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Let L be the ...

Question

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

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Solution

Here, $R = {(L_{1}, L_{2}) : L_{1} i s p a r a l l e l t o L_{2}}$
R is reflexive as any line $L_{1}$ is parallel to itself i.e., $(L_{1}, L_{1}) \in R .$
Now, let $(L_{1}, L_{2}) \in R$
$\Rightarrow L_{1}$ is parallel to $L_{2} \Rightarrow L_{2}$ is parallel to $L_{1} \Rightarrow (L_{2}, L_{1}) \in R$
Therefore, R is symmetric.
Now, let $(L_{1}, L_{2}), (L_{2}, L_{3}) \in R$
$\Rightarrow L_{1}$ is parallel to $L_{2}$ , also $L_{2}$ is parallel to $L_{3} .$
$\Rightarrow L_{1}$ is parallel to $L_{3}$
$\Rightarrow (L_{1}, L_{3}) \in R$ . Therefore, 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 slope.
The line parallel to the given line is of the form y=2x+c, where $c \in R .$
Hence, the set of all lines related to the given line is given by y=2x+c, where $c \in R .$

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