Answer to Question #203291 in Abstract Algebra for Raghav

Question

Give an example, with justification, of a function with domain Z \{2,3}and codomain N.Is this function 1 –1?Is it onto ?Give reasons for your answers.

Answer 1

Consider the function $f:\mathbb Z\setminus\{2,3\}\to\mathbb N,\ \ f(n)=\begin{cases} -n\ \text{ if }n< 0\\ n+1\ \text{ if }n\in\{ 0,1\} \text{ or } n\ge4 \end{cases}.$ By defenition of this function, the domain of $f$ is $\mathbb Z\setminus\{2,3\}$ and codomain is $\mathbb N.$

Taking into account that $f(-1)=1$ and $f(0)=1,$ we conclude that $f$ is not 1-1. Since for any $n\in\mathbb N$ we have that $f(-n)=-(-n)=n,$ the function $f$ is onto.