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  \textbf{CS 151 Complexity Theory} \hfill \textbf{Spring 2015}
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    {\Large Sample \LaTeX\ Template} \\[3mm]
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\today \hfill (Your Name)
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\begin{enumerate}
    \item Please remember that homework solutions should be:
    \begin{itemize}
      \item Clear
      \item Concise
      \item Precise
      \item Legible (if handwritten).
    \end{itemize}

    \item An example\footnote{Taken from \emph{Mathematical Writing} by Knuth, Larrabee, and Roberts.}:
    \begin{itemize}
\item[\bf{(Yes)}]``According to the `fundamental theorem of arithmetic' (proved in
ex.\ 1.2.4-21), each positive integer $u$ can be expressed in the form \[u =
2^{u_2}3^{u_3}5^{u_5}7^{u_7}11^{u_{11}}\ldots = \prod_{p \mbox{ prime}} p^{u_p}
\] where the exponents $u_2, u_3,\ldots$ are uniquely determined nonnegative
integers, and where all but a finite number of the exponents are zero.''

\item[\bf{(No)}]``If $\mathbf{L}^+(P,N_0)$ is the set of functions $f: P \rightarrow N_0$ with the property that
\[ \mathop{\exists}_{n_0\in N_0}\;\mathop{\forall}_{p\in P}\;p \ge n_0 \Rightarrow f(p) = 0\] then there exists a bijection $N_1 \rightarrow
\mathbf{L}^+(P,N_0)$ such that if $n \rightarrow f$ then \[n = \prod_{p\in P}
p^{f(p)}.\] Here $P$ is the prime numbers and $N_1 = N_0 \sim \{0\}$.''
    \end{itemize}

    \item Show that $\mathbf{P}=\mathbf{NP}$.
      \begin{enumerate}
        \item If $\mathbf{N}=1$ then $\mathbf{P}=\mathbf{P}$.
        \item If $\mathbf{P}=0$ then $0=0$.
        \item Collect \$1 million.
      \end{enumerate}

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