# Show that the relation R, defined on the set A of all polygons asR = {(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?

R = {(P1, P2): P1 and P2 have same the number of sides}

Proof :

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity: For Reflexivity, we need to prove that-

(a, a) R

R is reflexive since (P1, P1) R as the same polygon has the same number of sides with itself.

Symmetric: For Symmetric, we need to prove that-

If (a, b) R, then (b, a) R

Let (P1, P2) R.

P1 and P2 have the same number of sides.

P2 and P1 have the same number of sides.

(P2, P1) R

R is symmetric.

Transitive: For Transitivity, we need to prove that-

If (a, b) R and (b, c) R, then (a, c) R

Now, (P1, P2), (P2, P3) 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) R

R is transitive.

Hence, R is an equivalence relation.

And, now the elements in A related to the right-angled triangle (T) with sides 3, 4 and 5 are those polygons which have three sides (since T is a polygon with three sides).

Hence, the set of all elements in A related to triangle T is the set of all triangles.

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