Let $L$ be the set of all lines in $XY$ plane and $R$ be the relation in $L$ defined as $R =\{\left( L _{1}, L _{2}\right): L _{1} $ is parallel to $L _{2}\} .$ Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y=2 x+4$

$R =\{\left( L _{1}, L _{2}\right):$ L1 is parallel to $L _{2}\}$

$R$ is reflexive as any line $L_1$ is parallel to itself i.e., $ (L_{1},\, L _{1})\, \in R$

Now, let $\left( L _{1}, \,L _{2}\right) \in R$

$\Rightarrow L _{1}$ is parallel to $L _{2}\, \Rightarrow \,L _{2}$ is parallel to $L _{1}$.

$\Rightarrow \left( L _{2}, \,L _{1}\right) \in R$

$\therefore \,R$ 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 $L _{3}$

$\therefore R$ is transitive.

Hence, $R$ is an equivalence relation.

The set of all lines related to the line $y=2 x+4$ is the set of all lines that are parallel to the line $y=$ $2 x+4$

Slope of line $y=2 x+4$ 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 $c \in R$

Hence, the set of all lines related to the given line is given by $y=2 x+c,$ where $c \in R$