Question

Let L be the set of all straight lines in the Euclidean plane. Two lines ${l_1}$ and ${l_2}$ are said to be related by the relation R iff ${l_1}$ is parallel to ${l_2}$ . Then the relation R is
A) Only reflexive
B) Only symmetric
C) Only transitive
D) Equivalence relation

Answer 1

A relation R on a set A is said to be an equivalence relation if and only if the relation R is reflexive, symmetric and transitive.
Reflexive: A relation is said to be a reflexive relation, if $(a,a) \in R$ for every $a \in R$ .
Symmetric: A relation is said to be a symmetric relation, if $(a,b) \in R$ then $(b,a) \in R$ .
Transitive: A relation is said to be a transitive relation if $(a,b) \in R$ and $(b,c) \in R$ then $(a,c) \in R$ .

Complete answer: Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of a set are equivalent to each other, if and only if they belong to the same equivalence class.
Reflexivity:
In $xy$ coordinate plane consider a line ${l_1}$ .
We know that every line is parallel to itself.
Therefore ${l_1}$ is parallel to ${l_1}$ .
Hence R is a Reflexive relation.
Symmetry:
In $xy$ coordinate plane consider lines ${l_1}$ and ${l_2}$ .
Let ${l_1}$ be parallel to ${l_2}$ . This implies that ${l_2}$ is parallel to ${l_1}$ .
Hence R is a Symmetric relation.
Transitivity:
In $xy$ coordinate plane consider lines ${l_1}$ , ${l_2}$ and ${l_3}$ .
Let ${l_1}$ be parallel to ${l_2}$ and ${l_2}$ be parallel to ${l_3}$.
This implies that ${l_1}$ is parallel to ${l_3}$ .
Hence R is a Transitive relation.
Since R is Reflexive, Symmetric and Transitive , therefore it is an equivalence relation.
Hence the correct option is (D).

Note:
A relation R on a set A is said to be an equivalence relation if and only if the relation R is reflexive, symmetric and transitive. If any of these properties does not hold true then the relation R is never an equivalence relation.