S is a relation over the set R of all real numbers and it is given by $(a, b) ∈ S⇔ab ≥0$. Then, S is

an equivalence relation

Reflexivity: For any $a ∈ R$, we have

$a^2 = aa ≥0⇒ (a, a) ∈ S$

Thus, $(a, a) ∈ S$ for all $a ∈ R$.

So, S is a reflexive relation on R.

Symmetry: Let $(a, b) ∈ S$. Then,

$(a, b) ∈ S⇒ab ≥0⇒ ba ≥0⇒ (b, a) ∈ S$

Thus, $(a, b) ∈ S⇒ (b, a) ∈ S$ for all $a, b ∈ R$.

So, S is a symmetric relation on R.

Transitivity: Let $a, b, c ∈ R$ such that

$⇒(a, b) ∈S$ and $(b, c) ∈R$

$⇒ab ≥ 0$ and $bc ≥0$

$⇒a, b, c$ are of the same sign.

$⇒ac ≥ 0$

$⇒(a, c) ∈ R$.

Thus, $(a, b) ∈ S, (b, c) ∈S⇒ (a, c) ∈ S$.

So, S is a transitive relation on R.

Hence, S is an equivalence relation on R.