# Q.13 Show that the relation R defined in the set A of all polygons as $R = \{(P _1 , P _2 ) : P_1 \;and\; P_2 \;have \;same\; number \;of\; sides\}$, 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?

$R = \{(P _1 , P _2 ) : P_1 \;and\; P_2 \;have \;same\; number \;of\; sides\}$

Same polygon has same number of sides with itself,i.e. $(P_1,P_2) \in R$, so it is reflexive.

Let,

$(P_1,P_2) \in R$  i.e.P1 have same number of sides as  P2

P1 have same number of sides as Pis same as P2 have same number of sides as P1 i.e. $(P_2,P_1) \in R$

Hence,it is symmetric.

Let,

$(P_1,P_2) \in R$  and  $(P_2,P_3) \in R$  i.e. P1 have same number of sides as P2  and Phave same number of sides as P3

$\Rightarrow$ Phave same number of sides as P3   i.e. $(P_1,P_3) \in R$

Hence, it is transitive,

Thus,  $R = \{(P _1 , P _2 ) : P_1 \;and\; P_2 \;have \;same\; number \;of\; sides\}$, is an equivalence relation.

The elements in A related to the right angle triangle T with sides 3, 4 and 5 are those polygons which have 3 sides.

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

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