Question

# If A={7,8,9} then the relation R={(8,9)} in A is${\text{A}}{\text{.}}$ Symmetric only${\text{B}}{\text{.}}$ Symmetric and transitive only${\text{C}}{\text{.}}$ Transitive only${\text{D}}{\text{.}}$ Equivalence

Hint- Here, we will proceed by checking whether the given relation is symmetric or not, transitive or not, equivalence or not with the help of the general conditions which are used.

Given, set A={7,8,9} and relation R={(8,9)} in A
Any relation is said to be equivalence relation when that relation is reflexive, symmetric as well as transitive.
For a relation in set A to be reflexive, if $\left( {a,a} \right) \in R$ for every $a \in R$
For a relation R in set A to be symmetric, if $\left( {a,b} \right) \in R$ then $\left( {b,a} \right) \in R$
For a relation R in set A to be transitive, if $\left( {a,b} \right) \in R$ and $\left( {b,c} \right) \in R$ then $\left( {a,c} \right) \in R$
Clearly $\left( {8,9} \right) \in R$ but $\left( {9,8} \right) \notin R$, so the given relation R in set A does not satisfy the necessary condition for symmetric relation. Hence, relation R is not symmetric.
Since for the relation R in set A to be an equivalence relation, it is necessary for that relation to be symmetric also. Hence, the given relation R is not an equivalence relation.
The given relation R in set A contains only one element i.e., (8,9) so it is said to be transitive.
Therefore, the given relation R in set A is only transitive.
Hence, option C is correct.

Note- In this particular problem, we havenâ€™t checked whether the given relation R in set A is reflexive or not because only reflexive is not anywhere there in the options and it has already proved that the given relation isnâ€™t an equivalence relation. The above given transitive condition can only be applied if there is more than one element in the given relation but here there is only one element.