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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 $R=\left\{L_{1}, L_{2}\right): L_{1}$ is parallel to $\left.L_{2}\right\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y=2 x+4$.

Solution:

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 $c \in R\}$