Solution (a)
Step 1 (Definitions and Governing Laws):
- A relation $R$ on a set $L$ is symmetric if $(l_1, l_2) \in R \implies (l_2, l_1) \in R$.
- A relation $R$ on a set $L$ is transitive if $(l_1, l_2) \in R$ and $(l_2, l_3) \in R \implies (l_1, l_3) \in R$.
- Two lines are parallel if they have the same slope ($m_1 = m_2$).
Step 2 (Step-by-Step Working):
(i) Symmetric: Let $(l_1, l_2) \in R$. This implies $l_1 \parallel l_2$. Since parallelism is commutative, $l_2 \parallel l_1$. Therefore, $(l_2, l_1) \in R$. Hence, $R$ is symmetric.
(ii) Transitive: Let $(l_1, l_2) \in R$ and $(l_2, l_3) \in R$. This implies $l_1 \parallel l_2$ and $l_2 \parallel l_3$. By the property of parallel lines, if $l_1$ is parallel to $l_2$ and $l_2$ is parallel to $l_3$, then $l_1 \parallel l_3$. Therefore, $(l_1, l_3) \in R$. Hence, $R$ is transitive.
(iii) Set of related lines: The given line is $y = 3x + 2$. Any line parallel to this must have the same slope $m = 3$. Thus, the set of all lines related to $y = 3x + 2$ is the set of all lines of the form $y = 3x + c$, where $c \in \mathbb{R}$.
Step 3 (Final Answer):
The set of rail lines is: $\{y = 3x + c : c \in \mathbb{R}\}$.
Solution (b)
Step 1 (Definitions):
Relation $S = \{(l_1, l_2) : l_1 \perp l_2\}$.
Step 2 (Step-by-Step Working):
- Symmetric: If $l_1 \perp l_2$, then $l_2 \perp l_1$. Since the perpendicularity condition is mutual, $(l_1, l_2) \in S \implies (l_2, l_1) \in S$. Thus, $S$ is symmetric.
- Transitive: If $l_1 \perp l_2$ and $l_2 \perp l_3$, then $l_1$ and $l_3$ are both perpendicular to $l_2$. This implies $l_1 \parallel l_3$. Therefore, $(l_1, l_3) \notin S