Relations and Functions
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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 T with sides 3, 4 and 5?

A is the set of all polygons
R = {(P1, P2) : P1 and P2 have same number of sides }
Since P and P have the same number of sides
∴ (P.P) ∈ R ∀ P ∈ A.
∴ R is reflexive.
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
∴ (P1, P2) ∈ R ⇒ (P2, P1) ∈ R ∴ R is symmetric.
Let (P1, P2) ∈ R and (P2, P3) ∈ R.
⇒ P1 and P2 have the same number of sides and P2 and P3 have same number of sides
⇒ P1 and P3 have the same number of sides
⇒ (P1, P3) ∈ R
∴ (P1, P2), (P2, P3) ∈ R ∈ (P1, P3) ∈ R ∴ R is transitive.
∴ R is an equivalence relation.
Now T is a triangle.
Let P be any element of A.
Now P ∈ A is related to T iff P and T have the same number of sides P is a triangle
required set is the set of all triangles in A.

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