Question

Let $R=\left\{ \left( 1,3 \right),\left( 4,2 \right),\left( 2,4 \right),\left( 2,3 \right),\left( 3,1 \right) \right\}$ be a relation on the $Set\ A=\left\{ 1,2,3,4 \right\}$. The relation R is,
A. Not symmetric
B. Transitive
C. A function
D. Reflexive

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 solution -
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 and we will check the given relation for these.
Now, we have been given a relation $R=\left\{ \left( 1,3 \right),\left( 4,2 \right),\left( 2,4 \right),\left( 2,3 \right),\left( 3,1 \right) \right\}$ be on $Set\ A=\left\{ 1,2,3,4 \right\}$. Now, we have to check it for symmetric, reflexive, transitive.
Now, for the relation to be reflexive for all $a\in A\left( a,a \right)$ must belong to R but since $R=\left\{ \left( 1,3 \right),\left( 4,2 \right),\left( 2,4 \right),\left( 2,3 \right),\left( 3,1 \right) \right\}$. Therefore, \[\left( a,a \right)\notin R\ \ \forall a\in A\]. Hence the relation is not reflexive.
Now for the relation to be symmetric for all (a, b) that belongs to R, (b, a) must also belongs to R but in R for (2,3) there is no (3,2) to make R symmetric for (2,3) therefore the relation is not symmetric
Hence, the correct option is (A) not symmetric.

Note: To solve these types of questions it is important to note that we have checked only symmetric and since by solving this we get the one option same. Therefore, we have not tested further for transitive and reflexive as it is a single correct question.