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 $\{y=2x+c;c \in R\}$