Problem Statement
Let $L$ be the set of all straight lines in the $xy$-plane and $R$ be the relation in $L$ defined by $R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.
Verified Solution & Marking Scheme
Check Reflexivity
Every line $L_1$ is parallel to itself ($L_1 \parallel L_1$).
Therefore, $(L_1, L_1) \in R$ for all $L_1 \in L$. Hence, $R$ is reflexive.
Check Symmetry and Transitivity
**Symmetry:** If $(L_1, L_2) \in R$, then $L_1 \parallel L_2 \implies L_2 \parallel L_1 \implies (L_2, L_1) \in R$. Hence, $R$ is symmetric.
**Transitivity:** If $(L_1, L_2) \in R$ and $(L_2, L_3) \in R$, then $L_1 \parallel L_2$ and $L_2 \parallel L_3 \implies L_1 \parallel L_3 \implies (L_1, L_3) \in R$. Hence, $R$ is transitive.
Since $R$ is reflexive, symmetric, and transitive, it is an equivalence relation.
Find equivalence class of y = 2x + 4
Lines related to $y = 2x + 4$ are lines parallel to it, which must have the same slope $m = 2$.
Thus, the set of all related lines is given by:
$\{ y = 2x + c : c \in \mathbb{R} \}$