Question

In the set $\mathbf{ℝ}\mathbf{\left[}\mathit{x}\mathbf{\right]}$of all polynomials with real coefficients define $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{~}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}$if and only if ${\mathit{f}}^{\mathbf{\text{'}}}\left(x\right)\mathbf{=}\mathit{g}\mathbf{\text{'}}\left(x\right)$, where ‘ denotes the derivative. Prove that$\mathbf{~}$is an equivalence relation on$\mathbf{ℝ}\mathbf{\left[}\mathit{x}\mathbf{\right]}$.

Short answer

We proved that$~$ is an equivalence relation.

Step-by-step solution

To mention given data

Let $~$be defined on $\mathrm{ℝ}\left[x\right]$by:

localid="1659174882722" $f\left(x\right)~g\left(x\right)$if and only if $f^{\text{'}}\left(x\right)=g^{\text{'}}\left(x\right)$.

We have to prove that $~$is an equivalence relation.

In order to prove that$~$is an equivalence relation, we need to prove the following conditions:

1. ‘$~$’ is reflexive.
2. ‘$~$’ is symmetric.
3. ‘$~$’ is transitive.

To prove that ~ is reflexive

Let $f\left(x\right)\in \mathrm{ℝ}\left[x\right].$

The,

$$\begin{gathered} f\left(x\right)~f\left(x\right) \\ \iff f^{\text{'}}\left(x\right)=f^{\text{'}}\left(x\right) \end{gathered}$$

Hence,$~$is reflexive.

To prove that ~ is symmetric

Let$f\left(x\right),g\left(x\right)\in \mathrm{ℝ}\left[x\right]$.

Then suppose that

$$\begin{gathered} f\left(x\right)~g\left(x\right) \\ \iff f^{\text{'}}\left(x\right)=g^{\text{'}}\left(x\right) \\ \iff g^{\text{'}}\left(x\right)=f^{\text{'}}\left(x\right) \\ \iff g\left(x\right)~f\left(x\right) \end{gathered}$$

Hence,$~$is symmetric.

To prove that ~ is transitive

Let$f\left(x\right),g\left(x\right),h\left(x\right)\in \mathrm{ℝ}\left[x\right]$.

Then suppose that

$$\begin{gathered} f\left(x\right)~g\left(x\right),g\left(x\right)~h\left(x\right) \\ \iff f^{\text{'}}\left(x\right)=g^{\text{'}}\left(x\right),g^{\text{'}}\left(x\right)=h^{\text{'}}\left(x\right) \\ \iff f^{\text{'}}\left(x\right)=h^{\text{'}}\left(x\right) \\ \iff f\left(x\right)~h\left(x\right) \end{gathered}$$

Hence, $~$is transitive.

Hence, $~$is an equivalence relation since all the conditions are satisfied.