Let L be the set of all lines in XY-plane and R be the relation in L defined as $R = {(L_{1}, L_{2}) : L_{1} i s p a r a l l e l t o L_{2}}$ . Show that R is an equivalence relation. Find the set of all lines related to the line $y = 2 x$

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Solution

Since every line $l \in L$ is parallel to itself, therefore $(l, l) \in R \forall l \in L .$
$∴$ R is reflexive
$(L_{1}, L_{2}) \in R \Rightarrow L_{1} | | L_{2}$
$\Rightarrow (L_{2}, L_{1}) \in R .$
$∴ R$ is symmetric.
Next, let $(L_{1}, L_{2}) \in R$ and also $(L_{2}, L_{1}) \in R$
$\Rightarrow L_{1} | | L_{2} a n d L_{2} | | L_{3}$
$\Rightarrow L_{1} | | L_{3} \Rightarrow (L_{2}, L_{3}) \in R$
$∴ R$ is transitive.
Hence, R is an equivalence relation. Required set
={ 1:1 is a line related to the line $y = 2 x + y$ }={ 1:1 is a line parallel to $y = 2 x + 4$ }
={ 1:1 is a line whose equation is $y = 2 k + k,$ k being any real.}

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