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\begin{center}
{\huge  \sc   Homework 9} \\
\smallskip
{ Due Date: 11/23/2021}\\
\smallskip
{Name:  }\\
{\large  }
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\vspace{1cm}

\begin{enumerate}

\item[19.3] Two functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ are such that for all $x\in\mathbb{R}$
\[
g(x)=x^2+x+3, \text{\ and \ } (g\circ f)(x)=x^2-3x+5.
\]
Find all possibilities for $f$.

\newpage
\item[19.4] Let $X,Y,Z$ be sets, and let $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ be functions.
\begin{enumerate}
    \item Given that $g\circ f$ is onto, can you deduce that $f$ is onto? Give a proof or a counterexample. 
    
    \item Given that $g\circ f$ is onto, can you deduce that $g$ is onto? 
    
    \item Given that $g\circ f$ is 1-1, can you deduce that $f$ is 1-1?
    
    \item Given that $g\circ f$ is 1-1, can you deduce that $g$ is 1-1?

\end{enumerate}

\newpage 

    \item[19.5] Use the Pigeonhole Principle to prove the following statements involving a positive integer $n$:
    
    \begin{enumerate}
        \item In any set of 6 integers, there must be two whose difference is divisible by 5.
        
        \item In any set of $n+1$ integers, there must be two whose difference is divisible by $n$.
        
        \item Given any $n$ integers $a_1, a_2,\ldots, a_n$, there is a non-empty subset of these whose sum is divisible by $n$.
        
        \item Given any set $S$ consisting of ten distinct integers between 1 and 50, there are two different 5-element subsets of $S$ with the same sum. 
        
        \item Given any set $T$ consisting of nine distinct integers between 1 and 50, there are two disjoint subsets of $T$ with the same sum. 
        
        \item If any set of 101 integers chosen from the set $\{1,2\ldots,200\}$, there must be two integers such that one divides the other. 
    \end{enumerate}

\newpage

\item [19.6] 

\begin{enumerate}
    \item Find an onto function from $\mathbb{N}$ to $\mathbb{Z}$.
    
    \item Find a 1-1 function from $\mathbb{Z}$ to $\mathbb{N}$.
\end{enumerate}

\end{enumerate}

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