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 ≤ b²}
It can be observed that (1/2,1/2)∉R, since, 1/2>(1/2)²
∴ 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.