# 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?

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 which have 3 sides.

Therefore, the set of all elements in A related to triangle T is the set of all triangels.

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