Question

Let L be the set of all the 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 the lines related to the line y = 2x + 4.

Answer 1

L = Set of all the lines in XY-plane.

R = {(L1, L2): L1 is parallel to L2}

(i) (a) L1 is parallel to itself ⇒ R is reflexive.

(b) L1 is parallel to L2 ⇒ L2 is parallel to L1.

⇒ R is symmetric.

(c) Let L1 is parallel to L2 and L2 is parallel to L3. Obviously, L1 is parallel to L3.

⇒ R is transitive.

Hence, R is an equivalence relation.

(ii) Set of parallel lines related to y = 2x + 4 is y = 2x + c, where c is an arbitrary constant.