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\title{CSC165 Assignment 2 
  {\large (due November 13th, 10 pm)}}
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\newcommand{\Q}{\mathbb{Q}}
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This assignment will give you practice producing proofs and disproofs.
A significant part of your work will involve understanding and
applying definitions, so you should be sure to start this assignment
early.  Your proofs must use the proof structure from this course:
although symbolic notation may be replaced by precise English prose,
you must indicate the steps and scope of your proof using sequences
and indentation as we do in class.  Each conclusion should be
justified using the steps preceding it, given definitions, or
elementary mathematical facts (some of which are given in the attached
mathematical prerequisites).

Your assignment must be submitted electronically as a PDF file called
\verb+a2.pdf+.  Although you are welcome to use any word-processing
software you choose (no handwriting please) to produce the PDF, you
are also welcome to download the \LaTeX source for this assignment and
fill in your answers (I'll provide some \LaTeX documentation and
tutorial pointers).  You should submit your PDF file early and often
to: \verb+https://www.cdf.toronto.edu/students+.  You have to use your
CDF userid and password for this, so make sure all this is in order
well before the due date.

You may work with up to one other person on your assignment, provided
you establish your partnership by November 9th.  Partnerships are
established by each partner submitting a file called \verb+partnerA2+
containing the CDF userid of the other partner, to the
\verb+https://www.cdf.toronto.edu/students/+ web site.  Please submit
just one copy of the assignment per partnership.

Some questions will use the non-negative rational numbers, denoted
$\mathbb{Q}^{\geq 0}$, as the domain. These are ratios of natural numbers where
the denominator is non-zero.  The positive rational numbers, denoted
$\mathbb{Q}^+$ are $\mathbb{Q}^{\geq 0}-\{0\}$.  In symbols:
\begin{eqnarray*}
  \mathbb{Q}^{\geq 0} &=& \{n_1/n_2 \mid n_1, n_2 \in \mathbb{N},
  n_2 \not= 0\} \\
  \mathbb{Q}^+ &=& \{n_1/n_2 \mid n_1, n_2 \in \mathbb{N},
  n_1, n_2 \not= 0\} 
\end{eqnarray*}
\begin{enumerate}
\item Prove or disprove the following claims, which are variations on
  statements about the continuity of $2x$.  A good way to begin is to
  write out the outline of proof structures and disproof structures
  for each statement, and then try to see which is feasible to
  complete.  If you get stumped writing out the structures, it's time
  to talk to your instructor or a TA.
  \begin{enumerate}
  \item $\exists x \in\Q^{\geq 0}, \exists \epsilon \in \Q^+, \forall \delta
    \in \Q^+, \exists y \in\Q^{\geq 0}, |x-y|<\delta \wedge |2x-2y|\geq\epsilon$.
  \item $\exists x \in\Q^{\geq 0}, \forall \delta \in \Q^+, \exists \epsilon
    \in \Q^+, \exists y \in\Q^{\geq 0}, |x-y|<\delta \wedge |2x-2y|\geq\epsilon$.
  \item $\forall \epsilon\in\Q^+,\exists\delta \in\Q^+,\forall
    x\in\Q^+, \forall y\in\Q^+, |x-y|<\delta \Rightarrow
    |2x-2y|<\epsilon$.
  \end{enumerate}
\item Use the definitions of $\lfloor x\rfloor$, and $\lceil x\rceil$
  to prove or disprove the following statements.
  \begin{enumerate}
  \item $\forall x \in \Q^+, \forall y \in\Q^+, \lfloor x\rfloor
    \cdot \lceil y\rceil \geq \lfloor xy\rfloor$.
  \item $\exists x \in \Q^+, \exists y \in\Q^+, \lfloor x\rfloor
    \cdot \lceil y\rceil > \lceil xy\rceil$.
  \item $\forall x\in Q^+, \forall y \in\Q^+, \lfloor x\rfloor\cdot
    \lfloor y\rfloor \leq \lfloor xy\rfloor$.
  \end{enumerate}
\item Use the definitions of $m\bmod 3$ and $n\bmod 5$ from the sheet
  of mathematical prerequisites to prove or disprove the following
  statements.
  \begin{enumerate}
  \item $\exists n \in\mathbb{N}, n^2\bmod 5 = 2$.
  \item $\forall n\in\mathbb{N}, \forall r\in \{0,1,2\}, (n^3\bmod 3 =
    r) \Rightarrow (n \bmod 3 = r)$.
  \end{enumerate}
\end{enumerate}
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