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Show that the relation $\mathrm{R}$ defined in the set A of all polygons as $R=\left\{\left(P_1, P_2\right): P_1\right.$ 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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Let $\mathrm{n}$ be the number of sides of polygon $\mathrm{P}_1$. $\mathrm{R}=\left\{\left(\mathrm{P}_1, \mathrm{P}_2\right): \mathrm{P}_1\right.$ and $\mathrm{P}_2$ are $\mathrm{n}$ sides polygons $\}$
(i) (a) Any polygon $\mathrm{P}_1$ has $\mathrm{n}$ sides $\Rightarrow \mathrm{R}$ is reflexive
(b) If $\mathrm{P}_1$ has $n$ sides, $\mathrm{P}_2$ also has $\mathrm{n}$ sides then if $\mathrm{P}_2$ has $\mathrm{n}$ sides $\mathrm{P}_1$ also has $\mathrm{n}$ sides.
$\Rightarrow \mathrm{R}$ is symmetric.
(c) Let $\mathrm{P}_1, \mathrm{P}_2 ; \mathrm{P}_2, \mathrm{P}_3$ are $\mathrm{n}$ sided polygons. $\mathrm{P}_1$ and $\mathrm{P}_3$ are also $n$ sided polygons.
$\Rightarrow \mathrm{R}$ is transitive. Hence $\mathrm{R}$ is an equivelance relation.
(ii) The set $\mathrm{A}=$ set of all the triangles in a plane.
(i) (a) Any polygon $\mathrm{P}_1$ has $\mathrm{n}$ sides $\Rightarrow \mathrm{R}$ is reflexive
(b) If $\mathrm{P}_1$ has $n$ sides, $\mathrm{P}_2$ also has $\mathrm{n}$ sides then if $\mathrm{P}_2$ has $\mathrm{n}$ sides $\mathrm{P}_1$ also has $\mathrm{n}$ sides.
$\Rightarrow \mathrm{R}$ is symmetric.
(c) Let $\mathrm{P}_1, \mathrm{P}_2 ; \mathrm{P}_2, \mathrm{P}_3$ are $\mathrm{n}$ sided polygons. $\mathrm{P}_1$ and $\mathrm{P}_3$ are also $n$ sided polygons.
$\Rightarrow \mathrm{R}$ is transitive. Hence $\mathrm{R}$ is an equivelance relation.
(ii) The set $\mathrm{A}=$ set of all the triangles in a plane.
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