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10E

Expert-verified
Found in: Page 520

Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

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

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

See the step by step solution

Step 1: 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 $\mathrm{ℤ}$ is the set of integers, therefore, $\mathrm{ℤ}=\left\{-1,2,-3,4\right\}$.

Now, functions f and g are defined as;

$\mathrm{f}\left(-1\right)=-3\phantom{\rule{0ex}{0ex}}\mathrm{f}\left(2\right)=2\phantom{\rule{0ex}{0ex}}\mathrm{f}\left(-3\right)=4\phantom{\rule{0ex}{0ex}}\mathrm{f}\left(4\right)=-1$

And,

$\mathrm{g}\left(-1\right)=2\phantom{\rule{0ex}{0ex}}\mathrm{g}\left(2\right)=-3\phantom{\rule{0ex}{0ex}}\mathrm{g}\left(-3\right)=-1\phantom{\rule{0ex}{0ex}}\mathrm{g}\left(4\right)=4$

Step 2: 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;

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

Step 3: 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.