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.