JEE Main 2005 — Sets And Relations Question with Solution

We have to examine whether the relation $R$ satisfies the properties of reflexivity, symmetry, and transitivity.

Relation R : $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ on set $A=\{3,6,9,12\}$.

We will evaluate each of the three properties :

• Reflexivity : A relation is reflexive if all elements in the set are related to themselves. Here, (3,3), (6,6), (9,9), and (12,12) are all present in $R$, so $R$ is reflexive.




• Symmetry : A relation is symmetric if for every pair (a,b) in the relation, the pair (b,a) is also in the relation. In $R$, we have pairs such as (6,12) and (3,6), but their corresponding reverse pairs (12,6) and (6,3) are not in $R$, so $R$ is not symmetric.




• Transitivity : A relation is transitive if for every pair of pairs ((a,b), (b,c)) in the relation, the pair (a,c) is also in the relation. For the pairs (3,6) and (6,12) in $R$, we have (3,12) in $R$, and for the pairs (3,6) and (6,6) in $R$, we also have (3,6) in $R$. Therefore, $R$ is transitive.

Therefore, the relation $R$ is reflexive and transitive, but not symmetric.

So, the correct answer is Option D : $R$ is reflexive and transitive only.