Question

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L 1, L2): L1 is parallel to _L 2 _}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Answer 1

R = {(L 1, L2): L 1 is parallel to _L 2 _}
R is reflexive as any line L 1 is parallel to itself i.e., (_L 1, L1 _) ∈ R.
Now,
Let (_L 1, L2 _) ∈ R.
⇒ L 1 is parallel to L 2.
⇒ L2 is parallel to L 1.
⇒ (_L 2, L1 _) ∈ R
∴ R is symmetric.

Now,
Let (_L 1, L2 _), (_L 2, L3 _) ∈R.
⇒ L 1 is parallel to L 2. Also, L 2 is parallel to L 3.
⇒ L 1 is parallel to L 3.
∴R is transitive.

Hence, 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
It is known that parallel lines have the same slopes.
The line parallel to the given line is of the form y = 2x + c, where c ∈R.

Hence, the set of all lines related to the given line is given by y = 2x + c, where c ∈ R.