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\noindent
\parbox{2.25in}{\bf Math 2110}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $30^\mathrm{th}$, 2020}

\vspace{0.1in}

%\noindent {\large\bf Name:} \underline{\Large\quad My Name Goes Here \quad} \hfill 

\force \hfill Handwritten solutions are fine. \smiley  % uses wasysym package. :)
\hfill \force

\vspace{0.2in}

\begin{description}
\item[\#1] In class we showed $2+2=4$ and $3\cdot 2=6$ more-or-less using Axioms 11--14 on our E and PA handout. 

\begin{enumerate}[(a)]
\item Show $4+3=7$.
\item Show $5 \cdot 3=15$.
\end{enumerate}

Use the same formatting and level of detail as my ``proofs'' shown below. You may ``bury the details'' of addition when working through the multiplication problem.

\begin{minipage}{2.5in}
Showing $2+2=4$:

\begin{enumerate}[1:]
\item $2+(1')$ \hfill Definition of $2=1'$
\item $(2+1)'$ \hfill Axiom 12
\item $(2+0')'$ \hfill Definition of $1=0'$
\item $((2+0)')'$ \hfill Axiom 12
\item $((2)')'$ \hfill Axiom 11
\item $4$ \hfill Definition of $4=2''$
\end{enumerate}
\end{minipage}
\hfill
\begin{minipage}{2.5in}
Showing $3 \cdot 2 =6$:

\begin{enumerate}[1:]
\item $3 \cdot (1')$ \hfill Definition of $2=1'$
\item $(3 \cdot 1)+3$ \hfill Axiom 14
\item $(3 \cdot 0')+3$ \hfill Definition of $1=0'$
\item $(3 \cdot 0 + 3)+3$ \hfill Axiom 14
\item $(0+3)+3$ \hfill Axiom 13
\item $(3)+3$ \hfill Addition
\item $6$ \hfill Addition 
\end{enumerate}
\end{minipage}

\item[\#2] Formalize and justify. I'll start off with an example:

\noindent \fbox{\bf Proposition:} If $a \leq b$ and $c \leq d$, then $ac \leq bd$.

\vspace{0.05in}

\noindent {\bf Formal Statement:} $\forall a\,\forall b\, \forall c\, \forall d\, ((\exists k\,(b=a+k) \wedge \exists \ell\,(d=c+\ell)) \to \exists m\,(cd=bd+m))$

\vspace{0.05in}

\noindent {\bf Fairly-Formal Proof:}

\vspace{0.05in}

%\begin{minipage}{3.5in}
\begin{enumerate}[1:]
\item $a \leq b$ \hfill Given
\item $c \leq d$ \hfill Given
\item There exists $k$ such that $b=a+k$ \hfill Definition of inequality using 1
\item There exists $\ell$ such that $d=c+\ell$ \hfill Definition of inequality using 2
\item $bd=(a+k)(c+\ell)$ \hfill Equal values give equal answers
\item $(a+k)(c+\ell) = a(c+\ell)+k(c+\ell)$ \hfill Distributive law
\item $a(c+\ell)+k(c+\ell) = ac+a\ell+kc+k\ell$ \hfill Distributive laws
\item $bd=ac+(a\ell+kc+k\ell)$ \hfill Transitivity of equality (and associativity) using 5--7
\item $ac \leq bd$ \hfill Definition of inequality using 8
\end{enumerate}
%\end{minipage}
%\hfill

Now fill in the formal statement and give justifications for the steps in my ``Fairly-Formal'' proof. I won't be too picky about your justifications. Use proper names where we have them.

\vspace{0.05in}

\noindent \fbox{\bf Proposition:} If $a \,\Big|\,  b$ and $a \,\Big|\, c$, then $a \,\Big|\, (b+c)$.

\vspace{0.05in}

\noindent {\bf Formal Statement:} \qquad ???

\vspace{0.05in}

\noindent {\bf Fairly-Formal Proof:}

\vspace{0.05in}

\begin{minipage}{2.5in}
\begin{enumerate}[1:]
\item $a \,\Big|\,  b$ 
\item $a \,\Big|\,  c$ 
\item There exists $k$ such that $b=a\cdot k$ 
\item There exists $\ell$ such that $c=a\cdot \ell$ 
\end{enumerate}
\end{minipage}
\hfill
\begin{minipage}{2.5in}
\begin{enumerate}[1:]
\setcounter{enumi}{4} % start next at 5
\item $b+c=ak+ a\ell$ 
\item $ak+a\ell = a(k+\ell)$ 
\item $b+c = a(k+\ell)$ 
\item $a \,\Big|\, (b+c)$
\end{enumerate}
\end{minipage}

\newpage

\item[\#3] Prove that an odd number times an odd number is an odd number.

\item[\#4] Prove that $a \leq b$ implies that $a+x \leq b+x$.

\item[\#5] Prove that $a \,\Big|\, b$ and $b \,\Big|\, c$ implies that $a \,\Big|\, c$.

\item[\#6] Use mathematical induction to prove that the sum of the first $n$ odd numbers is $n^2$.

 In particular, show that $\displaystyle \sum\limits_{i=1}^n (2i-1) = n^2$.

\item[\#7] When $n \geq \,???$, we have $n^2<n!$ where $n!=n(n-1)\cdots 3\cdot 2 \cdot 1$ (i.e., the factorial function). Find the smallest value for ??? that makes this true. Then prove this inequality using mathematical induction.

{\it Hint:} When $n$ is positive, $1 \leq n \leq n^2$.


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