Let R be the relation over the set A of all straight lines in a plane such that l₁ Rl₂ ⇐⇒ l1 is parallel to l₂. Then R is:

The correct option is: An Equivalence relation.

To determine the nature of the relation $R$ defined on the set $A$ of all straight lines in a plane, we need to test if it satisfies the properties of reflexivity, symmetry, and transitivity.

1. Reflexivity:
A relation is reflexive if every element is related to itself. For the relation $R$ to be reflexive, every line $l$ in the set $A$ must satisfy:

$lRl$

By definition, a line is always parallel to itself. Therefore, $l$ is parallel to $l$ is always true. Thus, the relation is reflexive.

2. Symmetry:
A relation is symmetric if for any two elements, if the first is related to the second, then the second is also related to the first. For any two lines l₁ and l₂ in $A$, if:

$l_{1}R{l}_2$

then l₁ is parallel to l₂. This implies that l₂ is also parallel to l₁. Hence:

$l_{2}R{l}_1$

This shows that the relation is symmetric.

3. Transitivity:
A relation is transitive if, for any three elements, when the first is related to the second and the second is related to the third, the first is also related to the third. For any three lines l₁, l₂, and l₃ in $A$, if:

$l_{1}R{l}_2\text{and}{l}_{2}R{l}_3$

then l₁ is parallel to l₂, and l₂ is parallel to l₃. Consequently, l₁ must be parallel to l₃. Thus, we have:

$l_{1}R{l}_3$

This shows that the relation is transitive.

Conclusion:
Since the relation $R$ is reflexive, symmetric, and transitive, it satisfies all the conditions of an equivalence relation.