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

Hint: 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.