Let L be the set of all lines in XY plane and R be the relation in L defined as `R={(L_1,""""L_2): L_1(" i s p a r a l l e l t o L")_2}` . Show that R is an equivalence relation. Find the set of all lines related to the line `y" "=" "2x" "+" "4` .

Explanation

Here L = set of all lines in XY - plane and
`R = {( L_1, L_2): L_1` is parallel to `L_2}`
`therefore ` Every line is parallel to itself.
`therefore R` is reflexive.
Let `L_1, L_2 in L and (L_1, L_2) in R`
`rArr L_1` is parallel to `L_2`.
`rArr L_2 ` is parallel to `L_1`.
`rArr (L_2, L_1) in R`
`therefore R` is symmetric.
Let `L_1, L_2, L_3 in L and (L_1, L_2) in R and (L_2, L_3) in R`
`rArr L_1` is parallel to `L_2 and L_2` is parallel to `L_3`.
`rArr L_1` is parallel to `L_3`.
` rArr (L_1, L_3) in R`
`therefore R` is transitive.
`because R` is reflexive, symmetric and transtive.
`therefore R` is an equivalence relation.
Slope of given line `y = 2x + 4 ` is `=2`
`therefore ` A line parallel to this line is
`" " y = 2x +c`
Where c = constant.