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

Answer 1

The correct option is D

Equivalence

Explanation for the correct option:

Finding the the relation $R$:

In $xy$ co-ordinate plane.

Consider three lines $l_1,l_2\&l_3$​
Therefore $I\parallel I$ hence $R$ is reflexive
If $l_1\parallel l_2$ then this implies that $l_2\parallel l_1$​ hence $R$ is symmetric and
If $l_1\parallel l_2\text{\&}{l}_2\parallel l_3\text{⇒}{l}_1\parallel l_3$hence $R$ is transitive

Hence the relation is symmetric, reflexive and transitive. Therefore it is an equivalence relation.

Therefore, the correct answer is option (D).