now,

(5, 3), (3, 2) ∈ R

(as 5 < 3² = 9 and 3 < (2) ² = 4)

But, 5 > (2) ²= 4

∴ (5, 2) ∉ R

∴ R is not transitive.

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