Question

Show that the relation R, defined on the set A of all polygons as
R = {(P₁, P₂) : P₁ and P₂ 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?

Answer 1

We observe the following properties on R.

$$\begin{gathered} \mathrm{Reflexivity}:\mathrm{Let}{P}_1\mathrm{be}\mathrm{an}\mathrm{arbitrary}\mathrm{element}\mathrm{of}A. \\ \mathrm{Then},\mathrm{polygon}{P}_1\mathrm{and}{P}_1\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{number}\mathrm{of}\mathrm{sides},\mathrm{since}\mathrm{they}\mathrm{are}\mathrm{one}\mathrm{and}\mathrm{the}\mathrm{same}. \\ \Rightarrow \left(P_1,P_1\right)\in R\mathrm{for}\mathrm{all}{P}_1\in A \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}A. \\ \mathrm{Symmetry}:\mathrm{Let}\left(P_1,P_2\right)\in R \\ \Rightarrow P_1\mathrm{and}{P}_2\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{number}\mathrm{of}\mathrm{sides}. \\ \Rightarrow P_2\mathrm{and}{P}_1\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{number}\mathrm{of}\mathrm{sides}. \\ \Rightarrow \left(P_2,P_1\right)\in R\mathrm{for}\mathrm{all}{P}_1,P_2\in A \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}A. \\ \mathrm{Transitivity}:\mathrm{Let}\left(P_1,P_2\right),\left(P_2,P_3\right)\in R \\ \Rightarrow P_1\mathrm{and}{P}_2\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{number}\mathrm{of}\mathrm{sides}\mathrm{and}{P}_2\mathrm{and}{P}_3\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{number}\mathrm{of}\mathrm{sides}. \\ \Rightarrow P_1,P_2\mathrm{and}{P}_3\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{number}\mathrm{of}\mathrm{sides}. \\ \Rightarrow P_1\mathrm{and}{P}_3\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{number}\mathrm{of}\mathrm{sides}. \\ \Rightarrow \left(P_1,P_3\right)\in R\mathrm{for}\mathrm{all}{P}_1,P_{3}A \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}A. \end{gathered}$$

Hence, R is an equivalence relation on the set A.

Also, the set of all the triangles$\in$A is related to the right angle triangle T with the sides 3, 4, 5.