UPSC NDA Math Model Paper 4

Let's check the given relation for its type one by one.
Reflexive: Every line is parallel to itself. It means that lRl for all l ∈ L. Therefore, R is reflexive.
Symmetric: If a line l is parallel to m, then m is parallel to l. i. e. if lRm ⇒ mRl, ∀ l, m ∈ L. Therefore, R is symmetric.
Transitive: If a line l is parallel to m and m is parallel to k, then l is also parallel to k. i. e. if lRm and mRk ⇒ lRk, ∀ l, m, k ∈ L. Therefore, R is transitive.
Since the relation R is reflexive, symmetric and transitive as well, it is an equivalence relation.