# Problem 13 Solution

This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 13 Solution. Solutions for other problems are available at Exercise 1.1 Solutions
Exercise 1.1 Problem 13 Solution
13. Show that the relation R defined in the set A of all polygons as R = {(P1, P1) : 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?
The given relation is R = {(P1, P1) : P1 and P2 have same number of sides}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a)R, for every aA.
Any polygon L has the same number of sides as itself.
⇒ (L, L) ∈ R
R is reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2)R implies that (a_2, a_1)R, for all a_1, a_2A
If a polygon L has the same number of sides as the polygon M, then the polygon M also has the same number of sides as the polygon L
⇒ If (L, M) ∈ R then (M, L) ∈ R
∴ Relation R is symmetric
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2)R and (a_2, a_3)R implies that (a_1, a_3)R, for all a_1, a_2, a_3A
If a polygon L has the same number of sides as the polygon M and polygon M has the same number of sides as the polygon N, then the poloygon L has the same number of sides as the polygon N.
⇒ If (L, M) ∈ R and (M, N) ∈ R then (L, N) ∈ R
R is transitive
∴ As the relation R is reflexive, symmetric and transitive, the relation R 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?
A triangle is also a polygon with 3 sides, the elements in A related to this triangle are all the polygons that have 3 sides i.e. all the triangles.
So, the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5 is the set of all the triangles (Note here that the triangle being right-angled and the length of the sides is redundant or not useful information)