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

Hint: 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.