Question

Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have 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

Given , R = R = {(P1, P2): P1 and P2 have same number of sides}
here , (P1, P2) ∈ R as the same polygon has the same number of of sides with itself.
so, R is reflexive .

Let (P1 , P2)∈ R
then, P1 and P2 have the same number of sides,
P2 and P1 have the same number of sides,
so, (P2, P1) ∈ R
therefore, R is symmetric.
now,
Let (P1,P2), (P2,P3) ∈ R
here, P1 and P2 have the same number of sides, also P2 and P3 have the same number of sides.
=> P1 and P3 have the same number of sides.
e.g., (P1,P3) ∈ R
so, R is transitive.
hence, R is an equivalence relation.

The elements in A related to the right - angled triangle (To) with sides 3, 4, 5 are those polygon which have 3 sides.(because T is a polygon with 3 sides.)
hence, the set of all elements in A related to triangle T is the set of all triangles.