\documentclass[12pt]{article}
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\newcommand{\lf}{\left\lfloor}
\newcommand{\rf}{\right\rfloor}
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\newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
\newcommand{\Rpos}{{\sf R}^+}

\usepackage{amsmath}
\begin{document}

  \centerline{\bf Questions from Past 250 MIDTERMS}

\newpage

\begin{enumerate}

\item
In this problem (1) all symbols have their usual meaning, and
(2) a domain is a subset of $\R$.

\begin{enumerate}
\item[I)]
Consider the sentence

\bigskip

(A)$\qquad$ $(\forall x)(\exists y)[x+y=0]$.

\bigskip

a) Give an infinite domain where A is TRUE OR prove there
is no infinite domain where A is TRUE.

\bigskip

b) Give an infinite domain where A  is FALSE OR prove there
is no infinite domain where A is FALSE.

\bigskip

c) Give a finite domain with \textbf{at least three elements} where A is TRUE OR prove there
is no finite domain with at least three elements where A is TRUE.

\bigskip

d) Give a finite domain with \textbf{at least three elements} where A is FALSE OR prove there
is no finite domain with at least three elements where A is FALSE.

\newpage 


\item[II)]
Consider the sentence

\bigskip

(B)$\qquad$ $(\forall x)(\exists y)[xy=1]$.

\bigskip

a) Give an infinite domain where B is TRUE OR prove there
is no infinite domain where B is TRUE.

\bigskip

b) Give an infinite domain where B  is FALSE OR prove there
is no infinite domain where B is FALSE.

\bigskip

c) Give a finite domain with \textbf{at least three elements} where B is TRUE OR prove there
is no finite domain with at least three elements where B is TRUE.

\bigskip

d) Give a finite domain with \textbf{at least three elements} where B is FALSE OR prove there
is no finite domain with at least three elements where B is FALSE.
\end{enumerate}

\newpage

\item
Consider the following boolean function:

\begin{equation}
f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=
\begin{cases}
T & \text{if exactly ONE of the inputs is T} \\
F & \text{otherwise}\\
\end{cases}
\end{equation}

\begin{enumerate}
\item
How many rows are in the truthtable for $f$?
\item
How many rows in the truthtable for $f$ evalute to TRUE? 
\item
Write down all of the rows that make $f$ evaluate to TRUE.
\item 
Write a formula for $f$. (Do not write a truth table to do this.) 
\item
Let 
\begin{equation}
f_n(x_1,\ldots,x_n)=
\begin{cases}
T & \text{if exactly ONE of the inputs is T} \\
F & \text{otherwise}\\
\end{cases}
\end{equation}

Name a function $g$ such that the following is true:


{\it The technique used in problem d can be extended to show that there is a formula for $f_n$ of 
length $O(g(n))$ }.
\end{enumerate}

\newpage


\item
In this problem the domain is the natural numbers and the language has the
usual logical symbols and arithmetic operations.
\begin{enumerate}
\item
A number is {\it cool} if it can be written as the sum of $\le 3$ cubes.
Let $COOL(x)$ mean that $x$ is cool.
Write a formula for $COOL(x)$.
\item
(You may use $COOL(x)$ in this problem.)
Write a sentence to express the following:
{\it There exists an infinite number of numbers that are NOT cool}
\item
Give 2 examples of cool numbers. Prove that they are cool.
\item
Give 2 examples of numbers that are not cool. Prove that they are not cool.
\end{enumerate}

\newpage

\item
(20 points)
\begin{enumerate}
\item 
Compute the following mod 16:
$0^4, 1^4, 2^4,\ldots, 15^4$.

(You may use the following shortcut: $(16-a)^4 \equiv a^6 \pmod {16}$.)

\item
Use the results of part a to find a number $N$ and an infinite set $X$ such that
the following is true

\bigskip

{\it If $x\in X$ then $x$ cannot be written as the sum of $N$ fourth powers.}

\bigskip

Make $X$ as large as possible.

\end{enumerate}

\newpage

\item
Let $T(n)$ be defined by
$T(1)=0$

$$(\forall n\ge 1)\biggl [ T(n) = T\biggl ( \floor{\frac{n}{11}} \biggr ) + T\biggl (\floor{\frac{2n}{11}}\biggr ) + 
T\biggl (\floor{\frac{3n}{11}}\biggr ) + 2n\biggr ]$$

Use constructive induction to find a constant $A\in \N$  such that

$$(\forall n\ge 0)[ T(n) \le An].$$

\newpage

\item
\begin{enumerate}
\item
(0 points but it will be helpfu)
Compute all of the following mod 7:

$2^0$, $2^1$, $2^2$, $2^3$, $2^4$, $2^5$, $2^6$

\item
State a true conjecture about the value of $2^n \pmod 7$.
It should be of the form (and this is NOT true)

\begin{equation}
2^n \bmod 7 = 
\begin{cases}
1 & \text{if $n\equiv 0,1 \pmod 5$} \\
4 & \text{if $n\equiv 2,3 \pmod 5$}\\
6 & \text{if $n\equiv 4 \pmod 5$}\\
\end{cases}
\end{equation}

\item
Prove your conjecture by induction. 
\end{enumerate}





\end{enumerate}


\end{document}