Question

S is a relation over the set R of all real numbers and it is given by $\left( {a,b} \right) \in S \Leftrightarrow ab \geqslant 0$. Then, S is
A. Symmetric and transitive only
B. Reflexive and symmetric only
C. Anti-symmetric relation
D. An equivalence relation

Answer 1

Before attempting this question, one should have prior knowledge about the concept of relations and also remember that when a relation is reflexive, symmetric and transitive at the same time such relation is equivalence relation, use this information to approach the solution.

Complete step-by-step answer:
Now, it’s been given that S is a relation over the set R of all real numbers and it is given by $\left( {a,b} \right) \in S \Leftrightarrow ab \geqslant 0$, we have to tell about S and the 4 options are been given to us.
Now let’s check for S to be reflexive relation:
If we have $\left( {a,b} \right) \in S \Leftrightarrow ab \geqslant 0$ such that $\left( {a,a} \right) \in S \Leftrightarrow a \times a \geqslant 0$satisfies that it is reflexive
So $\left( {a,a} \right) \in S \Leftrightarrow {a^2} \geqslant 0$ is absolutely true as the square of any number is always greater than 0, hence the relation on S is reflexive.
Now let’s check for Symmetric relation:
If $\left( {a,b} \right) \in S \Leftrightarrow ab \geqslant 0$is true than if $\left( {b,a} \right) \in S \Leftrightarrow ba \geqslant 0$holds true than the relation is said to be symmetric.
If $ab \geqslant 0$then $a \geqslant 0$, $b \geqslant 0$or $a \leqslant 0$, $b \geqslant 0$
Hence clearly for both the conditions if $ab \geqslant 0$is true then $ba \geqslant 0$will always hold true, hence the relation is symmetric.
Now let’s check for transitive relation:
If $\left( {a,b} \right) \in S \Leftrightarrow ab \geqslant 0$ and $\left( {b,c} \right) \in S \Leftrightarrow bc \geqslant 0$ implies that $\left( {a,c} \right) \in S \Leftrightarrow ac \geqslant 0$then the relation is said to be transitive
So $\left( {b,c} \right) \in S \Leftrightarrow bc \geqslant 0$ (equation 1)
$\left( {a,b} \right) \in S \Leftrightarrow ab \geqslant 0$ (equation 2)
Taking the product of both the equations
$a{b^2}c \geqslant 0$ Now ${b^2}$is always greater than 0 so this left us with $ac \geqslant 0$
Clearly $\left( {a,c} \right) \in S \Leftrightarrow ac \geqslant 0$Thus S is a Transitive relation.
Clearly relation S is reflexive, symmetric and transitive hence it is an equivalence relation.

So, the correct answer is “Option (d)”.

Note: In the above solution we came across the term “relation” which can be explained relation between sets the types of relation are reflexive relation in which each element relates to itself, symmetric relation which says that if a relates to b then b also relates to a, transitive relation which says that if (a, b) belongs to R, (b, c) belongs to R then (a, c) also should belong to R, there are some more types of relation such as empty relation, universal relation, identity relation, etc.