Question

Let r be a relation from R(set of real numbers) to R defined by $r = \{ (a,b)|a,b \in R$and $a - b + \sqrt 3 $ is an irrational number}. The relation r is
A. An equivalence relation
B. Reflexive only
C. Symmetric only
D. Transitive only

Answer 1

Hint: In these types of questions, think of the basic definition of the types of relations given in and try to check whether the relation mentioned in the questions satisfies the condition for any type of relation or not. Also, use the property that multiplication is commutative in nature.

Complete step by step Solution:
Before starting with the solution, let us discuss the different types of relations we use in mathematics. There are 8 types of relations, out of which important ones are reflexive, symmetric, transitive, and equivalence relations.
Reflexive relations are those relation in which each and every element is mapped to itself, i.e., $(a,a) \in R$. Symmetric relations are those for which, if $(a,b) \in R$
 then $(b,a)$ must also belong to R. This can be represented as $aRb \Rightarrow bRa$
Transitive relation is those if $(a,b)$and $(b,c) \in R$ and then $(a,c)$must also belong to R, i.e., $(a,b)and(b,c) \in R \Rightarrow (a,c) \in R$
Now let’s start with the solution to the above question
$a \in R$
$ \Rightarrow a - a + \sqrt 3 = \sqrt 3 $ which is an irrational number.
$(a,a) \in R$ Hence it is a reflexive relation
$(a,b) \in R$
$ \Rightarrow a - b + \sqrt 3 $
$ \Rightarrow b - a + \sqrt 3 $
They both are irrational so it is a symmetric relation.
$(a,b)$and $(b,c) \in R$
$ \Rightarrow a - b + \sqrt 3 $ is an irrational number
$ \Rightarrow b - c + \sqrt 3 $ is an irrational number
$ \Rightarrow a - c + \sqrt 3 $ is an irrational number
It is a transitive relation.
Now, if there exists a relation in which it is reflexive, symmetric, and transitive at the same time, then the relation is called an equivalence relation.
Hence, the correct option is A.

Note: Try to understand that a relation can also be called a transitive relation if there exists $aRb$, but doesn’t exist any relation $bRc$. Also, most of the questions we solved above are either solved by using statements based on observation or by our experience, as we did in the above question. Care should be taken while selecting options. If it is multiple choice then also the answer will be an equivalence relation.