Question

Check whether the relation R in R defined as R = {(a, b): a ≤ b 3 } is reflexive, symmetric or transitive.

Answer 1

$\text{\bf{reflexive relation}}$ A relation is said to be reflexive relation, if every element of A is related to itself.
thus,(a,a) ∈ R , for all a∈ A => R is reflexive

$\text{\bf{symmetric relation}}$ A relation is said to be symmetric relation, if
(a, b) ∈ R => (b, a) ∈ R , for all a, b ∈ A
e.g., aRb => bRa, for all a, b ∈ A => R is symmetric.

$\text{\bf{transitive relation}}$ A relation is said to be transitive relation if
(a, b) ∈ R and (b, c) ∈ R => (a,c) ∈ R for all a,b,c ∈ A

now, let's check all relations :
given, R = {(a,b) : a ≤ b³ }
it can observed that (1/3, 1/3) ∉ R as 1/3 $\nleq$1/3²
therefore, given relation R is not reflexive.

Now, (1,3) ϵ R (as 1 < 3³ = 27)
But, (3,1) ∉ R (as 3³ > 1)
Therefore,given relation R is not symmetric.

now if we have (3, 3/2) , (3/2, 6/5) ∈ R
as 3 < (3/2)³ and (3/2) < (6/5)³
but (3, 6/5) ∉ R as 3 > (6/5)³
therefore, given relation R is not transitive

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