Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I1₁, I₂):I₁ is parallel to I₂}. What is the type of given relation?

The correct option is Equivalence relation.

To determine the type of the relation $R$ defined on the set $L$ (noting a typographical correction from "I" to $L$ for the set of lines and $L_1,L_2$ for the lines) in a two-dimensional $XY$ plane as:
$R=\left\{\right(L_1,L_2):L_1\text{is parallel to}{L}_2\}$,
we need to check if the relation satisfies three properties: reflexivity, symmetry, and transitivity. An equivalence relation is one that satisfies all three properties.

Step 1: Reflexivity
A relation $R$ on a set $L$ is reflexive if every element is related to itself, i.e., $(L_1,L_1)\in R$ for all $L_1\in L$.
Since any line $L_1$ is always parallel to itself (coincident lines are considered parallel as they share the same slope), we have:
$L_1||L_1$
Therefore, $(L_1,L_1)\in R$. The relation is reflexive.

Step 2: Symmetry
A relation $R$ is symmetric if $(L_1,L_2)\in R$ implies $(L_2,L_1)\in R$.
If a line $L_1$ is parallel to a line $L_2$, then it is geometrically true that $L_2$ is also parallel to $L_1$. Written mathematically:
$L_1||L_2\Rightarrow L_2||L_1$
Thus, if $(L_1,L_2)\in R$, then $(L_2,L_1)\in R$. The relation is symmetric.

Step 3: Transitivity
A relation $R$ is transitive if $(L_1,L_2)\in R$ and $(L_2,L_3)\in R$ implies $(L_1,L_3)\in R$.
Let $L_1,L_2,L_3$ be three lines in the plane. If $L_1$ is parallel to $L_2$, and $L_2$ is parallel to $L_3$, then $L_1$ must be parallel to $L_3$.
$L_1||L_2\text{and}{L}_2||L_3\Rightarrow L_1||L_3$
Thus, $(L_1,L_3)\in R$. The relation is transitive.

Since the relation $R$ is reflexive, symmetric, and transitive, it is defined as an equivalence relation.