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\noindent {\large\bf Exam 1 Extra Credit \hfill 21-127 Concepts of Math \hfill due 07.20.2012}\\
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You have the opportunity to earn back some of the points you lost on Exam 1. Each lettered problem below is assigned a specific percentage weight. You may choose {\bf up to 3 of them} to write up and submit for credit. Based on the sum of the corresponding percentage weights (say $Y$) and how well you do on them, you can earn some points back. This number depends on how well you did on the exam in the following way: the more points you lost on the exam, the greater potential you have to earn points back. This still depends on you trying and hard and doing well on these submitted problems.\vspace{0.3cm}\\
(Note: $Y$ is capped at 100. Even if you submit three problems weighted at 35\%, 35\%, and 40\%, you can only earn back up to whatever number of points you can possibly earn back. If you want a more detailed mathematical description of that number, ask me in person.)\vspace{0.3cm}\\
I also request that you do {\bf not} submit three problems of the same type. That is, if you're already writing up 2 induction problems, choose A/G/H/J as your third problem.\vspace{0.3cm}\\
It is important that you {\bf DO NOT COLLABORATE} on these problems during this week.\vspace{0.3cm}\\
I have chosen to do this to help you out and show you that you {\it can} and {\it will} earn better grades. I am giving up my free time to grade this extra assignment to {\it literally} give you free points. If I find out that you are just helping each other out $\ldots$ that would defeat the purpose, and you should not expect me to offer this again.

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\begin{enumerate}[A.]
\item A rigorous definition of the relation ``divides'' is given as follows:

Let $a,b\in\Z$. Then we say $a\mid b \iff \exs k\in\Z\st b=ak$

Prove or disprove each of the following claims, using this definition.
\begin{itemize}
\item $\fa a,b,c\in\Z\st\bigl((a\mid b \oand b\mid c)\imp a\mid c\bigr)$
\item $\fa a,b,c\in\Z\st\bigl((a\mid c\oand b\mid c)\imp a+b\mid c\bigr)$
\item $\fa a,b,c,d\in\Z\st\bigl((a\mid b\oand c\mid d)\imp ab\mid cd\bigr)$
\item $\fa a,b,c,d\in\Z\st\bigl((a\mid b\oand c\mid d)\imp a+b\mid c+d\bigr)$
\end{itemize}
{\bf[35\%]}
\item Prove the following claim by induction: $\fa n\in\N\st 3\mid n^3-n$\vspace{0.2cm}\\{\bf[40\%]}
\item Prove the following claim by induction: $\fa n\in\N-\{2,3,4\}\st n^2<2^n$\vspace{0.2cm}\\{\bf[30\%]}
\item Prove the following claim by induction: $\fa n\in\N\st\disp{\sum_{k=1}^n k^3 = \of{\sum_{k=1}^n k}^2}$\vspace{0.2cm}\\{\bf[25\%]}
\item Define a sequence recursively as follows: $a_0 = 1$ and $a_{n} = 4n + a_{n-1}$ for all $n\in \N$ with $n>0$.

Prove that $a_n = 2n^2 + 2n + 1$ for every $n\in\N\cup\{0\}$.\vspace{0.2cm}\\{\bf[25\%]}
\item Define the sequence of Fibonacci numbers by $f_0=0$ and $f_1=1$ and
\[ \fa n\in\N-\{1\}\st\; f_n=f_{n-1}+f_{n-2} \]
That is, $f_0=0,f_1=1,f_2=1,f_3=2,f_4=3,f_5=5$, and so on.

Prove the following claim about the Fibonacci numbers by induction:
\[ \fa n\in\N\cup\{0\}\st \sum_{k=0}^n f_k = f_{n+2}-1 \]
{\bf[35\%]}
\item Write the logical negation of the following claim in symbolic form. Then, prove or disprove the original claim.
\[\fa n\in\N\st (n^3\text{ is even }\iff n\text{ is even})\]
{\bf [25 \%]}
\item Write the logical negation of the following claim in symbolic form. Then, prove or disprove the original claim.
\[\fa x,y,z,w\in\R\st (xy+zw)^2\leq(x+z)^2(y+w)^2\]
{\bf[25\%]}
\item Let $S,T$ be sets. Prove that
\[ \PP(S\cap T)=\PP(S)\cap\PP(T) \]
{\bf[20\%]}
\item Consider the function $f(x)=3x-2$, defined on the real numbers, $\R$. Let $P=\{y\in\R\mid y>0\}$ be the set of positive real numbers. Prove the following claim:
\[ \fa\ve\in P\st \exs\delta\in P\st\fa x,y\in\R\st |x-y|<\delta\imp|f(x)-f(y)|<\ve \]
\vspace{0.2cm}\\{\bf[25\%]}
\end{enumerate}

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