Question

If a relation R on the set \left\\{ {1,2,3} \right\\} be defined by R = \left\\{ {\left( {1,2} \right)} \right\\} , the R is
$\left( 1 \right)$ reflexive
$\left( 2 \right)$ transitive
$\left( 3 \right)$ symmetric
$\left( 4 \right)$ none of these

Answer 1

We have to find that the given relation R satisfies the conditions of which type of relation . We solve this question using the knowledge of the types of relation and the conditions which are required by the relation to satisfy that particular type . First we will write all the conditions of the types of relations and then we will check that either the given relation R satisfies the conditions of the relations or not . The condition which the relation satisfies is the type of the given relation .

Complete step-by-step solution:
Given :
A relation $R$ on the set \left\\{ {1,2,3} \right\\} be defined by R = \left\\{ {\left( {1,2} \right)} \right\\}
Now , we have to check the conditions for the relation to be reflexive , symmetric or transitive .
Now , taking the three conditions as three cases .
$Case{\text{ }}1{\text{ }}:$
In the given question the given relation is said to be reflexive if and only if it has all the elements of the set \left\\{ {1,2,3} \right\\} related to itself in the relation set $R$ . I.e. The set of relations $R$ should have elements $\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right)$ in it .
As the given set of relation $R$ doesn’t have the elements stated above , so the relation $R$ is not reflexive .
Hence , The relation $R$ is not reflexive .
$Case{\text{ }}2{\text{ }}:$
In the given question the given relation is said to be transitive if and only if it has all the elements of the set \left\\{ {1,2,3} \right\\} related to each other in the relation set $R$ as $\left( {1,2} \right)$ and $\left( {2,3} \right)$ then the relation $R$ should have $\left( {1,3} \right)$ in the set of the relation $R$ . I.e. The set of relation $R$ should have elements $\left( {1,2} \right),\left( {2,3} \right),\left( {1,3} \right)$ and so on in it .
As the given set of relation $R$ doesn’t have the elements stated above , so the relation $R$ is not transitive .
Hence , The relation $R$ is not transitive .
$Case{\text{ }}3{\text{ }}:$
In the given question the given relation is said to be symmetric if and only if it has all the elements of the set \left\\{ {1,2,3} \right\\} related to each other in the relation set $R$ as $\left( {1,2} \right)$ then the relation $R$ should have $\left( {2,1} \right)$ in the set of
the relation $R$ . I.e. The set of relations $R$ should have elements $\left( {1,2} \right),\left( {2,1} \right)$ and so on in it .
As the given set of relation $R$ doesn’t have the elements stated above , so the relation $R$ is not symmetric .
Hence , The relation $R$ is not symmetric .
As , the given set of relation $R$ does not satisfy any of the three conditions . Thus , the relation $R$ is not symmetric , not transitive and not reflexive .
Hence , the correct option is (4) .

Note: For the relation to be reflexive : A relation $R$ across a set $A$ is reflexive only if each and every element of the set $X$ is related to itself i.e. $\left( {a,a} \right)$ belongs to $R$ for all values of $a \in A$.
For the relation to be transitive : A relation $R$ across a set $A$ is transitive only if every element a , b , c in the set $X$ relates as a to b and b to c, then the relation R should also have an elements that relates as a to c.
For the relation to be symmetric : A relation $R$ across a set $A$ is symmetric only if it has the element in the set $X$ related to each other as a to b, then the relation $R$ should have an element that relates as b to a.