Question

Let \[R\] and \[S\] be two relations on set \[A\] . Then
\[1)R\] and \[S\] are transitive, then \[R \cup S\] is also transitive
\[2)R\] and \[S\] are transitive, then \[R \cap S\] is also transitive
\[3)R\] and \[S\] are reflexive, then \[R \cap S\] is also reflexive
\[4)R\] and \[S\] are symmetric then \[R \cup S\] is also symmetric
\[5)2,3,4\] are Correct

Answer 1

Hint: We have to find the relation R . We solve this using the concept of relation and function . We should have knowledge of the relation of the function . We should have the knowledge of types of relations and we should know how to prove the types of relations . We should also know about equivalence relations .

Complete step-by-step answer:
Let us consider that \[A,R\] and \[S\] be sets such that \[A = \left\{ {1,2,3} \right\},R = \left\{ {1,2} \right\}\] and \[S = \left\{ {2,3} \right\}\] .
\[1)R\] and \[S\] are transitive, then \[R \cup S\] is also transitive
As we know, the transitive property of functions is not valid for the unions of two sets .
Hence , we conclude that \[R \cup S = \left\{ {1,2,3} \right\}\] is not transitive .

\[2)R\] and \[S\] are transitive, then \[R \cap S\] is also transitive
Let \[\left\{ {\left( {a,b} \right),\left( {b,c} \right)} \right\} \in R \cap S\]
From the intersection, we infer that \[\left( {a,b} \right),\left( {b,c} \right) \in R\] and \[\left( {a,b} \right),\left( {b,c} \right) \in S\]
Eliminating the common component, we get \[\left( {a,c} \right) \in R,\left( {a,c} \right) \in S\]
Combing the above two results we get, \[\;\left( {a,c} \right) \in R \cap S\]
 [ and is also stated as and , we know this from the concept of sets ]
Hence , we get that
If \[R,S\] are transitive then \[R \cap S\] is transitive.

\[3)R\] and \[S\] are reflexive, then \[R \cap S\] is also reflexive
If set \[R,S\] are reflexive ,
Such that ,
\[\left( {a,a} \right) \in R\] and \[\left( {a,a} \right) \in S\]
We get ,
\[R \cap S\] both are reflexive.

\[4)R\] and \[S\] are symmetric then \[R \cup S\] is also symmetric
If set \[R,S\] are symmetric
Such that ,
\[\left( {a,b} \right) \in Ror\left( {b,a} \right) \in S\]
We get ,
\[R \cup S\] both are symmetric .
[ and is also stated as and , we know this from the concept of sets ]
Hence , the correct option is \[\left( 5 \right)\] .
So, the correct answer is “Option 5”.

Note: A relation \[R\] in a set \[X\] is called an empty relation , if no element of \[X\] is related to any element of \[X\] I.e. \[R = \phi \subset X \times X\] .
A relation \[R\] in a set \[X\] is called a universal relation , if each element of \[X\] is related to any element of \[X\] I.e. \[R = X \times X\] .
Both the empty relation and the universal relation are also known as trivial relations .