Question

Show that the relation R, defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have the same number of sides} is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Answer 1

Let n be the number of sides of a polygon P1.

R = {(P1, P2): P1 and P2 are n sided polygons}.

(i) (a) Any polygon P1 has n sides ⇒ R is reflexive.

(b) If P1 has n sides, P2 also has n sides, then if P2 has n sides, P1 also has n sides.

⇒ R is symmetric.

(c) Let P1, P2; P2, P3 are pairs of n sided polygons. Thus, P1 and P3 are also n sided polygons.

⇒ R is transitive.

Hence, R is an equivalence relation.

(ii) The set A = set of all the triangles in a plane.