# Let L be the set of all lines in XY - planes and R be the relation in L defined as R = $${ (L_1, L_2 ) : L_1 \: is \: parallel\: to\: L_2}$$. Show the R is an equivalence relation. Find the set of all the lines related to the line $$y=2x+4.$$

Solution :
Let L be the given set of all lines in a plane.
Reflexive : For each line $l \in L$ we have
$l \parallel l \to L \parallel L$
Hence R is reflexive
Symmetric : Let $L_1$ and $L_2 \in L$, Such that $(L_1,L_2) \in R$
Then $(L_1,L_2) \in R => L_1 \parallel L_2$ and $L_2 \parallel$
Hence they are symmetric on L
Transitive let $L_1,L_2,L_3 \in L$ Such that
$(L_1,L_2) \in R => L_1 \parallel L_2$ and $L_2 \parallel L_3$
$\qquad => L_1 \parallel L_3$
$\qquad=> (L_1,L_2) \in R$
Hence R is transitive on L
Given line $y= 2x+4$ such that $(x,y ) \in R$ , where $y= 2x+4$ is a straight line
Where C is the y intercept .
$y= 2x+4$ is any straight line card hence the set of all straight lines related to the above line will differ only by the Y intercept 'C'
Hence set of all the lines related to the line $y= 2x+4$ is $\{y=2x+c;c \in R\}$