Let $L$ be the set of all lines in $X Y$ plane and $R$ be the relation in $L$ defined as $R=\left\{\left(L_{1}, L_{2}\right)\right.$ : $L_{1}$ is parallel to $\left.L_{2}\right\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y=2 x+4$.
$R=\left\{\left(L_{1}, L_{2}\right): L_{1}\right.$ is parallel to $\left.L_{2}\right\}$
$R$ is reflexive as any line $L_{1}$ is parallel to itself i.e., $\left(L_{1}, L_{1}\right) \in R$.
Now,
Let $\left(L_{1}, L_{2}\right) \in \mathrm{R}$.
$\Rightarrow L_{1}$ is parallel to $L_{2}$.
$\Rightarrow L_{2}$ is parallel to $L_{1}$.
$\Rightarrow\left(L_{2}, L_{1}\right) \in \mathrm{R}$
∴ R is symmetric.
Now,
Let $\left(L_{1}, L_{2}\right),\left(L_{2}, L_{3}\right) \in R$.
$\Rightarrow L_{1}$ is parallel to $L_{2}$. Also, $L_{2}$ is parallel to $L_{3}$.
$\Rightarrow L_{1}$ is parallel to $L_{3}$.
∴R is transitive.
Hence, R is an equivalence relation.
The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x + 4.
Slope of line y = 2x + 4 is m = 2
It is known that parallel lines have the same slopes
The line parallel to the given line is of the form $y=2 x+c$, where $c \in \mathbf{R}$.
Hence, the set of all lines related to the given line is given by $y=2 x+c$, where $c \in \mathbf{R}$.