Let $L$ be the set of all lines in the $XY$ plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.
(D) $1$. Reflexivity: For any line $L_1 \in L$,$L_1$ is parallel to itself. Thus,$(L_1, L_1) \in R$. So,$R$ is reflexive.
$2$. Symmetry: Let $(L_1, L_2) \in R$. This means $L_1$ is parallel to $L_2$. Since $L_1 \parallel L_2$ implies $L_2 \parallel L_1$,we have $(L_2, L_1) \in R$. So,$R$ is symmetric.
$3$. Transitivity: Let $(L_1, L_2) \in R$ and $(L_2, L_3) \in R$. This means $L_1 \parallel L_2$ and $L_2 \parallel L_3$. Since $L_1 \parallel L_2$ and $L_2 \parallel L_3$ implies $L_1 \parallel L_3$,we have $(L_1, L_3) \in R$. So,$R$ is transitive.
Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.
$4$. Set of lines: The set of all lines related to $y = 2x + 4$ consists of all lines parallel to it. Parallel lines have the same slope. The slope of $y = 2x + 4$ is $m = 2$. Thus,any line parallel to it is of the form $y = 2x + c$,where $c \in \mathbb{R}$.
$2$. Symmetry: Let $(L_1, L_2) \in R$. This means $L_1$ is parallel to $L_2$. Since $L_1 \parallel L_2$ implies $L_2 \parallel L_1$,we have $(L_2, L_1) \in R$. So,$R$ is symmetric.
$3$. Transitivity: Let $(L_1, L_2) \in R$ and $(L_2, L_3) \in R$. This means $L_1 \parallel L_2$ and $L_2 \parallel L_3$. Since $L_1 \parallel L_2$ and $L_2 \parallel L_3$ implies $L_1 \parallel L_3$,we have $(L_1, L_3) \in R$. So,$R$ is transitive.
Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.
$4$. Set of lines: The set of all lines related to $y = 2x + 4$ consists of all lines parallel to it. Parallel lines have the same slope. The slope of $y = 2x + 4$ is $m = 2$. Thus,any line parallel to it is of the form $y = 2x + c$,where $c \in \mathbb{R}$.
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