Question

Let $R$ be the relation over the set $A$ of all straight lines in a plane such that $l_1 \, R \, l_2 \iff l_1$ is parallel to $l_2$. Then $R$ is:

• A. Symmetric
• B. An Equivalence relation
• C. Transitive
• D. Reflexive

Answer 1

Answer: (B)

An Equivalence relation

Answer 2

The relation R is defined as l 1R l2 ↔ l 1 is parallel to l 2. To check if R is an equivalence relation, we need to verify the following properties:

Reflexivity: A line is parallel to itself, so l 1R l1 holds for all l 1, so the relation is reflexive.

Symmetry: If l 1 is parallel to l 2, then l 2 is parallel to l 1, so the relation is symmetric.

Transitivity: If l 1 is parallel to l 2, and l 2 is parallel to l 3, then l 1 is parallel to l 3, so the relation is transitive.

Since all three properties hold, the relation R is an equivalence relation.