Let R be the relation over the set A of all straight lines in a plane such that $l_{1} \mathrm{R} l_{2} \Leftrightarrow l_{1}$ is parallel to $l_{2}$. Then R is :

Let $R$ be the relation over the set $A$ of all straight lines in a plane such that $l_{1} R l_{2} \Leftrightarrow l_{1}$ is parallel to $l_{2}$.

To determine what kind of relation $R$ is, I need to check its properties.

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Checking reflexivity:

Is every line parallel to itself?

Yes, any straight line $l$ is parallel to itself, so $l R l$ is true for all lines.

Therefore, $R$ is reflexive.

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Checking symmetry:

If line $l_1$ is parallel to line $l_2$, is line $l_2$ parallel to line $l_1$?

Yes, if $l_1$ is parallel to $l_2$, then $l_2$ is also parallel to $l_1$ by definition.

Therefore, $R$ is symmetric.

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Checking transitivity:

If line $l_1$ is parallel to line $l_2$, and line $l_2$ is parallel to line $l_3$, is line $l_1$ parallel to line $l_3$?

Yes, this is a fundamental property of parallel lines.

Therefore, $R$ is transitive.

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Since relation $R$ is reflexive, symmetric, and transitive, $R$ is an equivalence relation.

Answer: The relation $R$ is an equivalence relation.