Question

Let S be the set of all polynomial with real coefficient ,if f, g€S, define f~g if f¹=g¹ where f¹ is the derivative of f, show that ~ is an equivalence relation. Describe the equivalence classes of S

Answer 1

Since $f'=f',$ we conclude that $f\sim f,$ and hence the relation is reflexive.

Let $f\sim g.$ Then $f'=g',$ and hence $g'=f'.$ We conclude that $g\sim f,$ and hence the relation is symmetric.

Let $f\sim g$ and $g\sim h.$ Then $f'=g'$ and $g'=h'.$ It follows that $f'=h',$ and hence the relation is transitive.

Therefore, this relation is an equivalence relation.

Let us describe the equivalence classes of $S.$

It follows that

$[f]=\{g\in S:g\sim f\}=\{g\in S:g'=f'\} \\=\{g\in S:g=f+C,C\in \R\}=\{f+C:C\in \R\}$