Question

Show that the relation $R$ defined in the set $A$ of all polygons as $R=\left\{\right(P_1,P_2):P_1\text{and}{P}_2\text{have same number of sides}\}$, is an equivalence relation. What is the set of all elements in $A$ related to the right-angle triangle $T$ with sides $3,4\&5$$?$

Answer 1

Checking reflexivity of a relation.
$R=\left\{\right(P_1,P_2):P_1\text{and}{P}_2\text{have same number of sides}\}$
Check reflexive
$P_1\&P_1$ are the same polygon
So, $P_1\&P_1$ have the same number of sides
$\therefore (P_1,P_1)\in R$
So, $R$ is reflexive.

Checking given relation is symmetric or not.
For symmetric,
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 $(P_2,P_1)\in R$
$\therefore R$ is symmetric

Checking transitivity of a relation.
For transitive,
If $P_1\&P_2$ have the same number of sides, and $P_2\&P_3$ have the same number of sides,
So, if $(P_1,P_2\in R\&(P_2,P_3)\in R$ then $(P_1,P_3)\in R$
$\therefore R$ is transitive.
Since, $R$ is reflexive, symmetric and transitive.
Hence, $R$ is an equivalence relation.

Finding the set of all elements in $A$ related to the right-angle triangle $T$ with sides $3,4$ and $5$.
$R=\left\{\right(P_1,P_2):P_1\text{and}{P}_2\text{have same number of sides}\}$
Here, $P_1=T$
So, $(T,P_2)$ are in relation $R$
So, $T\&P_2$ have same number of sides.
So, $P_2$ is set of all triangles.
Hence, the set of all elements in $A$ related to triangle $T$ is the set of all triangles.
Hence, $R$ is an equivalence relation and the set of all elemnets in $A$ related to triangle $T$ is the set of all triangles.