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\begin{center}
{\Huge 21-128 and 15-151 problem sheet 6}

Solutions to the following seven exercises and optional bonus problem are to be submitted through 
blackboard by 8:30AM on

\textbf{Thursday 20th October 2016.}

There are also some practice problems, not to be turned in, for those seeking more practice
and also for review prior to the exam.

\end{center}


\subsubsection*{Problem 1}
Given a three-digit decimal number $n$, written as $\alpha \beta \gamma$ in base $10$, the \textit{reverse} of $n$ is the 
number written as $\gamma \beta \alpha$ in base $10$. (For example, the reverse of $235$ is $532$, and the reverse of $3$ is $300$.)

Let $n$ be an integer between $1$ and $999$ written as $abc$ in base $10$. Suppose $a \ne c$. Let $x$ be the (positive) difference 
between $n$ and its reverse. Prove that the sum of $x$ and its reverse is $1089$.


\subsubsection*{Problem 2}
Let $a,b,c,d \in \mathbb{R}$ be constants with $a \ne 0$ and $c \ne 0$. Suppose that $f,g : \mathbb{R} \to \mathbb{R}$ with 
$f(x)=ax+b$ and $g(x)=cx+d$ for all $x \in \mathbb{R}$. Prove that $f$ and $g$ are injective and surjective, but that the 
function $h : \mathbb{R} \to \mathbb{R}$ defined by $h = g \circ f - f \circ g$ is neither injective nor surjective.


\subsubsection*{Problem 3}
Verify that the function $f : (0,1) \to \mathbb{R}$ defined by
$$f(x) = \frac{2x-1}{2x(1-x)} \quad \text{for all } x \in (0,1)$$
is a bijection.


\newpage
\subsubsection*{Problem 4}
Consider three functions $f,g,h : \mathbb{R} \to \mathbb{R}$, defined for all $x \in \mathbb{R}$ by
$$f(x) = \frac{x}{1+x^2}, \quad g(x) = \frac{x^2}{1+x^2}, \quad h(x) = \frac{x^3}{1+x^2}$$
\begin{enumerate}[(a)]
\item Determine which of these functions are injective.
\item Prove that $f$ and $g$ are not surjective.
\end{enumerate}


\subsubsection*{Problem 5}
Given real numbers $a,b,c,d$, let $f : \mathbb{R}^2 \to \mathbb{R}^2$ be defined by $f(x,y) = (ax+by,cx+dy)$ 
for all $(x,y) \in \mathbb{R}^2$. Prove that $f$ is injective if and only if $f$ is surjective.


\subsubsection*{Problem 6}
Let $f : A \to B$ and $g : B \to C$ be functions, and define $h = g \circ f$. Determine which of the following statements are true, giving proofs for the true statements and counterexamples for the false statements:
\begin{enumerate}[(a)]
\item If $h$ is injective, then $f$ is injective.
\item If $h$ is injective, then $g$ is injective.
\item If $h$ is surjective, then $f$ is surjective.
\item If $h$ is surjective, then $g$ is surjective.
\end{enumerate}


\subsubsection*{Problem 7}
Consider functions $f : A \to B$ and $g : B \to A$. Prove that
\begin{enumerate}[(a)]
\item If $f \circ g$ is the identity function on $B$, then $f$ is surjective.
\item If $g \circ f$ is the identity function on $A$, then $f$ is injective.
\end{enumerate}

\textit{To remind you: given a set $X$, the identity function on $X$ is the function $\mathrm{id}_X : X \to X$ 
defined by $\mathrm{id}_X(x)=x$ for all $x \in X$.}


\subsubsection*{Bonus Problem - 2 points}
If the common difference $d$ of an arithmetic progression starting from 1 is relatively prime to 10, show that the
sequence $1,\, 1 + d,\, 1 + 2d,\, \ldots$ contains infinitely many powers of 10. Is this still true if the arithmetic progression 
starts from 2?


\subsubsection*{Extra Problem 1}
Consider a function $f : A \to A$. Prove that if $f \circ f$ is injective, then $f$ is injective.


\subsubsection*{Extra Problem 2}
Let $f : A \to B$ be a function.
\begin{enumerate}[(a)]
\item Prove that there exists a set $X$ and functions $p : A \to X$ and $i : X \to B$, with $p$ surjective and $i$ injective, 
such that $f = i \circ p$.
\item Prove that there exists a set $Y$ and functions $j : A \to Y$ and $q : Y \to B$, with $j$ injective and $q$ surjective, 
such that $f = q \circ j$.
\end{enumerate}


\subsubsection*{Extra Problem 3}
Recall that if $A,B \subseteq \mathbb{R}$, a function $f : A \to B$ is \textbf{increasing} if, for all $x,y \in A$, if $x<y$ 
then $f(x) < f(y)$.

Let $A$ and $B$ be subsets of $\mathbb{R}$ and let $f : A \to B$ be a bijection. Prove that if $f$ is increasing, then $f^{-1}$ 
is increasing.


\end{document}

