Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.

Method - 1

Answer

R = {(a, b): a ≤ b2}

It can be observed that

 

∴R is not reflexive.

Now, (1, 4) ∈ R as 1 < 4^2

But, 4 is not less than 1^2.

∴(4, 1) ∉ R

∴R is not symmetric. Now,

(3, 2), (2, 1.5) ∈ R

(as 3 < 2^2 = 4 and 2 < (1.5)^2 = 2.25) But, 3 > (1.5)^2 = 2.25

∴(3, 1.5) ∉ R

∴ R is not transitive.

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

Method - 2