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\begin{center}
{\Huge 21-128 and 15-151 problem sheet 2}

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

\textbf{Thursday 15th September 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}
For what conditions on sets $A$ and $B$ does $A\setminus B=B\setminus A$ hold?

\subsubsection*{Problem 2}
Assuming only arithmetic (not the quadratic formula or calculus), prove that
$$\{ x \in \mathbb{R} : x^2-2x-3 < 0 \} = \{ x \in \mathbb{R} : -1 < x < 3 \}$$

\subsubsection*{Problem 3}
Let $S = [3] \times [3]$ (the Cartesian product of $\{ 1, 2, 3 \}$ with itself). 
Let $T$ be the set of ordered pairs $(x,y) \in \mathbb{Z} \times \mathbb{Z}$ such that $0 \le 3x+y-4 \le 8$. 
Prove that $S \subseteq T$. Does equality hold?

\subsubsection*{Problem 4}
Prove the following identities involving complementation of sets.
\begin{enumerate}[(a)]
\item $(A \cup B)^c = A^c \cap B^c$;
\item $A \cap [(A \cap B)^c] = A\setminus B$;
\item $A \cap [(A \cap B^c)^c] = A \cap B$;
\item $(A \cup B) \cap A^c = B\setminus A$.
\end{enumerate}

\subsubsection*{Problem 5}
Let $f : A \to B$ be a function and let $C,D \subseteq A$.
\begin{enumerate}[(a)] \vspace{-10pt}
\item Prove that $f(C \cap D) \subseteq f(C) \cap f(D)$.
\item Give an example to demonstrate that equality need not hold in (a).
\end{enumerate}

\subsubsection*{Problem 6}
Let $\mathbb{N^{+}}$ denote the set of positive integers and consider
the function $f : \mathbb{N^{+}} \times \mathbb{N^{+}} \to \mathbb{R}$ defined by
$$f(a,b) = \frac{(a+1)(a+2b)}{2}$$
\begin{enumerate}[(a)]
\item Show that the image of $f$ is a subset of $\mathbb{N^{+}}$.
\item Determine exactly which positive integers are elements of the image of $f$. 
(\textbf{Hint:} Formulate a hypothesis by trying values.)
\end{enumerate}

\subsubsection*{Problem 7}
For $a \in \mathbb{R}$ and $f : \mathbb{R} \to \mathbb{R}$, show that (a) and (b) below have different meanings.
\begin{enumerate}[(a)]
\item $(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x \in \mathbb{R})[ |x-a| < \delta \Rightarrow |f(x) - f(a)| < \varepsilon]$.
\item $(\exists \delta > 0)(\forall \varepsilon > 0)(\forall x \in \mathbb{R})[ |x-a| < \delta \Rightarrow |f(x) - f(a)| < \varepsilon]$.
\end{enumerate}
(\textbf{Hint:} Find a function $f : \mathbb{R} \to \mathbb{R}$ and an element $a \in \mathbb{R}$ for which 
(a) and (b) have different truth values.)

\subsubsection*{Bonus Problem  (2 points)}
Let $S$ be a non-empty set of people in a bar. Express the following statement symbolically:
\begin{quote}
\textit{There is a person in the bar such that, if that person is drinking, then everyone else in the bar is drinking.}
\end{quote}
Prove that it is true.

\subsubsection*{Extra Problem 1}
Let $A = \{ \text{January}, \text{February}, \dots, \text{December} \}$. 
Given $x \in A$, let $f(x)$ be the number of days in $x$. Does $f$ define a function from $A$ to $\mathbb{N}$?

\subsubsection*{Extra Problem 2}
Let $A, B \subseteq \mathbb{R}$, let $f : \mathbb{R} \to \mathbb{R}$ and let $P$ denote the set of positive real numbers. 
Without using negation words (e.g. `not'), negate the following expressions:
\begin{enumerate}[(a)]
\item For all $x \in A$, there is a $b \in B$ such that $b > x$.
\item There is an $x \in A$ such that, for all $b \in B$, $b > x$.
\item For all $x,y \in \mathbb{R}$, if $f(x) = f(y)$ then $x=y$.
\item For all $b \in \mathbb{R}$, there is an $x \in \mathbb{R}$ such that $f(x) = b$.
\item For all $x,y \in \mathbb{R}$ and all $\varepsilon \in P$, there is a $\delta \in P$ such that 
$|x-y| < \delta$ implies $|f(x)-f(y)| < \varepsilon$.
\item For all $\varepsilon \in P$, there is a $\delta \in P$ such that, for all $x,y \in \mathbb{R}$, 
$|x-y| < \delta$ implies $|f(x)-f(y)| < \varepsilon$.
\end{enumerate}

\subsubsection*{Extra Problem 3}
A function $g : \mathbb{R} \to \mathbb{R}$ is \textit{even} if $g(-x) = g(x)$ for all $x \in \mathbb{R}$, 
or \textit{odd} if $h(-x)=-h(x)$ for all $x \in \mathbb{R}$.

Let $f : \mathbb{R} \to \mathbb{R}$.
\begin{enumerate}[(a)] \vspace{-10pt}
\item Prove that there exists a unique pair of functions $g,h : \mathbb{R} \to \mathbb{R}$ such that $g$ is even, 
$h$ is odd, and $f=g+h$. (\textbf{Hint:} Express both $f(x)$ and $f(-x)$ in terms of $g(x)$ and $h(x)$, and solve 
the resulting system of equations.)
\item When $f$ is a polynomial function, express $g$ and $h$ as in (a) in terms of the coefficients of $f$.
\end{enumerate}


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