# 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 $$5$$?

Toolbox:
• A relation R in a set A is a equivalance relation if it is symmetric, reflexive and transitive.
• A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for every $\; a\in\;A$
• A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$
• If a triangle $T_1$ is similar to $T_2$ then $T_2$ will be similar to $T_1$
Given $R=\{(P_1,P_2):$ $P_1$ and$\; P_2$ have same number of sides$\}$:
For any Polygon w/ $P$ number of sides, $(P,P)\in R$ as same polygon has same number of sides with itself. Hence $R$ is reflexive.
If $P_1$ has same number of sides as $P_2$ the $P_2$ also has same number of sides as $P_1$
Hence it follows that $(P_1,P_2) \in R \Rightarrow (P_2P_1) \in R$ and $R$ is symmetric.
If $P_1$ has same number of sides as $P_2$ and $P_2$ has same number of sides as $P_3$ then $P_1$ and $P_3$ have the same number of slides.
Hence it follows that $R$ is transitive also.
Therefore, $R$ is an equivalence relation since its reflexive, symmetric and transitive.
Consider a rightangle triangle $T$ with the sides $3,4,5$. Since $T$ is a polygon with 3 sides, the the set of all elements in $A$ related to the right angle triangle $T$ is the set of all triangles.
edited Mar 8, 2013