% ============================================================
%  Venn Diagram Shading Worksheet — SOLUTIONS
%  J. Murphy  |  JMaths  |  Sets & Probability
%  MYP Year 9
% ============================================================
\documentclass[11pt]{article}

\usepackage[a4paper,margin=1cm,includeheadfoot,headsep=5pt,footskip=14pt]{geometry}
\usepackage{amsmath,amssymb}
\usepackage{enumitem}
\usepackage[table]{xcolor}
\usepackage{fancyhdr}
\usepackage{tikz}
\usepackage{mdframed}
\usepackage{array}
\usepackage{booktabs}

\usetikzlibrary{calc, intersections}

% ── House colours ────────────────────────────────────────────────────────────
\definecolor{ibblue}{HTML}{0F4C81}
\definecolor{accent}{HTML}{0F766E}
\definecolor{shadeblue}{HTML}{B3CCE8}
\definecolor{correct}{HTML}{15803D}
\definecolor{warmgray}{RGB}{245,246,248}

% ── Header / footer ──────────────────────────────────────────────────────────
\pagestyle{fancy}
\fancyhf{}
\renewcommand{\headrulewidth}{0.6pt}
\renewcommand{\headrule}{\color{ibblue}\hrule width\headwidth height 0.6pt}
\renewcommand{\footrulewidth}{0pt}
\lhead{\footnotesize\color{ibblue}\bfseries MYP Year 9 Mathematics}
\chead{\footnotesize\color{ibblue}\bfseries Venn Diagram Shading — SOLUTIONS}
\rhead{\footnotesize\color{accent}\bfseries Teacher Copy}
\cfoot{\footnotesize\color{ibblue}\thepage}
\rfoot{\footnotesize\color{accent} J. Murphy}

% ── Spacing ──────────────────────────────────────────────────────────────────
\setlength{\parindent}{0pt}
\setlength{\parskip}{2pt}
\setlist[enumerate]{nosep,leftmargin=*,labelsep=0.5em}
\setlist[itemize]{nosep,leftmargin=*,labelsep=0.4em}

% ── Section style ────────────────────────────────────────────────────────────
\newcommand{\jmsection}[1]{%
  \vspace{6pt}%
  \noindent\colorbox{ibblue}{\parbox{\dimexpr\linewidth-2\fboxsep}{%
    \color{white}\bfseries\large\hspace{4pt}#1\hspace{4pt}}}%
  \vspace{4pt}\par}

% ============================================================
%  SHADING MACROS
%  Circles (scale 0.72):
%    A: centre (2.15,2) radius 1.45
%    B: centre (3.85,2) radius 1.45
%  Rectangle: (0,0)-(6,4)
%
%  Three-set (scale 0.78):
%    A: (2.7,4.3) r=1.85
%    B: (5.3,4.3) r=1.85
%    C: (4.0,2.1) r=1.85
%  Rectangle: (0,0)-(8,7)
% ============================================================

% ---------- Two-set base (draws rect + circles + labels, no fill) ----------
\newcommand{\twoVennBase}{%
  \draw[ibblue!60, line width=0.7pt] (0,0) rectangle (6,4);
  \node[ibblue!70, font=\small] at (0.35,0.35) {$U$};
  \draw[ibblue, line width=1.2pt] (2.15,2) circle (1.45);
  \draw[ibblue, line width=1.2pt] (3.85,2) circle (1.45);
  \node[ibblue, font=\bfseries] at (1.1,3.4) {$A$};
  \node[ibblue, font=\bfseries] at (4.9,3.4) {$B$};
}

% ---------- Three-set base ----------
\newcommand{\threeVennBase}{%
  \draw[ibblue!60, line width=0.7pt] (0,0) rectangle (8,7);
  \node[ibblue!70, font=\small] at (0.4,0.4) {$U$};
  \draw[ibblue, line width=1.2pt] (2.7,4.3) circle (1.85);
  \draw[ibblue, line width=1.2pt] (5.3,4.3) circle (1.85);
  \draw[ibblue, line width=1.2pt] (4.0,2.1) circle (1.85);
  \node[ibblue, font=\bfseries] at (1.2,6.5) {$A$};
  \node[ibblue, font=\bfseries] at (6.8,6.5) {$B$};
  \node[ibblue, font=\bfseries] at (4.0,0.3) {$C$};
}

% ============================================================
%  TWO-SET SHADED VENNS
% ============================================================

% (a) A ∪ B  — shade everything inside A or B
\newcommand{\solAuB}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}
    \clip (0,0) rectangle (6,4);
    \fill[shadeblue] (2.15,2) circle (1.45);
    \fill[shadeblue] (3.85,2) circle (1.45);
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (b) A ∩ B  — shade only the lens overlap
\newcommand{\solAiB}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}
    \clip (2.15,2) circle (1.45);
    \fill[shadeblue] (3.85,2) circle (1.45);
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (c) A′  — shade everything NOT in A  (entire rect minus circle A)
\newcommand{\solAc}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}[even odd rule]
    \clip (0,0) rectangle (6,4) (2.15,2) circle (1.45);
    \fill[shadeblue] (0,0) rectangle (6,4);
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (d) B′  — shade everything NOT in B
\newcommand{\solBc}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}[even odd rule]
    \clip (0,0) rectangle (6,4) (3.85,2) circle (1.45);
    \fill[shadeblue] (0,0) rectangle (6,4);
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (e) A ∩ B′  — only A, excluding the overlap
\newcommand{\solAiBC}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}[even odd rule]
    \clip (2.15,2) circle (1.45) (3.85,2) circle (1.45);
    \fill[shadeblue] (2.15,2) circle (1.45);
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (f) A′ ∩ B  — only B, excluding the overlap
\newcommand{\solACiB}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}[even odd rule]
    \clip (3.85,2) circle (1.45) (2.15,2) circle (1.45);
    \fill[shadeblue] (3.85,2) circle (1.45);
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (g) (A ∪ B)′  — shade outside both circles
\newcommand{\solAuBc}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}[even odd rule]
    \clip (0,0) rectangle (6,4) (2.15,2) circle (1.45) (3.85,2) circle (1.45);
    \fill[shadeblue] (0,0) rectangle (6,4);
  \end{scope}
  % re-fill to cover even-odd artifact between circles
  \begin{scope}[even odd rule]
    \clip (0,0) rectangle (6,4) (2.15,2) circle (1.45) (3.85,2) circle (1.45);
    \fill[shadeblue] (0,0) rectangle (6,4);
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (h) (A ∩ B)′  — everything except the lens
\newcommand{\solAiBc}{%
\begin{tikzpicture}[scale=0.72]
  % Fill everything, then white-out the lens
  \begin{scope}
    \clip (0,0) rectangle (6,4);
    \fill[shadeblue] (0,0) rectangle (6,4);
    % cut out intersection
    \begin{scope}
      \clip (2.15,2) circle (1.45);
      \fill[white] (3.85,2) circle (1.45);
    \end{scope}
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (i) A′ ∪ B  — complement of A, union with B
%   = everything outside A, plus all of B
%   = universe minus (A only) = universe minus (A∩B')
\newcommand{\solACuB}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}
    \clip (0,0) rectangle (6,4);
    % fill everything
    \fill[shadeblue] (0,0) rectangle (6,4);
    % remove A-only (A minus overlap)
    \begin{scope}[even odd rule]
      \clip (2.15,2) circle (1.45) (3.85,2) circle (1.45);
      \fill[white] (2.15,2) circle (1.45);
    \end{scope}
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (j) A′ ∩ B′  = (A ∪ B)′  — same as (g)
\newcommand{\solACiBC}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}[even odd rule]
    \clip (0,0) rectangle (6,4) (2.15,2) circle (1.45) (3.85,2) circle (1.45);
    \fill[shadeblue] (0,0) rectangle (6,4);
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (k) A ∪ B′  — all of A, plus everything outside B
%   = everything except (B only)
\newcommand{\solAuBC}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}
    \clip (0,0) rectangle (6,4);
    \fill[shadeblue] (0,0) rectangle (6,4);
    % remove B-only
    \begin{scope}[even odd rule]
      \clip (3.85,2) circle (1.45) (2.15,2) circle (1.45);
      \fill[white] (3.85,2) circle (1.45);
    \end{scope}
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% (l) A′ ∪ B′  = (A ∩ B)′  — everything except the lens
\newcommand{\solACuBC}{%
\begin{tikzpicture}[scale=0.72]
  \begin{scope}
    \clip (0,0) rectangle (6,4);
    \fill[shadeblue] (0,0) rectangle (6,4);
    \begin{scope}
      \clip (2.15,2) circle (1.45);
      \fill[white] (3.85,2) circle (1.45);
    \end{scope}
  \end{scope}
  \twoVennBase
\end{tikzpicture}}

% ============================================================
%  THREE-SET SHADED VENNS
%  A:(2.7,4.3)r=1.85  B:(5.3,4.3)r=1.85  C:(4.0,2.1)r=1.85
% ============================================================

% (a) A ∩ B ∩ C — central region only
\newcommand{\threeAiBiC}{%
\begin{tikzpicture}[scale=0.78]
  \begin{scope}
    \clip (2.7,4.3) circle (1.85);
    \clip (5.3,4.3) circle (1.85);
    \fill[shadeblue] (4.0,2.1) circle (1.85);
  \end{scope}
  \threeVennBase
\end{tikzpicture}}

% (b) A ∪ B ∪ C — all three circles
\newcommand{\threeAuBuC}{%
\begin{tikzpicture}[scale=0.78]
  \begin{scope}
    \clip (0,0) rectangle (8,7);
    \fill[shadeblue] (2.7,4.3) circle (1.85);
    \fill[shadeblue] (5.3,4.3) circle (1.85);
    \fill[shadeblue] (4.0,2.1) circle (1.85);
  \end{scope}
  \threeVennBase
\end{tikzpicture}}

% (c) (A ∪ B ∪ C)′ — outside all three circles
\newcommand{\threeAuBuCc}{%
\begin{tikzpicture}[scale=0.78]
  \begin{scope}[even odd rule]
    \clip (0,0) rectangle (8,7)
          (2.7,4.3) circle (1.85)
          (5.3,4.3) circle (1.85)
          (4.0,2.1) circle (1.85);
    \fill[shadeblue] (0,0) rectangle (8,7);
  \end{scope}
  \threeVennBase
\end{tikzpicture}}

% (d) A ∩ B ∩ C' = A∩B minus C
%   = lens of A&B, minus the central triple overlap
\newcommand{\threeAiBminusC}{%
\begin{tikzpicture}[scale=0.78]
  % A∩B minus C: clip to A∩B, then cut out C
  \begin{scope}[even odd rule]
    % first define A∩B region, then subtract C
    \begin{scope}
      \clip (2.7,4.3) circle (1.85);
      \begin{scope}[even odd rule]
        \clip (5.3,4.3) circle (1.85) (4.0,2.1) circle (1.85);
        \fill[shadeblue] (5.3,4.3) circle (1.85);
      \end{scope}
    \end{scope}
  \end{scope}
  \threeVennBase
\end{tikzpicture}}

% (e) (A ∪ B) ∩ C′
%   = everything in A or B, but NOT in C
\newcommand{\threeAuBiCc}{%
\begin{tikzpicture}[scale=0.78]
  \begin{scope}
    % Fill A
    \begin{scope}[even odd rule]
      \clip (2.7,4.3) circle (1.85) (4.0,2.1) circle (1.85);
      \fill[shadeblue] (2.7,4.3) circle (1.85);
    \end{scope}
    % Fill B
    \begin{scope}[even odd rule]
      \clip (5.3,4.3) circle (1.85) (4.0,2.1) circle (1.85);
      \fill[shadeblue] (5.3,4.3) circle (1.85);
    \end{scope}
  \end{scope}
  \threeVennBase
\end{tikzpicture}}

% (f) A ∩ B′ ∩ C′
%   = only the A-alone region (inside A, outside B and C)
\newcommand{\threeAiBCcCc}{%
\begin{tikzpicture}[scale=0.78]
  \begin{scope}[even odd rule]
    \clip (2.7,4.3) circle (1.85)
          (5.3,4.3) circle (1.85);
    \begin{scope}[even odd rule]
      \clip (2.7,4.3) circle (1.85)
            (4.0,2.1) circle (1.85);
      \fill[shadeblue] (2.7,4.3) circle (1.85);
    \end{scope}
  \end{scope}
  \threeVennBase
\end{tikzpicture}}

% ============================================================
%  CONTEXT VENN (D1) — large, F and G circles, scale 1.0
% ============================================================
% F:(2.8,2.75)r=2.1   G:(5.2,2.75)r=2.1   Box:(0,0)-(8,5.5)
\newcommand{\contextVennBase}{%
  \draw[ibblue!60, line width=0.8pt] (0,0) rectangle (8,5.5);
  \node[ibblue!70] at (0.4,0.4) {$U$};
  \draw[ibblue, line width=1.4pt] (2.8,2.75) circle (2.1);
  \draw[ibblue, line width=1.4pt] (5.2,2.75) circle (2.1);
  \node[ibblue, font=\bfseries\large] at (1.1,5.0) {$F$};
  \node[ibblue, font=\bfseries\large] at (6.9,5.0) {$G$};
}

% ============================================================
\begin{document}

% ── Title block ──────────────────────────────────────────────────────────────
\begin{center}
  {\color{ibblue}\LARGE\bfseries Shading Venn Diagrams — \textsc{Solutions}}\\[3pt]
  {\color{accent}\small Sets $\cdot$ Complements $\cdot$ Intersections $\cdot$ Unions}
\end{center}
{\color{ibblue}\rule{\linewidth}{0.8pt}}
\vspace{4pt}

% ============================================================
\jmsection{Section A \quad Notation Warm-Up --- Answers}
% ============================================================

\textbf{A1.}\; $U = \{1,2,3,4,5,6,7,8,9,10\}$,\; $P = \{2,4,6,8,10\}$,\; $Q = \{1,2,3,4,5\}$.

\begin{enumerate}[label=(\alph*), itemsep=3pt]
  \item $P \cup Q = \{1,2,3,4,5,6,8,10\}$
  \item $P \cap Q = \{2,4\}$
  \item $P' = \{1,3,5,7,9\}$
  \item $P' \cap Q = \{1,3,5\}$
  \item $(P \cup Q)' = \{7,9\}$
\end{enumerate}

\vspace{6pt}

% ============================================================
\jmsection{Section B \quad Two-Set Shading --- Solutions}
% ============================================================

{\small Shaded regions shown in blue. Verify by checking which regions satisfy the set expression.}

\vspace{5pt}

% ── Row 1 ──────────────────────────────────────────────────────────────────
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (a)\quad $A \cup B$}\\[2pt]
  \solAuB\\
  {\color{correct}\scriptsize\itshape Everything inside A or B}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (b)\quad $A \cap B$}\\[2pt]
  \solAiB\\
  {\color{correct}\scriptsize\itshape Lens (overlap) only}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (c)\quad $A'$}\\[2pt]
  \solAc\\
  {\color{correct}\scriptsize\itshape Everything outside A}
\end{minipage}

\vspace{10pt}

% ── Row 2 ──────────────────────────────────────────────────────────────────
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (d)\quad $B'$}\\[2pt]
  \solBc\\
  {\color{correct}\scriptsize\itshape Everything outside B}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (e)\quad $A \cap B'$}\\[2pt]
  \solAiBC\\
  {\color{correct}\scriptsize\itshape A only (left crescent)}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (f)\quad $A' \cap B$}\\[2pt]
  \solACiB\\
  {\color{correct}\scriptsize\itshape B only (right crescent)}
\end{minipage}

\vspace{10pt}

% ── Row 3 ──────────────────────────────────────────────────────────────────
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (g)\quad $(A \cup B)'$}\\[2pt]
  \solAuBc\\
  {\color{correct}\scriptsize\itshape Outside both circles}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (h)\quad $(A \cap B)'$}\\[2pt]
  \solAiBc\\
  {\color{correct}\scriptsize\itshape Everything except the lens}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (i)\quad $A' \cup B$}\\[2pt]
  \solACuB\\
  {\color{correct}\scriptsize\itshape All of B, plus outside A}
\end{minipage}

\vspace{10pt}

% ── Row 4 ──────────────────────────────────────────────────────────────────
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (j)\quad $A' \cap B'$}\\[2pt]
  \solACiBC\\
  {\color{correct}\scriptsize\itshape $=(A\cup B)'$: outside both}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (k)\quad $A \cup B'$}\\[2pt]
  \solAuBC\\
  {\color{correct}\scriptsize\itshape All except B-only region}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (l)\quad $A' \cup B'$}\\[2pt]
  \solACuBC\\
  {\color{correct}\scriptsize\itshape $=(A\cap B)'$: all except lens}
\end{minipage}

\newpage

% ============================================================
\jmsection{Section C \quad Three-Set Shading --- Solutions}
% ============================================================

{\small $A$: top-left circle, $B$: top-right circle, $C$: bottom circle.}

\vspace{5pt}

% Row 1
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (a)\quad $A \cap B \cap C$}\\[2pt]
  \threeAiBiC\\
  {\color{correct}\scriptsize\itshape Centre only}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (b)\quad $A \cup B \cup C$}\\[2pt]
  \threeAuBuC\\
  {\color{correct}\scriptsize\itshape All 3 circles}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (c)\quad $(A \cup B \cup C)'$}\\[2pt]
  \threeAuBuCc\\
  {\color{correct}\scriptsize\itshape Outside all three circles}
\end{minipage}

\vspace{12pt}

\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (d)\quad $A \cap B \cap C'$}\\[2pt]
  \threeAiBminusC\\
  {\color{correct}\scriptsize\itshape A\&B lens, excluding centre}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (e)\quad $(A \cup B) \cap C'$}\\[2pt]
  \threeAuBiCc\\
  {\color{correct}\scriptsize\itshape A and B above C boundary}
\end{minipage}
\hfill
\begin{minipage}[t]{0.31\linewidth}
  \centering
  {\color{accent}\bfseries\small (f)\quad $A \cap B' \cap C'$}\\[2pt]
  \threeAiBCcCc\\
  {\color{correct}\scriptsize\itshape A-only: left petal}
\end{minipage}

\vspace{10pt}

\textbf{C7. Eight-region identification — Answers:}

\vspace{3pt}

\begin{enumerate}[label=\textbf{Region \arabic*:}, leftmargin=3.8cm, itemsep=2pt]
  \item $A \cap B' \cap C'$ \quad {\color{correct}\small (elements only in $A$)}
  \item $A \cap B \cap C'$ \quad {\color{correct}\small (elements in $A$ and $B$ but not $C$)}
  \item $A' \cap B \cap C'$ \quad {\color{correct}\small (elements only in $B$)}
  \item $A \cap B' \cap C$ \quad {\color{correct}\small (elements in $A$ and $C$ but not $B$)}
  \item $A \cap B \cap C$ \quad {\color{correct}\small (elements in all three sets)}
  \item $A' \cap B \cap C$ \quad {\color{correct}\small (elements in $B$ and $C$ but not $A$)}
  \item $A' \cap B' \cap C$ \quad {\color{correct}\small (elements only in $C$)}
  \item $A' \cap B' \cap C'$ \quad {\color{correct}\small (elements in none of the three sets)}
\end{enumerate}

\newpage

% ============================================================
\jmsection{Section D \quad Context Problem --- Solutions}
% ============================================================

\textbf{D1.}\; $U$ = 40 students, $F$ = French, $G$ = German.
$n(F) = 28$, $n(G) = 19$, $n(F \cap G) = 7$.

\begin{enumerate}[label=(\alph*), itemsep=8pt]

\item \textbf{Consistency check:}\\
Using inclusion-exclusion: $n(F \cup G) = 28 + 19 - 7 = 40$.\\
Since every student studies at least one language, the total should equal 40. $\checkmark$

\item \textbf{Completed Venn diagram:}

\vspace{5pt}
\begin{center}
\begin{tikzpicture}[scale=1.0, line width=1pt]
  \contextVennBase
  % F only region
  \node[font=\bfseries\large] at (1.8,2.75) {$21$};
  % F∩G overlap
  \node[font=\bfseries\large] at (4.0,2.75) {$7$};
  % G only region
  \node[font=\bfseries\large] at (6.2,2.75) {$12$};
  % outside (neither)
  \node[font=\large\itshape, ibblue!70] at (0.8,1.0) {$0$};
\end{tikzpicture}
\end{center}

{\color{correct}\small
  $F$ only: $28 - 7 = 21$. \quad
  $G$ only: $19 - 7 = 12$. \quad
  Neither: $40 - 40 = 0$.
}

\item $P(F \text{ only}) = P(F \cap G') = \dfrac{21}{40}$

\item $P(\text{exactly one language}) = \dfrac{21 + 12}{40} = \dfrac{33}{40}$

\item $P(F \mid G) = \dfrac{n(F \cap G)}{n(G)} = \dfrac{7}{19}$

\item \textbf{Shaded Venn for $F' \cap G$:}

\vspace{5pt}
\begin{center}
\begin{tikzpicture}[scale=1.0, line width=1pt]
  % Shade F'∩G = G only crescent
  \begin{scope}[even odd rule]
    \clip (5.2,2.75) circle (2.1) (2.8,2.75) circle (2.1);
    \fill[shadeblue] (5.2,2.75) circle (2.1);
  \end{scope}
  \contextVennBase
\end{tikzpicture}
\end{center}

{\color{correct}\small $F' \cap G$ represents students who study German \textbf{but not} French ($n = 12$).}

\end{enumerate}

\vspace{12pt}

\begin{mdframed}[backgroundcolor=warmgray, linecolor=accent, linewidth=0.8pt,
                 innertopmargin=5pt, innerbottommargin=5pt,
                 innerleftmargin=6pt, innerrightmargin=6pt]
{\small\color{ibblue}
\textbf{Key identities to remind students:}
\begin{itemize}[itemsep=1pt]
  \item $A' \cap B' = (A \cup B)'$ \quad (De Morgan's first law)
  \item $A' \cup B' = (A \cap B)'$ \quad (De Morgan's second law)
  \item $|A \cup B| = |A| + |B| - |A \cap B|$ \quad (inclusion-exclusion)
\end{itemize}
}
\end{mdframed}

\vspace{8pt}

\begin{center}
  {\color{ibblue!50}\rule{0.6\linewidth}{0.5pt}}\\[2pt]
  {\footnotesize\color{ibblue!70} \textit{JMaths} \;$\cdot$\; Sets \& Venn Diagrams \;$\cdot$\; MYP Year 9 \;$\cdot$\; \textsc{Solutions}}
\end{center}

\end{document}
