\documentclass[12pt]{article}
\usepackage{ammaths}
\usepackage{wasysym}

\begin{document}

\module{Introduction to logic}{01}{AP07}{1 period}

\prereq{None.}

\object{\begin{itemize}
        \item Discover the concepts of implication, converse and contrapositive.
	\end{itemize}}

\mater{\begin{itemize}
	\item Implications on cards.
	\item Answer sheet.
	\item Exercise sheet.
	\item Slideshow.
	\end{itemize}}

\modpart{Matching game}{10 mins}

Each student in the class is given a card with an implication on it. 
Students mingle to find :
\begin{enumerate}
\item first, the converse of their implication ;
\item second, the contrapositive of their implication.
\end{enumerate}

Using a slideshow, the teacher introduces the vocabulary of logic.

\modpart{True or not}{10 mins}

Students gather in groups of 4 : an implication, its converse, its contrapositive and the converse of the
contrapositive. Together, they discuss which ones of the four sentences are true.

\modpart{Exercises}{Remaining time}

Still working in groups, students have to solve a few exercises about implication.


\pagebreak

\begin{tabularx}{\linewidth}{|l|X|l|X|l|p{2cm}|}
\hline
Surname && First name && Form & \rule{0pt}{15pt} \\[5pt]
\hline
\end{tabularx}

\vspace*{1.5cm}

\moddocdis{Introduction to logic}{01}{AP07}{Answer sheet}

\sectionblack{Your four implications}

\bigskip

\begin{tabularx}{\linewidth}{r|X|c|}
\cline{2-3}
\bf\textsf{Type} & \bf\textsf{Sentence} & \bf\textsf{True ?} \\
\cline{2-3}
& & \\
Implication & & \\
& & \\
\cline{2-3}
& & \\
Contrapositive & & \\
& & \\
\cline{2-3}
& & \\
Converse & & \\
& & \\
\cline{2-3}
Contrapositive& & \\
 of the  & & \\
converse & & \\
\cline{2-3}
\end{tabularx}

\bigskip


\sectionblack{Exercises}

\subsection*{Exercise 1}

For each of the following implications, write its converse then decide if the inital implication is true and if
the converse is true.

\bigskip

\begin{tabularx}{\linewidth}{r|X|c|}
\cline{2-3}
\bf\textsf{Type} & \bf\textsf{Sentence} & \bf\textsf{True ?} \\
\cline{2-3}
Implication & If a triangle $ABC$ is inscribed in a circle of diameter $[AB]$, & \\
& then it's right-angled in $C$. & \\
\cline{2-3}
& & \\
Converse & & \\
& & \\
\cline{2-3}
Implication & If a line passes through the midpoints of two sides of a triangle, & \\
& then it's parallel to the third side. & \\
\cline{2-3}
& & \\
Converse & & \\
& & \\
\cline{2-3}
\end{tabularx}

\begin{tabularx}{\linewidth}{r|X|c|}
\cline{2-3}
\bf\textsf{Type} & \bf\textsf{Sentence} & \bf\textsf{True ?} \\
\cline{2-3}
Implication & If a triangle has an altitude that is also a median, & \\
& then it's isosceles. & \\
\cline{2-3}
& & \\
Converse & & \\
& & \\
\cline{2-3}
Implication & If two altitudes of a triangle meet in a point $H$, & \\
& then the third altitude also passes through $H$. & \\
\cline{2-3}
& & \\
Converse & & \\
& & \\
\cline{2-3}
Implication & If a product is equal to $0$, & \\
& then at least one of the terms is equal to 0. & \\
\cline{2-3}
& & \\
Converse & & \\
& & \\
\cline{2-3}
\end{tabularx}

\subsection*{Exercise 2}

For each pair of statements $p$ and $q$, tick the propositions that are true.

\bigskip


\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} I live in France. &
\textbf{$q$ :} I live in Europe. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}\bigskip

\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} I am overage. &
\textbf{$q$ :} I'm $19$. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}\bigskip

\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} $CDEF$ is a parallelogram. &
\textbf{$q$ :} $CDEF$ is a square. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}\bigskip

\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} $x \in \N$. &
\textbf{$q$ :} $x \in \Z$. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}\bigskip

\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} $MNP$ is right-angled in $M$. & 
\textbf{$q$ :} $MP^2+MN^2=NP^2$. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}\bigskip

\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} $x\ge -2$. & 
\textbf{$q$ :} $x\ge -1$. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}\bigskip

\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} $a+b=5$. &
\textbf{$q$ :} $a=2$ and $b=3$. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}\bigskip

\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} $4x-(x+5)=7$. & 
\textbf{$q$ :} $x=4$. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}\bigskip

\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} $n$ is prime. & 
\textbf{$q$ :} $n$ is not a multiple of $3$. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}\bigskip

\begin{tabular}{p{6cm}p{6cm}}
\textbf{$p$ :} $(ax+b)(cx+d)=0$. & 
\textbf{$q$ :} $ax+b=0$ or $cx+d=0$. \\
\end{tabular}

\begin{tabular}{lp{2cm}lp{2cm}lp{2cm}lp{2cm}lp{2cm}}
$\Square$ & $p\Rightarrow q$ & $\Square$ & $q\Rightarrow p$ & $\Square$ & $p\Leftrightarrow q$&
$\Square$ & $\neg p\Rightarrow \neg q$ & $\Square$ & $\neg q\Rightarrow \neg p$ \\
\end{tabular}


% For each line of the table below, tick the implications and equivalence that are true.
% \begin{center}\footnotesize
% \begin{tabular}{|p{4cm}|p{4cm}|c|c|c|c|c|}
% \hline 
% \bf Proposition $p$ &\bf  Proposition $q$ &  $p\Rightarrow q$& $q\Rightarrow p$ & $p\Leftrightarrow q$&
% $\neg p\Rightarrow \neg q$ & $\neg q\Rightarrow \neg p$ \\ \hline
% I live in France & I live in Europe & & & & &\\ \hline
% I am overage & I'm $19$ & & & & &\\ \hline
% $CDEF$ is a parallelogram & $CDEF$ is a square & & & & &\\ \hline
% $x \in \N$ & $x \in \Z$ & & & & &\\ \hline
% $x=2\sqrt{3}$ & $x^2=12$ & & & & &\\ \hline
% $MNP$ is right-angled in $M$ & $MP^2+MN^2=NP^2$ & & & & &\\ \hline
% $x\ge -2$ & $x\ge -1$ & & & & &\\ \hline
% $a+b=5$ & $a=2$ and $b=3$ & & & & &\\ \hline
% $4x-(x+5)=7$ & $x=4$ & & & & &\\ \hline
% $n$ is prime & $n$ is not a multiple of $3$ & & & & &\\ \hline
% $(ax+b)(cx+d)=0$ & $ax+b=0$ or $cx+d=0$ & & & & &\\ \hline\end{tabular}
% \end{center}






\pagebreak

\moddoc{Implications}
\begin{center}\large

If the triangle $ABC$ is right-angled in $A$, then $BC^2=AB^2+AC^2$.

\bigskip
\hrulefill
\bigskip

If $BC^2=AB^2+AC^2$, then the triangle $ABC$ is right-angled in $A$.

\bigskip
\hrulefill
\bigskip

If the triangle $ABC$ is not right-angled in $A$, then $BC^2\neq AB^2+AC^2$.

\bigskip
\hrulefill
\bigskip

If $BC^2\neq AB^2+AC^2$, then the triangle $ABC$ is not right-angled in $A$.

\bigskip
\hrulefill
\bigskip

If a quadrilateral is a square,\\ then its diagonals have the same length.

\bigskip
\hrulefill
\bigskip

If the diagonals of a quadrilateral have the same length,\\ then it's a square.

\bigskip
\hrulefill
\bigskip

If a quadrilateral is not a square,\\ then its diagonals don't have the same length.

\bigskip
\hrulefill
\bigskip

If the diagonals of a quadrilateral don't have the same length,\\ then it's not a square.

\bigskip
\hrulefill
\bigskip

If $x$ is a real number such that $x^2=9$, then $x=3$.

\bigskip
\hrulefill
\bigskip

If $x$ is a real number such that $x=3$, then $x^2=9$.

\bigskip
\hrulefill
\bigskip

If $x$ is a real number such that $x^2\neq 9$, then $x\neq 3$.

\bigskip
\hrulefill
\bigskip

If $x$ is a real number such that $x\neq 3$, then $x^2\neq 9$.

\bigskip
\hrulefill
\bigskip

If the median of a statistical set of data is $12$,\\ then at least $50\%$ of the values are greater than or equal to
$12$.

\bigskip
\hrulefill
\bigskip

If at least $50\%$ of the values are greater than or equal to $12$,\\ then the median of a statistical set of data is
$12$.

\bigskip
\hrulefill
\bigskip

If the median of a statistical set of data is not equal to $12$,\\ then less than $50\%$ of the values are greater than
or equal to $12$.

\bigskip
\hrulefill
\bigskip

If  less than $50\%$ of the values are greater than or
equal to $12$,\\ then the median of a statistical set of data is not equal to $12$.

\bigskip
\hrulefill
\bigskip

If a triangle is equilateral, then it's right-angled.

\bigskip
\hrulefill
\bigskip

If a triangle is right-angled, then it's equilateral.

\bigskip
\hrulefill
\bigskip

If a triangle is not equilateral, then it's not right-angled.

\bigskip
\hrulefill
\bigskip

If a triangle is not right-angled, then it's not equilateral.

\bigskip
\hrulefill
\bigskip

\end{center}



\end{document}
