Show that the relation R defined in the set A of all polygons as $R = {(P_{1}, P_{2}) : P_{1} a n d P_{2}$ 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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Solution

$R =$ { $(P_{1}, P_{2})$ : $P_{1}$ and $P_{2}$ have same number of sides}
Reflexivity :
$P_{1}$ & $P_{2}$ are the same polygon.
So, $P_{1}$ & $P_{2}$ have the same number of sides.
Therefore,
$(P_{1}, P_{1}) \in R$
So, R is reflexive.

Symmetry :
If $P_{1}$ & $P_{2}$ have the same number of sides, then $P_{2}$ & $P_{1}$ have the same number of sides.
So, if $(P_{1}, P_{2}) \in R$ , then $(P_{2}, P_{1}) \in R$ .
So, R is symmetric.

Transitivity :
If $P_{1}$ & $P_{2}$ have the same number of sides, and $P_{2}$ & $P_{3}$ have the same number of sides, then $P_{1}$ & $P_{3}$ have the same number of sides.
So, if $(P_{1}, P_{2}) \in R$ & $(P_{2}, P_{3}) \in R$ , then $(P_{1}, P_{3}) \in R$ .
So, R is transitive.

Hence, R is an equivalence relation.
The set of all elements in A related to triangle T is the set of all triangles.

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