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.
$R = \{ ({L_1},\;{L_2}):{L_1}$ is parallel to ${L_2}\}$
(i) Reflexive
Let ${L_1} \in L$ ${L_1} \in L \cdot {L_1}\parallel {L_1}\;\;\;i.e.,\;\;\;({L_1},\;{L_1}) \in R.$ Thus R, is reflexive.
(ii) Symmetric
${L_1},\;{L_2} \in L$
Let $({L_1},\;{L_2}) \in R \Rightarrow {L_1}\parallel {L_2} \Rightarrow {L_2}\parallel {L_1} \Rightarrow ({L_2},\;{L_1}) \in R$
Thus, R is symmetric.
(iii) Transitive
${L_1},\;{L_2},\;{L_3} \in L.$ Let $({L_1},\;{L_2}) \in R\;\;\;and\;\;{L_2},\;{L_3} \in R$
$\Rightarrow$ ${L_1}\parallel {L_2}\;\;and\;\;{L_2}\parallel {L_3} \Rightarrow {L_1}\parallel {L_3}$
Thus, R is transitive. Hence, R is equivalence relation.
All lines related to the line y $=$ 2x + 4 are y $=$ 2x + c, where c is a real number.
$L = \{ (y = 2x + 4,\;y = 2x + c):x,\;y \in R\} .$
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