Question

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.

Answer 1

R = {(a, b): a $\leq$ b2}
It can be observed that $\bigg(\frac{1}{2},\frac{1}{2}\bigg)$∉ R, since $\frac{1}{2}>\bigg(\frac{1}{2}\bigg)^2=\frac{1}{4}$
∴R is not reflexive.

Now, (1, 4) ∈ R as 1 < 42
But, 4 is not less than 12.
∴(4, 1) ∉ R
∴R is not symmetric.

Now, (3, 2), (2, 1.5) ∈ R
(as 3 < 22 = 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.