Let L be the set of all lines in XY plane and R be the relation in L defined as R

Question:

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 $y=2 x+4$.

Solution:

$R=\left\{\left(L_{1}, L_{2}\right): L_{1}\right.$ 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 $c \in \mathbf{R}$.