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\begin{document}

\head{Worksheet 3}{Week 3: 2024-09-10}

These problems are to be completed in class on Tuesday and Thursday. Work in groups! And come to
office hours if you have further questions! They're great! \vspace{5mm}

\begin{framed}
  \textit{``My methods are really methods of working and thinking; this is why they have crept
    in everywhere anonymously.''} ---Amalie Emmy Noether (1882-1935)

  Emmy Noether was a German mathematician who made foundational contributions to abstract
  algebra and physics despite the profound professional barriers to women at the time. Her
  career began when David Hilbert and Felix Klein, noticing her ingenuity in the field of
  invariant theory, invited her to the University of G\"{o}ttingen in 1915. Despite their high
  praise (Hilbert and Klein were enormously influential mathematicians), the university refused
  outright to offer any position, saying ``what will our soldiers think when they return to the
  university and find that they are required to learn at the feet of a woman?''

  Noether worked as a lecturer at G\"{o}ttingen without pay for several years; eventually the
  university started to pay her, but she was never granted the title of full professor. Her
  mathematical work ranged from ring and field theory to algebraic topology, and laid the
  foundation for much of modern commutative algebra and algebraic geometry.

  \noindent\makebox[\textwidth][c]{
  \begin{minipage}[t]{0.5\linewidth}
    \vspace{-2ex}
    \centering
    \includegraphics[width=3cm]{noether}
  \end{minipage}}
\end{framed}

\begin{problem}
  Convert the following statements from English into formal logic. (Note that they are not
  necessarily true.)
  \begin{enumerate}
  \item There is an integer that is divisible by every integer. (Note: one can write
    \( k \divides n \) to mean ``\( k \) divides \( n \)'' or ``\( n \) is divisible by
    \( k \)''.)
  \item There is no greatest integer.
  \item There is no greatest odd integer.
  \end{enumerate}
\end{problem}

\vspace{7cm}

\begin{problem}
  Translate the following logical formula to plain English, where \( x \) represents a real
  number. Use as few variables as possible.
  \[ \exists m \in \ZZ, \exists n \in \ZZ, (n \neq 0 \wedge n x = m). \]
\end{problem}

\vspace{10cm}

\begin{problem*}{Repeat}
  Let \( x \) and \( y \) be real numbers. Prove that if both \( x + y \) and \( x - y \) are
  rational, then \( x \) and \( y \) are rational.
\end{problem*}

\vspace{7cm}

\begin{problem}
  Prove that, for all \( x, y \in \QQ \), if \( x < y \) then there exists some \( z \in \QQ \)
  so that \( x < z < y \).
\end{problem}

\vspace{3cm}

\begin{problem}
  Recall De Morgan's laws for quantifiers, which state:
  \[ \neg (\exists x : P(x)) \iff \forall x : (\neg P(x)),
    \qquad \qquad
    \neg (\forall x : P(x)) \iff \exists x : (\neg P(x)) \]
\end{problem}

\vspace{3cm}

\begin{problem}
  Prove that, if an integer \( n \) is \textit{not} divisible by 3, then there exists an integer
  \( k \) so that \( n^2 = 3 k + 1 \).

  Prove that, if \( r, s, t \) are \textit{all} not divisible by 3, then \( r^2 + s^2 + t^2 \)
  is divisible by 3.
\end{problem}

\vspace{7cm}

\begin{problem}
  Explain the difference between the following statements:
  \begin{enumerate}
  \item \( \exists x \in X \) such that \( \forall y \in Y \), \( P(x, y) \).
  \item \( \forall y \in Y \), \( \exists x \in X \) such that \( P(x, y) \).
  \end{enumerate}
\end{problem}

\vspace{7cm}

\begin{problem}
  Prove that, for any real number \( \delta > 0 \) and any \( M \in \RR \), there exists some
  \( n \in \ZZ \) so that \( n \delta > M \).
\end{problem}

\vspace{7cm}

\begin{problem}
  Prove that, for any real number \( t > 1 \) and any \( M \in \RR \), there exists some
  \( n \in \ZZ \) so that \( t^n > M \).
\end{problem}

\vspace{7cm}

\begin{problem}
  Prove that, for any \( 0 < t < 1 \) and any \( \varepsilon > 0 \), there exists some
  \( n \in \ZZ \) so that \( t^n < \epsilon \).
\end{problem}

\end{document}