Question

Do exercise 9 when $\mathbf{A}\mathbf{=}\mathbf{\mathbb{Z}}$.

Exercise 9: Let $\mathbf{A}\mathbf{=}\left\{1,2,3,4\right\}$.Exhibit functions f and g from A to A such that $\mathbf{f}\mathbf{}\mathbf{o}\mathbf{}\mathbf{g}\mathbf{\ne }\mathbf{g}\mathbf{}\mathbf{o}\mathbf{}\mathbf{f}$ .

Short answer

It can be concluded that the exhibit functions f and g from A to A, then $\mathrm{fog}\ne \mathrm{gof}$.

Step-by-step solution

Consider the given statement

It is given that the exhibit functions f and g from A to A, then $\mathrm{fog}\ne \mathrm{gof}$.

As it is known that $\mathrm{\mathbb{Z}}$is the set of integers, therefore, $\mathrm{\mathbb{Z}}=\left\{-1,2,-3,4\right\}$.

Now, functions f and g are defined as;

$$\begin{gathered} \mathrm{f}\left(-1\right)=-3 \\ \mathrm{f}\left(2\right)=2 \\ \mathrm{f}\left(-3\right)=4 \\ \mathrm{f}\left(4\right)=-1 \end{gathered}$$

And,

$$\begin{gathered} \mathrm{g}\left(-1\right)=2 \\ \mathrm{g}\left(2\right)=-3 \\ \mathrm{g}\left(-3\right)=-1 \\ \mathrm{g}\left(4\right)=4 \end{gathered}$$

Determine the values of functions fog and gof at A={-1,2} 

The value of fog and gof at $\mathrm{A}=\left\{-1,2\right\}$are;

$\begin{array}{rcl}\mathrm{fog}\left(-1\right)& =& \mathrm{f}\left(2\right)\\ & =& 2\\ \mathrm{gof}\left(-1\right)& =& \mathrm{f}\left(-3\right)\\ & =& -1\\ \mathrm{fog}\left(2\right)& =& \mathrm{f}\left(3\right)\\ & =& 4\\ \mathrm{gof}\left(2\right)& =& \mathrm{f}\left(2\right)\\ & =& 3\end{array}$

Determine the values of functions fog and gof at A={-3,4} 

The value of fog and gof at $\mathrm{A}=\left\{-3,4\right\}$are;

$\begin{array}{rcl}\mathrm{fog}\left(-3\right)& =& \mathrm{f}\left(-1\right)\\ & =& -3\\ \mathrm{gof}\left(-3\right)& =& \mathrm{f}\left(4\right)\\ & =& 4\\ \mathrm{fog}\left(4\right)& =& \mathrm{f}\left(4\right)\\ & =& -1\\ \mathrm{gof}\left(4\right)& =& \mathrm{f}\left(-1\right)\\ & =& 2\end{array}$

Therefore, it can be observed from all the cases that $\mathrm{fog}\ne \mathrm{gof}$. So, the provided statement has been proved.