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\begin{document}

\module{Set-builder notation}{2}{06}{2 periods}

\prereq{Logical operators, Venn diagrams and the concept of set}

\object{\begin{itemize}
        \item Master the set-builder-notations.
	\item Build a glossary of useful symbols.
	\item Discover Russell's paradox
	\end{itemize}}

\mater{\begin{itemize}
	\item Answer sheet : matching set definitions and lists.
	\item Glossary : symbols used to define sets.
	\item Beamer with the solutions, the glossary and Russel's paradox.
	\end{itemize}}

\modpart{Matching sets and definitions in teams}{55 mins}

Students work in groups of five. They are handed out a list of set definitions and lists of numbers that they have to match.

\modpart{Marking phase}{15 mins}

While the answers are given by the teacher on a beamer, each team marks the answer sheet of another one.

\modpart{Glossary}{25 mins}

Each team must build a glossary about sets. Theb, the main symbols used in the set-builder notation are shown and explained by the teacher.


\modpart{Russell's paradox}{15 mins}

Russell's paradox is explained by the teacher.

\pagebreak

\moddocdis{Set-builder notation}{2}{06}{Answer sheet}

\begin{flushright}
\Large Name : \underline{\hspace{6cm}}\\ \bigskip
Grade : \underline{\hspace{6cm}}\\
\end{flushright}


\vspace{1cm}

\setlength{\columnsep}{2cm}
\begin{multicols}{2}
\begin{flushright}
$\{2k+1 : k\in\Z\}$  $\Box$\medskip \\ %1
$\{n : n\in\Z\wedge \dis\frac{12}{n}\in\Z\}$  $\Box$\medskip \\  %2
$\{n^2 : n\in\N\cap[0,5]\}$ $\Box$\medskip \\  %3
$\{3k : k\in\N\}$ $\Box$\medskip \\  %4
$\{x\in\R : 3x\in\Z\}$ $\Box$\medskip \\  %5
$\{a+bi : a\in\R\wedge b\in\R\}$ $\Box$\medskip \\ %6
$\{7k+1 : k\in\Z\}$ $\Box$\medskip \\ %7
$\{n\in\N : \exists m\in\Z, m^2=n\}$ $\Box$\medskip \\ %8
$\{x\in\R : x^2=x\}$ $\Box$\medskip \\ %9
$\{x\in\R : \exists p,q\in\Z, q\neq 0 \wedge xq=p\}$ $\Box$\medskip \\ %10
$\{r\in\R : \exists k\in\Z, r^2=k\}$ $\Box$\medskip \\ %11
$\{n\in\Z : |n|\leq 5\}$ $\Box$\medskip \\ %12
$\{x\in\R : |x|\leq 5\}$ $\Box$\medskip \\ %13
$\{x\in\R : \forall y\in\N, x\leq y\}$ $\Box$\medskip \\ %14
$\{(x,y) : x\in\N \wedge y\in\N \wedge\ x+y=7\}$ $\Box$\medskip \\ %15
$\{x\in\R : x\leq -3 \vee\ x\geq 5\}$ $\Box$\medskip \\ %16
$\{n\in\Z : \forall m\in\Z, n>m\}$ $\Box$\medskip \\ %17
$\{n\in\N : \nexists m\in\N-\{1,n\}, \dis\frac{n}{m}\in\N  \}$ $\Box$\medskip \\ %18
$\{(x,y) : x\in\N \wedge y\in\N \wedge\ xy\leq 6\}$ $\Box$\medskip \\ %19
$\{(x,y) : x\in\N \wedge y\in\N \wedge\ y=2x\}$ $\Box$\medskip \\ %20
\end{flushright}

$\Box$ $6$, $21$, $336$, $4272$ \medskip\\   %4
$\Box$ $3+5i$, $\sqrt2-i$, $17$, $-1+\pi i$ \medskip \\ %6
$\Box$ $-5$, $-2\pi$, $-\frac47$, $-0.0001$ \medskip\\  %14
$\Box$ $0$, $1$, $9$, $16$ \medskip \\  %3
$\Box$ $-5$, $-\sqrt{19}$, $17.25$, $5$ \medskip \\  %16
$\Box$ $0$, $9$, $81$, $196$ \medskip\\ %8
$\Box$ $\frac{12}{7}$, $-\frac{2}{3}$, $42$, $\frac{-28}{2}$ \medskip\\  %10
$\Box$ $3$, $6$, $-12$, $1$ \medskip \\ %2
$\Box$ $17$, $5$, $43$, $199$ \medskip\\ %18
$\Box$ $5$, $\sqrt2$, $-13$, $\sqrt{13}$ \medskip\\ %11
$\Box$ $(2,2)$ ; $(1,5)$ ; $(3,1)$ ; $(1,1)$ \medskip\\ %19
$\Box$ $-4$, $-2$, $0$, $3$ \medskip\\ %12
$\Box$ $\frac73$, $-17$, $-\frac{52}3$, $0$  \medskip\\  %5
$\Box$ $0$, $1$ \medskip\\ %9
$\Box$ $-7$, $-3$, $5$, $17$ \medskip  \\  %1
$\Box$ $(3,6)$ ; $(12,24)$ ; $(5,10)$ ; $(131,262)$ \medskip\\  %20
$\Box$ $(2,5)$ ; $(1,6)$ ; $(4,3)$ ; $(3,4)$ \medskip\\  %15
$\Box$ $\sqrt2$, $-\pi$, $2.256$, $-4.99$ \medskip\\  %13
$\Box$ $-20$, $1$, $50$, $778$ \medskip\\  %7
$\Box$ $\varnothing$ \medskip\\  %17

\end{multicols}

\pagebreak

\moddocdis{Set-builder notation}{2}{06}{Lesson}

\section*{Glossary}

\begin{multicols}{2}
\subsection*{Set-builder notation}

\begin{description}
\item[$\{\ldots\}$ :] A set.
\item[$:$ or $|$ :] ``such that''.
\end{description}

\subsection*{Sets of numbers}

\begin{description}
\item[$\N$ :] The whole numbers (natural numbers and zero).
\item[$\Z$ :] The integers.
\item[$\D$ :] The decimal numbers.
\item[$\Q$ :] The rational numbers
\item[$\R$ :] The real numbers.
\item[$\R^\star$ :] The non-zero real numbers.
\item[$\R^+$ :] The positive real numbers.
\item[$\R^-$ :] The negative real numbers.
\end{description}

\subsection*{Set, subsets and elements}

\begin{description}
\item[$\cap$ :] The intersection of two sets.
\item[$\cup$ :] The union of two sets.
\item[$\in$ :] ``is an element of the set''
\item[$\subset$ :] ``is a subset of the set''
\end{description}


\subsection*{Logical operators}

\begin{description}
\item[$\neg$ :] The negation of a proposition, ``not''.
\item[$\wedge$ :] The conjunction of two propositions, ``and''.
\item[$\vee$ :] The disjunction of two propositions, ``or''.
\item[$\forall$ :] ``For all''
\item[$\exists$ :] ``There exists''
\item[$\nexists$ :] ``There doesn't exist''
\end{description}
\end{multicols}

\section*{Russell's paradox}

Let $R$ be the set of all sets that don't contain themselves, that
is $$R=\{X:X\mbox{ is a set and }X\not\in X\}.$$
Then $R$  either is or is not an element of itself. 
\begin{itemize}
\itemb If $R$ is not an element of itself, then $R$ is a set that doesn't contain itself, and so $R$ is an
element of $R$, which is precisely the contrary of what we
supposed.
\itemb Conversely, if $R$ is an element of itself, then $R$ does not
contain itself, so we also get to a contradiction.
\end{itemize}
So $R$ cannot be a set, or our definition of a set is not
satisfactory.

\end{document}
