Question

Given the relation on $R=\left\{ \left( a,b \right),\left( b,c \right) \right\}$ in the set A $\left\{ a,b,c \right\}$. Then the minimum number of ordered pairs which added to R make it an equivalence relation is,
A. 5
B. 6
C. 7
D. 8

Answer 1

Hint: We will be using the concepts of functions and relations to solve the problem. We will be using the definitions of reflexive relation, symmetric relations and transitive relations to verify if each relation holds or not and hence deduce the answer.

Complete step-by-step answer:

Now, we have been given a relation and we have to find whether the relation is reflexive, symmetric, transitive or a combination of these.
Now, we know that reflexive relations are those in which every element is mapped to itself i.e. $\left( a,a \right)\in R$ while symmetric relations are those for which if a R b then b R a. Also, holds and transitive are those relations in which if a R b and b R c then a R c must be held.
Now, we know different types of relations, we will check the given relation for these.
Now, we have been given a relation $R=\left\{ \left( a,b \right),\left( b,c \right) \right\}$ in the set A $\left\{ a,b,c \right\}$ and we have to find the minimum number of ordered pairs which added to R make it an equivalence relation.
Now, for R to be a reflexive relation on set A it must contain $\left( a,a \right),\left( b,b \right),\left( c,c \right)$. Also, we have $\left( a,b \right)\in R\ and\ \left( b,c \right)\in R$. So, for the relation to be symmetric, $\left( b,a \right)\in R\ and\ \left( c,b \right)\in R$ is also necessary.
Now, we have \[R=\left\{ \left( a,a \right),\left( b,b \right),\left( c,c \right),\left( a,b \right),\left( b,a \right),\left( b,c \right),\left( c,b \right) \right\}\]
Now, for R to be transitive if $\left( a,b \right)\in R\ and\ \left( b,c \right)\in R$ then (a, c) must also belong to R.
$\left( c,b \right)\in R\ and\ \left( b,a \right)\in R$ then (c, a) must also belong to R.
Hence, the relation R is \[\left\{ \left( a,a \right),\left( b,b \right),\left( c,c \right),\left( a,b \right),\left( b,a \right),\left( b,c \right),\left( c,b \right) \right\}\].
So, we have to add 7 ordered pairs. Hence, the correct option is (C).

Note: To solve these types of questions it is important to note that a R b means that a is related to b by a relation R. Also these types of questions are solved easily by giving examples and counterexamples. Also, we have to check the relation for reflexive, symmetric and transitive relation to check it for equivalence relation.