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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?

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Here A = set of all polygons
`and R = { (P_1, P_2) : P_1 and P_2` have same numbers of sides }
`therefore (P, P) in R` is true.
`rArr R` is reflexive.
Let `P_1, P_2 in A and (P_1, P_2) in R`
`rArr P_1 and P_2 ` have same number of sides.
`rArr P_2 and P_1` have same number of sides.
`rArr " " (P_2, P_1) in R`
`therefore R ` is symmetric.
Let `P_1, P_2, P_3 in A`
`and (P_1, P_2) in R and (P_2, P_3) in R`
`rArr P_1, P_2` have same niumber of sides and `P_2, P_3` have same number of sides.
`rArr P_1 and P_3` have same number of sides.
`rArr " " (P_1, P_3) in R`
` therefore R` is transitive.
`because R` is reflexive, symmetric and transitive.
`therefore R` is an equivalence relation.
Now that polygon will be related to a right angled triangles of sides 3, 4 and 5 which have three sides i.e., the polygon is a triangle.
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