Question

The function $\mathbf{f}\mathbf{:}\mathbf{\mathbb{Z}}\mathbf{\to }{\mathbf{\mathbb{Z}}}_{\mathbf{5}}$ given by$\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$ is a homomorphism by Example 13. Find ${\mathbf{k}}_{\mathbf{f}}$(notation as in Exercise 33).

Short answer

The $k_f$notation is given as follows:

${\mathrm{k}}_{\mathrm{f}}=\left\{\mathrm{a}\in \mathrm{\mathbb{Z}}|5\mathrm{divides}\mathrm{a}\right\}$

Step-by-step solution

Referring to the Exercise 33.

According to Exercise 33,

If $f:G\to H$is a homomorphism, then $k_f$can be given as:

$k_f=\left\{a\in G|f\left(a\right)=e_H\right\}$

Finding kf notation. 

Since function$\mathrm{f}:\mathrm{\mathbb{Z}}\to {\mathrm{\mathbb{Z}}}_5$

Therefore, $\left[\mathrm{x}\right]=\left[0\right]$ is possible if and only if $5|\mathrm{x}$.

Hence, we conclude that.

${\mathrm{k}}_{\mathrm{f}}=\left\{\mathrm{a}\in \mathrm{\mathbb{Z}}|5\mathrm{divides}\mathrm{a}\right\}$

Hence the${\mathrm{k}}_{\mathrm{f}}$ of the function is given as ${\mathrm{k}}_{\mathrm{f}}=\left\{\mathrm{a}\in \mathrm{\mathbb{Z}}|5\mathrm{divides}\mathrm{a}\right\}$