Show that the relation R defined in the set A of all polygons as if $R = \{ ({P_1},\;{P_2}):{P_1}$ and ${P_2}$ have same number of sides $\}$, is an equivalence relation. What is the set of the elements in A related to the right angle triangle T with sides 3, 4 and 5 ?

R $= \{(P,, P2) : P,$ and P2 have same number of sides$\}.$

(i) Reflexive

Let ${P_1} \in A.$ Consider the element$({P_1},\;{P_1}).$ It shows that ${P_1}$ and ${P_1}$ have same number of sides.
${P_1}$ have same number of sides.

$\Rightarrow$ $({P_1},\;{P_2}) \in R.$ Hence, R is reflexive.

(ii) Symmetric

Let ${P_1},\;{P_1} \in A.\;\;\;If\;\;({P_1},\;{P_2}) \in R,$

$\Rightarrow$ ${P_1}$ and ${P_2}$ have same number of sides.
$\Rightarrow$ ${P_2}$ and ${P_1}$ have same number of sides

$\Rightarrow$ $({P_2},\;{P_1}) \in R \Rightarrow R$ is symmetric.

(iii) Transitive

Let ${P_1},\;{P_2},\;{P_3} \in A,\;\;\;If\;\;({P_1},\;{P_2}) \in R$ and $({P_2},\;{P_3}) \in R,$

$\Rightarrow$ ${P_1}\;\;and\;\;{P_2}$ have same number of sides and ${P_2}$ and ${P_3}$ have same number of sides
$\Rightarrow$ ${P_1}$ and ${P_3}$ have same number of sides $\Rightarrow$ $({P_1},\;{P_3}) \in R$
Thus, R is transitive.

Hence, R is an equivalence relation.

We know that, if 3, 4, 5 are the sides of a triangle, then the triangle is right-angled( Pythagoras triplet) Now, the set of elements in A related to T is the set of right angled triangles.