Question

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 as follows;

It is given that the exhibit functions f and g from A to A such that $fog\ne gof$.

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.