Given in set of all lines in XY plane, $R=\{(L_1,L_2):L_1$ is parallel to $L_2\}$:

Since every line is parallel to itself, for any line $L$, $(L_1L_1) \in R$. Hence $R$ is reflexive.

If a line $L_1$ is parallel to another line $L_2$, the converse is true. i.e., $L_2$ is parallel to $L_1$.

Hence $ (L_1L_2) \in R \Rightarrow (L_2L_4) \in R$. Therefore $R$ is symmetric.

If a line $L_1$ is parellel to another line $L_2$ and $L_2$ is parallel to line $L_3$, then $L_1$ and $L_3$ must all be parallel. Therefore $R$ is transitive.

Hence $R$ is an equivalence relation as its reflexive, symmetric and transitive.

Since the given line $y=2x+4$ is of the form $y=mx+c \rightarrow$ slope $m=2$

We know that all parallel lines have the same slope. Therefore, the set of lines related to the given line is the set defined by $y=2x+c$, where $c \in R$.