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\title{CSC165 Fall 2014, Assignment \#2\\
{\large Due November 3rd, 10:00 p.m.}}
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\begin{document}
\maketitle
\noindent
The aim of this assignment is for you to practice devising and
presenting proofs.  You may work in groups of no more than three
students, and you should produce a single solution in a PDF file named
\texttt{a2.pdf}, submitted to
\href{https://markus.cdf.toronto.edu/csc165-2014-09}{MarkUs}.

You will receive 20\% of the marks for any question you either leave
blank, or write ``I cannot answer this.''  You will receive 0 for any
false claim you ``prove,'' or any true claim you ``disprove.''

\begin{enumerate}
\item For $x\in\mathbb{R}$, define  $\floor{x}$ by:
    \begin{equation*}
        \floor{x} \in\mathbb{Z} \wedge \floor{x}
        \leq x \wedge (\forall z \in\mathbb{Z}, z \leq x \Rightarrow z
        \leq \floor{x}).
    \end{equation*}
    \ldots where $\mathbb{Z}$ stands for the set of integers, and
    $\mathbb{R}$ stands for the set of real numbers.
    Use the definition of $\floor{x}$ to prove or disprove each of the
    following claims, using the structured proof technique from this
    course.  \textbf{Note:} You must use the definition given here,
    not some other (possibly equivalent) definition for $\floor{x}$.
  \begin{description}
  \item[Claim 1.1:] ~ 
    \begin{equation*}
      \forall x \in\mathbb{R}, \forall y \in\mathbb{R}, x > y \Rightarrow
      \floor{x} \geq \floor{y}.
    \end{equation*}
  \item[Claim 1.2:] ~
    \begin{equation*}
  \forall x\in\mathbb{R}, \forall e \in\mathbb{R}^+, \exists
  d \in\mathbb{R}^+, \forall w\in\mathbb{R}, |x-w|< d
  \Rightarrow |\floor{x}  - \floor{w}| < e
    \end{equation*}
  \item[Claim 1.3:] ~
    \begin{equation*}
        \exists x\in\mathbb{R}, \forall e \in\mathbb{R}^+, \exists
  d \in\mathbb{R}^+, \forall w\in\mathbb{R}, |x-w|< d
  \Rightarrow |\lfloor x\rfloor - \lfloor w\rfloor| < e
    \end{equation*}
  \item[Claim 1.4:] ~
    \begin{equation*}
      \exists x \in\mathbb{R}, \floor{x + 1} \neq \floor{x} + 1
    \end{equation*}
  \end{description}

\item \textbf{Prove} or \textbf{disprove} the claim, and
  \textbf{prove} or \textbf{disprove} the converse:
  \begin{description}
  \item[Claim 2.1:] ~
    \begin{equation*}
      \forall n \in\mathbb{N}, \left(\exists k\in\mathbb{N}, n =
        5k+2\right) \Rightarrow \left(\exists j\in\mathbb{N}, n^2 = 5j+4\right)
    \end{equation*}
  \item[Claim 2.2:] ~
    \begin{equation*}
      \forall m, n\in\mathbb{N}, \left(\exists k\in\mathbb{N}, m =
        7k+3\right) \wedge \left(\exists j\in\mathbb{N}, n = 7j + 4\right)
      \Rightarrow \left(\exists i\in\mathbb{N}, mn = 7i + 5\right)
    \end{equation*}
  \end{description}

\end{enumerate}

\end{document}

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