Let $L$ be the set of all lines in $X Y$-plane and $R$ be the relation in $L$ defined as $R=\left\{L_{1}, L_{2}\right): L_{1}$ is parallel to $\left.L_{2}\right\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y=2 x+4$.
We observe the following properties of R.
Reflexivity: Let $L_{1}$ be an arbitrary element of the set $L$. Then,
$L_{1} \in L$
$\Rightarrow L_{1}$ is parallel to $L_{1}$ [Every line is parallel to itself]
$\Rightarrow\left(L_{1}, L_{1}\right) \in R$ for all $L_{1} \in L$
So, $R$ is reflexive on $L$.
Symmetry : Let $\left(L_{1}, L_{2}\right) \in R$
$\Rightarrow L_{1}$ is parallel to $L_{2}$
$\Rightarrow L_{2}$ is parallel to $L_{1}$
$\Rightarrow\left(L_{2}, L_{1}\right) \in R$ for all $L_{1}$ and $L_{2} \in L$
So, $R$ is symmetric on $L$.
Transitivity : Let $\left(L_{1}, L_{2}\right)$ and $\left(L_{2}, L_{3}\right) \in R$
$\Rightarrow L_{1}$ is parallel to $L_{2}$ and $L_{2}$ is parallel to $L_{3}$
$\Rightarrow L_{1}, L_{2}$ and $L_{3}$ are all parallel to each other
$\Rightarrow L_{1}$ is parallel to $L_{3}$
$\Rightarrow\left(L_{1}, L_{3}\right) \in R$
So, $R$ is transitive on $L$.
Hence, R is an equivalence relation on L.
Set of all the lines related to $y=2 x+4$
$=L^{\prime}=\{(x, y): y=2 x+c$, where $c \in R\}$