Show that the relation S in the set R of real numbers, defined on S = {(a, b): a, b ∈ R and a ≤ b³} is neither reflexive, nor symmetric nor transitive.

We have the relation S, defined as

S = {(a, b): a, b ∈ R and a ≤ b³}

Here, R is a set of real numbers, this means that

a and b are real numbers.

Let us understand what reflexive relation is.

A binary relation R over a set X is reflexive if every element of X is related to itself. Formally, this may be written ∀x ∈ X: x R x.

So, if the relation was reflexive, then (a, a) ∈ R

That is,

a ≤ a³

Check: For a = 1.

1 ≤ 1³

⇒ 1 ≤ 1, is true.

For a = 2.

2 ≤ 2³

⇒ 2 ≤ 8, is true.

Let us test with fractions.

For .

, is not true.

Thus, a ≤ a³ is not true for all real values of a.

Hence, S is not reflexive.

Let us understand what symmetric relation is.

A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if a = b is true then b = a is also true.

So, if the relation was symmetric, then

If (a, b) ∈ R, then (b, a) ∈ R.

That is,

If a ≤ b³, then b≤ a³.

Check: For a = 2 and b = 3.

If 2 ≤ 3³, then 3 ≤ 2³

⇒ If 2 ≤ 27, then 3 ≤ 8, is true.

For a = 2 and b = 10.

If 2 ≤ 10³, then 10 ≤ 2³

⇒ If 2 ≤ 1000, then 10 ≤ 8, is not true.

Thus, b ≤ a³ is not true for all real values of a and b.

Hence, S is not symmetric.

Let us understand what transitive relation is.

A binary relation R over a set X is transitive if whenever an element a is related to an element b and b is related to an element c then a is also related to c.

So, if the relation is transitive, then

If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

That is,

If a ≤ b³ and b ≤ c³, then a ≤ c³.

Check: For a = 2, b = 3 and c = 4.

If 2 ≤ 3³ and 3 ≤ 4³, then 2 ≤ 4³

⇒ If 2 ≤ 27 and 3 ≤ 64, then 2 ≤ 64, is true.

For a = 3, b = 3/2 and c = 4/3.

If and , then

⇒ If 3 ≤ 3.37 and 1.5 ≤ 2.37, then 3 ≤ 2.37, is not true.

Thus, if a ≤ b³ and b ≤ c³, then a ≤ c³ is not true for all real values of a, b and c.

Hence, S is not transitive.

Thus, S is neither reflexive, nor symmetric nor transitive.