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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?



Class 12, NCERT Chapter 1,  Exercise 1.1, Q13


It is given that the relation R defined in the set A of all polygons as
R = {(P1, P2): P1 and P2 have same number of sides},
Then, R is reflexive since (P1, P2) ϵ R as the same polygon has the same number of sides with itself.
Let (P1, P2) ϵ R
⇒ P1 and P2 have the same number of sides.
⇒ P2 and P1 have the same number of sides.
⇒ (P2, P1) ϵ R
Therefore, R is symmetric.

Now, let (P1, P2), (P2, P3) ϵ R
⇒ 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.
⇒ (P1, P3) ϵ R
Therefore, R is transitive.

Thus, R is an equivalence relation.

The elements in A related to the right-angled triangle (T) with sides 3, 4 and 5 are those polygons that have 3 sides.
Therefore, the set of all elements in A related to triangle T is the set of all triangles.
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