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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?

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?

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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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