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\title{CSC165, Summer 2005,
  Assignment 1\\
  {\large Due: Thursday June 9th, 10 am}}
\author{Danny Heap}
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\begin{document}
\maketitle

\section*{Instructions}
\label{instructions}

Please work on all questions.  Turn in the outline and structure of a
solution, even if you cannot provide every step, and we will
try to assign some part marks.  However, if there is any question you
cannot see how to even begin, leave it blank you will receive 20\%
of the marks for that question.

Be sure to give full credit to any sources you consult (other than
course notes, TAs, and the instructor) in preparing this problem set.
If you try to pass off somebody else's work as your own for credit,
you are committing an academic offense, and that can entail serious
consequences.  Any ideas that you do not attribute to someone else are
assumed to be the ideas of the author(s) listed below, and will be
evaluated for grading.

Write your name(s) and student number(s) (maximum of two names and two
student numbers) in the space below.\vspace{2\baselineskip}

\noindent Name\hfill
\rule{0.8\textwidth}{1pt}\\[2\baselineskip]
\noindent Student \#\hfill
\rule{0.8\textwidth}{1pt}\\[2\baselineskip]
\noindent Name\hfill
\rule{0.8\textwidth}{1pt}\\[2\baselineskip]
\noindent Student \#\hfill
\rule{0.8\textwidth}{1pt}
\newpage

\begin{enumerate}
\item \textbf{Prerequisites}

The following fragments are from the U of T Arts \& Science Calendar,
specifying the prerequisites for CSC165: 

\begin{quotation}\noindent
CSC107H1/CSC108H1/(CSC148H1/CSC150H1 taken concurrently); U Advanced
Functions and Introductory Calculus, and one of U Geometry and
Discrete Mathematics or U Mathematics of Data Management OR OAC
Calculus and one of Algebra \& Geometry or Finite Mathematics\\
\vdots\\
\noindent
Explanation of Symbols The comma (,) the semi-colon(;) the ampersand
(\&) and the plus sign (+) all mean ``AND''. The slash (/) means
``OR''.
\end{quotation}
Devise symbolic notation for each of the predicates connecting a
single course to a student.  For example, $CSC107(x)$ may represent
``x has completed $CSC107H1$.''  Using the operators $\wedge$ and
$\vee$, combine your predicates into three \textbf{different}
predicates, each of which is \textbf{consistent} with the given
English description of the prerequisites.  By \textbf{different} I
mean the predicates specify different sets of students, and by
\textbf{consistent} I mean that there is a reasonable interpretation
of the English description that has the same meaning as your
predicate.  Justify your answers.

\item \textbf{The Venn of statements}
 
  Let $S$ represent the set of stories, and

  \begin{itemize}
  \item Let $A(s)$ represent ``$s$ is apocryphal.''
  \item Let $B(s)$ represent ``$s$ is blasphemous.''
  \item Let $C(s)$ represent ``$s$ is convoluted.''
  \end{itemize}
  Consider this Venn diagram, where shaded regions are exactly those
  containing one or more stories.
  \begin{center}
    \resizebox{0.3\textwidth}{!}{
      \includegraphics{abc.eps}
    }
  \end{center}

  Which of the following are true, which are false?  Justify your
  answers using the Venn diagram.
  \begin{enumerate}
  \item $\forall s \in S, A(s) \Rightarrow B(s)$.
  \item Every blasphemous story is apocryphal.
  \item $\exists s \in S, C(s) \Rightarrow A(s)$.
  \item Some story is both apocryphal and convoluted only if it is blasphemous.
  \item $\exists s \in S, B(s) \Rightarrow C(s)$.
  \item Any story that is both apocryphal and blasphemous must be convoluted.
  \end{enumerate}

\item \textbf{Various Venns}

For some domain $D$, consider the statement:
\begin{displaymath}
  \forall x\in D, E(x) \Rightarrow (F(x) \Rightarrow G(x)).
\end{displaymath}
Draw three unshaded Venn diagrams with domain $D$ and intersecting
sets $E$, $F$, and $G$ that create 8 regions analogous to the diagram
in question~2.  Now shade each diagram to indicate which regions are
non-empty, so that no diagram contradicts the above statement.  In
addition, each pair of diagrams must be distinct, having at least one
region that's shaded in one and unshaded in another.  Justify your
diagrams.

\item \textbf{Virtue rewarded}

The following piece of propaganda undertakes to show the correlation
between the number of hours per week (including lectures and
tutorials) spent on CSC165 and final mark.
\begin{center}
  \begin{tabular}[t]{|c|c|c|}
    \hline
    student & mark & hours/week \\ \hline \hline
    $A$ & 65 & 4 \\
    $B$ & 70 & 5 \\
    $C$ & 75 & 6 \\
    $D$ & 80 & 7 \\
    $E$ & 85 & 8 \\
    $F$ & 90 & 3 \\ \hline
  \end{tabular}
\end{center}

If our domain $D$ is $\{A,B,C,D,E,F\}$, then which of the following
conclusions are justified or not justified?  Explain your answers.

\begin{enumerate}
\item Any student who spends at least 7 hours per week on the course
  earns a mark of at least 80.
\item All students who earn a mark of at least 80 study at least 7
  hours per week.
\item All students who study at least 9 hours per week earn a mark of
  at least 90.
\item All students who study less than 3 hours per week earn a mark of
  at least 95.
\item Not all students who study less than 2 hours per week earn a
  mark of less than 65.
\end{enumerate}
\end{enumerate}
\end{document}