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 1^2.
∴(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.