# Let $$L$$ be the set of all lines in $$XY$$ plane and $$R$$ be the relation in $$L$$ defined as $$R = \{(L1, L2) : L1\:$$ is parallel to $$L2\}$$. Show that $$R$$ is an equivalence relation. Find the set of all lines related to the line $$y = 2x + 4.$$

Toolbox:
• A relation R in a set A is a equivalance relation if it is symmetric, reflexive and transitive.
• A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for every $\; a\in\;A$
• A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$
• All parallel lines have the same slope.
Given in set of all lines in XY plane, $R=\{(L_1,L_2):L_1$ is parallel to $L_2\}$:
Since every line is parallel to itself, for any line $L$, $(L_1L_1) \in R$. Hence $R$ is reflexive.
If a line $L_1$ is parallel to another line $L_2$, the converse is true. i.e., $L_2$ is parallel to $L_1$.
Hence $(L_1L_2) \in R \Rightarrow (L_2L_4) \in R$. Therefore $R$ is symmetric.
If a line $L_1$ is parellel to another line $L_2$ and $L_2$ is parallel to line $L_3$, then $L_1$ and $L_3$ must all be parallel. Therefore $R$ is transitive.
Hence $R$ is an equivalence relation as its reflexive, symmetric and transitive.
Since the given line $y=2x+4$ is of the form $y=mx+c \rightarrow$ slope $m=2$
We know that all parallel lines have the same slope. Therefore, the set of lines related to the given line is the set defined by $y=2x+c$, where $c \in R$.
answered Feb 22, 2013 by
edited Mar 8, 2013