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**NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 13 Solution**. Solutions for other problems are available at Exercise 1.1 SolutionsExercise 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 a ∈ A.

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_2 ∈ A

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_3 ∈ A

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)