Question

A relation R in a set A is called symmetric, if for all ${{a}_{1}},{{a}_{2}}\in A$
(a) $\left( {{a}_{1}},{{a}_{2}} \right)\in R\Rightarrow \left( {{a}_{2}},{{a}_{1}} \right)\in R$
(b) $\left( {{a}_{1}},{{a}_{2}} \right)\in R\Rightarrow \left( {{a}_{1}},{{a}_{1}} \right)\in R$
(c) $\left( {{a}_{1}},{{a}_{2}} \right)\in R\Rightarrow \left( {{a}_{2}},{{a}_{2}} \right)\in R$
(d) None of these

Answer 1

Hint: Think of the basic definition of the types of relations, especially the symmetric relation, and try to represent it mathematically using the set theory's notations.

Complete step-by-step answer:
To start with the solution, let us discuss different types of relations. There are a total of 8 types of relations that we study, out of which the major ones are reflexive, symmetric, transitive, and equivalence relation.
Reflexive relations are those in which each and every element is mapped to itself, i.e., $\left( a,a \right)\in R$ . Symmetric relations are those for which, if $\left( a,b \right)\in R\text{ }$ then $\left( b,a \right)$ must also belong to R. This can be represented as $aRb\Rightarrow bRa$ . Now, transitive relations are those for which, if $\left( a,b \right)\text{ and }\left( b,c \right)\in R$ then $\left( a,c \right)$ must also belong to R, i.e., $\left( a,b \right)\text{ and }\left( b,c \right)\in R\Rightarrow \left( a,c \right)\in R$ .
Now, if there is a relation, which is reflexive, symmetric, and transitive simultaneously, then the relation is said to be an equivalence relation. For example: let us consider a set A=(1,2). Then the relation {(1,2),(2,1),(1,1),(2,2)} is an equivalence relation.
Therefore, according to the definition of the symmetric set as mentioned above, we can say that a relation R in a set A is called symmetric, if for all ${{a}_{1}},{{a}_{2}}\in A$ , $\left( {{a}_{1}},{{a}_{2}} \right)\in R\Rightarrow \left( {{a}_{2}},{{a}_{1}} \right)\in R$ .
Hence the answer to the above question is option (a).

Note: Most of the questions as above are either solved by using statements based on observation or taking examples, as we did in the above question. If can think of the questions like above by assuming some examples which satisfy the constraints mentioned.