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.
It is given that the relation in L defined as
R = {(L1, L2): L1 is parallel to L2}
R is reflexive as any line L1 is parallel to itself
⇒ (L1, L2) ϵ R
Now, Let (L1, L2) ϵ R
⇒ L1 is parallel to L2.
⇒ L2 is parallel to L1.
⇒ (L2, L1) ϵ R
Therefore, R is symmetric.
Now, Let (L1, L2), (L2, L3) ϵ R
⇒ L1 is parallel to L2. Also, L2 is parallel to L3.
⇒ L1 is parallel to L3.
⇒ (L1, L3) ϵ R
Therefore, R is transitive.
Therefore, 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 m = 2
We know that parallel lines have the same slopes.
The line parallel to the given line is of the form y = 2x + c where, c ϵ R.
Therefore, the set of all lines related to the given line by y = 2x + c, where c ϵ R.