\documentclass[a4paper,10pt]{article}

\usepackage[Mickael]{ammaths}
\usepackage{alterqcm}

\begin{document}

\entete{European section, season 2}{Test \#1}{Monday, october the 16th, 2009}

\begin{center}
\begin{minipage}{12cm}
{\sf Some items of this test are multiple choice questions. A good answer earns 2 points and any wrong answer costs 1 point, a missing answer earns or costs nothing. Other items are free response questions, all of them worth 4 points, where any incomplete or imperfect answer will be rewarded.}
\end{minipage}
\end{center}

\medskip

\begin{center}
\begin{tabularx}{16cm}{XcX}
{\cellcolor{lightgray}} & {\cellcolor{lightgray}}\bfseries\sffamily Section 1 -- Syllogisms & {\cellcolor{lightgray}}\\
\end{tabularx}
\end{center}


\begin{alterqcm}[tone=\sf QUESTIONS,ttwo=\sf ANSWERS,symb=\dingsquare,corsymb=\dingchecksquare]
\AQquestion{Is the following syllogism valid, true, neither of both ?
\begin{quote}No horses are blue.\\ Some birds are blue.\\ No birds are horses.\end{quote}}{%
 {True},
 {Valid},
 {Neither},
 {Both}	}
\AQquestion{Is the following syllogism valid, true, neither of both ?
\begin{quote}Driving a big car uses a lot of gas.\\ Using a lot of gas is expensive.\\ Driving a big car is expensive.\end{quote}}{%
 {True},
 {Valid},
 {Neither},
 {Both}	}
\AQquestion{What is the middle term in the following syllogism ?
\begin{quote}All sunny days are great.\\ Some mondays are sunny days.\\ Some Mondays are great.\end{quote}}{%
 {Sunny days},
 {Great},
 {Mondays},
 {Are}	}
\AQquestion{What is the total number of valid types of syllogisms.}{%
 {256},
 {19},
 {4},
 {42}	}
\AQquestion{The letter A stands for}{%
 {Universal affirmative.},
 {Universal negative.},
 {Particular affirmative.},
 {Particular negative.}	}
\end{alterqcm}


\begin{enumerate}\setcounter{enumi}{5}
\item Write a ``Barbara'' syllogism of your own invention.
\begin{center}
\begin{tabular}{p{10cm}}
\ \\[10pt] \hline
\ \\[10pt] \hline
\ \\[10pt] \hline
\end{tabular}
\end{center}
\end{enumerate}

\pagebreak

\begin{center}
\begin{tabularx}{16cm}{XcX}
{\cellcolor{lightgray}} & {\cellcolor{lightgray}}\bfseries\sffamily Section 2 -- Formal logic and equivalences & {\cellcolor{lightgray}}\\
\end{tabularx}
\end{center}

\begin{enumerate}
\item Match each logical operator to the right truth table.
\begin{center}
\begin{tabular}{cccccc}
 & $p\wedge q$ & $p\vee q$ & $p\rightarrow q$ & $p\leftrightarrow q$ &  \\
 & \ding{114} & \ding{114} & \ding{114} & \ding{114} &  \\[20pt]
\ding{114} & \ding{114} & \ding{114} & \ding{114} & \ding{114} & \ding{114} \\
$\begin{array}{c|c|c}
p & q & \\ \hline
0 & 0 & 1\\
0 & 1 & 0\\
1 & 0 & 0\\
1 & 1 & 1\\
\end{array}$ & $\begin{array}{c|c|c}
p & q & \\ \hline
0 & 0 & 1\\
0 & 1 & 0\\
1 & 0 & 1\\
1 & 1 & 1\\
\end{array}$ & $\begin{array}{c|c|c}
p & q & \\ \hline
0 & 0 & 0\\
0 & 1 & 1\\
1 & 0 & 1\\
1 & 1 & 1\\
\end{array}$ & $\begin{array}{c|c|c}
p & q & \\ \hline
0 & 0 & 0\\
0 & 1 & 0\\
1 & 0 & 0\\
1 & 1 & 1\\
\end{array}$ & $\begin{array}{c|c|c}
p & q & \\ \hline
0 & 0 & 1\\
0 & 1 & 1\\
1 & 0 & 0\\
1 & 1 & 1\\
\end{array}$ & $\begin{array}{c|c|c}
p & q & \\ \hline
0 & 0 & 0\\
0 & 1 & 0\\
1 & 0 & 1\\
1 & 1 & 1\\
\end{array}$ \\
\end{tabular}

\end{center}

\end{enumerate}

\bigskip


\begin{alterqcm}[tone=\sf QUESTIONS,ttwo=\sf ANSWERS,symb=\dingsquare,corsymb=\dingchecksquare,numbreak=1]
\AQquestion{The proposition $p\rightarrow q$ is formally equivalent to}{%
 {$q\rightarrow p$},
 {$\neg q\rightarrow \neg p$},
 {$\neg p\rightarrow \neg q$},
 {$p\leftrightarrow q$}	}
\AQquestion{The proposition $\neg(p\vee q)$ is formally equivalent to}{%
 {$p\wedge q$},
 {$\neg p\wedge \neg q$},
 {$\neg p\vee \neg q$},
 {$p\vee q$}	}
\AQquestion{The proposition $\neg(p\wedge q)$ is formally equivalent to}{%
 {$p\wedge q$},
 {$\neg p\wedge \neg q$},
 {$\neg p\vee \neg q$},
 {$p\vee q$}	}
\AQquestion{The proposition $p\leftrightarrow q$ is formally equivalent to}{%
 {$(p\rightarrow q)\wedge (q\rightarrow p)$},
 {$(p\rightarrow q)\vee (q\rightarrow p)$},
 {$(p\rightarrow q)\wedge (p\rightarrow \neg q)$},
 {$(p\rightarrow q)\vee (p\rightarrow \neg q)$}	}
\end{alterqcm}


\begin{enumerate}\setcounter{enumi}{5}
\item Prove, by building their truth tables, that the following propositions are formally equivalent.
$$\neg(r\rightarrow p\vee q)\textrm{ and }\neg p\wedge\neg q\wedge r.$$
\end{enumerate}
\fbox{\parbox{15.7cm}{\ \vspace{6cm}}}

\pagebreak

\begin{center}
\begin{tabularx}{16cm}{XcX}
{\cellcolor{lightgray}} & {\cellcolor{lightgray}}\bfseries\sffamily Section 3 -- Venn diagrams & {\cellcolor{lightgray}}\\
\end{tabularx}
\end{center}

\begin{multicols}{2}
\begin{enumerate}
\item Draw a Venn diagram with two sets $A$ and $B$ such that $x\in A\rightarrow x\in B$.
\end{enumerate}
\fbox{\parbox{7.5cm}{\ \vspace{4cm}}}
\begin{enumerate}\setcounter{enumi}{1}
\item Draw a Venn diagram with three sets $A$, $B$ and $C$ such that $A\subset B$ and $C\subset\bar{B}$.
\end{enumerate}
\fbox{\parbox{7.5cm}{\ \vspace{4cm}}}
\end{multicols}

{\sf The following multiple choice questions are all about the configuration you represented in the question 2.}

\begin{alterqcm}[tone=\sf QUESTIONS,ttwo=\sf ANSWERS,symb=\dingsquare,corsymb=\dingchecksquare,numbreak=2]
\AQquestion{The intersection $A\cap B$ is equal to}{%
 {$\varnothing$},
 {$A$},
 {$B$},
  {Neither of these sets.}	}
\AQquestion{The intersection $A\cap C$ is equal to}{%
 {$\varnothing$},
 {$A$},
 {$B$},
  {Neither of these sets.}	}
\AQquestion{The intersection $B\cap C$ is equal to}{%
 {$\varnothing$},
 {$A$},
 {$B$},
 {Neither of these sets.}	}
\AQquestion{Is the equality $A\cup B=A$ true of false in this situation ?}{%
 {True},
 {False},
 {It's impossible to say}	}
\AQquestion{Is the equality $A\cup B=B$  true of false in this situation ?}{%
 {True},
 {False},
 {It's impossible to say}	}
\AQquestion{Is the inclusion $A\cup C\subset B$ true of false in this situation ?}{%
 {True},
 {False},
 {It's impossible to say}	}
\AQquestion{Is the equality $B\cup C=\Omega$ true of false in this situation ?}{%
 {True},
 {False},
 {It's impossible to say}	}
\AQquestion{Is the equality  $B\cup \bar{A}=\Omega$ true of false in this situation ?}{%
 {True},
 {False},
 {It's impossible to say}	}
\end{alterqcm}

\pagebreak

\begin{center}
\begin{tabularx}{16cm}{XcX}
{\cellcolor{lightgray}} & {\cellcolor{lightgray}}\bfseries\sffamily Section 4 -- Set-builder notation & {\cellcolor{lightgray}}\\
\end{tabularx}
\end{center}

\begin{enumerate}
\item Match each set definition with the right partial list of elements.
\begin{center}
\begin{tabular}{cccc}
$\{n^2 : n\in\N\cap[2,7]\}$ & $\{x\in\R : x^2=2x\}$ & $\{3k+1 : k\in\Z\}$ & $\{n : n\in\Z\wedge \dis\frac{15}{n}\in\Z\}$\\
\ding{114} & \ding{114} & \ding{114} & \ding{114} \\[20pt]
\ding{114} & \ding{114} & \ding{114} & \ding{114} \\
$2$, $0$ & $16$, $37$, $10$  & $16$, $36$, $9$ &  $1$, $3$, $15$\\
\end{tabular}
\end{center}
\end{enumerate}

\begin{alterqcm}[tone=\sf QUESTIONS,ttwo=\sf ANSWERS,symb=\dingsquare,corsymb=\dingchecksquare,numbreak=1]
\AQquestion{Which one of the following sets is empty ?}{%
 {$\{n\in\N : n^2=4\}$},
 {$\{n\in\N : n^2=5\}$},
 {$\{n\in\R : n^2=4\}$},
 {$\{n\in\R : n^2=5\}$}	}
\AQquestion{The notation $\R^+$ denotes what set ?}{%
 {$\N$},
 {$]0;+\infty[$},
 {$[0;+\infty[$},
 {$\Q^+$}	}
\AQquestion{The natural numbers are the elements of what set ?}{%
 {$\N$},
 {$\N^\star$},
 {$\Z$},
 {$\Z^\star$}	}
\AQquestion{Only one of the following statements is true. Which one ?}{%
 {$\forall x\in\R$, $x^2=5$},
 {$\forall x\in\Z$, $x^2=5$},
 {$\exists x\in\R$, $x^2=5$},
 {$\exists x\in\Z$, $x^2=5$}	}
\end{alterqcm}

\begin{enumerate}\setcounter{enumi}{5}
\item Explain in a few words Russell's paradox.
\end{enumerate}
\fbox{\parbox{15.8cm}{\ \vspace{7cm}}}


\end{document}
