# 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

Last updated date: 29th Mar 2023

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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 haven’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 isn’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.

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 haven’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 isn’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.

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