Show that the relation R defined in the set A of all polygons as `R={(P_1,P_2):P_1( and P)_2 h a v e s a m e n u m b e r o f s i d e s}`, is an equivalence relation. What is the set of all elements in A related to the right angle triangle

Explanation

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.