Question

Let `L` be the set of all lines in `X Y=p l a n e` and `R` be the relation in `L` defined as `R={(L_1,L_2): L_1` is parallel to `L_2}dot` Show that `R` is an equivalence relation. Find the set of all lines related to the line `y=2x+4.`

Answer 1

`R = {(L1, L2): L1` is parallel to `L2}`.
`R` is reflexive as any line `L1` is parallel to itself, `(L1, L1) in R`.
Now,
Let `(L1, L2) in R`.
`=> L1` is parallel to `L2` and `L2` is parallel to ` L1`
`=> (L2, L1) in R`.
`:. R` is symmetric.
Now,
Let `(L1, L2), (L2, L3) in R`
`L1` is parallel to `L2`. Also, `L2` is parallel to `L3`.
`:.L1` is parallel to `L3`.
`:. R` is transitive.
As `R` is reflexive, symmetric and transitive, `R` is an equivalence relation.
The set of all lines related to the line `y = 2x + 4` is the set of all lines that are parallel to the line `y = 2x + 4`.
Slope of line `y = 2x + 4` is `2`.
We know that parallel lines have the same slopes.