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%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
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% Document Info
\author{apowers313}
\pdfinfo{
  /Title (mereotopology.pdf)
  /Creator (Cheatography)
  /Author (apowers313)
  /Subject (Mereotopology Cheat Sheet)
}

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% Header and Footer
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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Mereotopology Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{apowers313} via \textcolor{DarkBackground}{\uline{cheatography.com/31528/cs/9577/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}apowers313 \\
  \uline{cheatography.com/apowers313} \\
        \uline{\seqsplit{ato}.ms}
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 26th October, 2016.\\
   Updated 24th October, 2016.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
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% www.emerson.emory.edu/services/latex/latex_169.html
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\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{x{1.9 cm} x{3.268 cm} x{2.432 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Ground Mereology Axioms}}  \tn
% Row 0
\SetRowColor{LightBackground}
\{\{bb\}\}axiom & \{\{bb\}\}meaning & \{\{bb\}\}defn. \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
𝗠 & Ground Mereology &  \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
Pxy & x is a part of y &  \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
\{\{nobreak\}\}Reflexivity & x is a part of itself & \{\{nobreak\}\}Pxx \tn 
% Row Count 7 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\{\{nobreak\}\}Antisymmetry & x and y can't be parts of each other, unless they are actually the same thing & \{\{nobreak\}\}Pxy ∧ Pyx → x=y \tn 
% Row Count 12 (+ 5)
% Row 5
\SetRowColor{white}
\{\{nobreak\}\}Transitivity & if x is a part of y, and y is a part of z, then x is a part of z & \{\{nobreak\}\}Pxy ∧ Pyz → Pxy \tn 
% Row Count 16 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{1.52 cm} x{2.28 cm} x{3.8 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Ground Mereology Definitions}}  \tn
% Row 0
\SetRowColor{LightBackground}
\{\{bb\}\}sym. & \{\{bb\}\}meaning & \{\{bb\}\}defn. \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
PP & Proper Part & PPxy := Pxy ∧ ¬Pyx \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
O & Overlap & Oxy := ∃z (Pzx ∧ Pzy) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
U & Underlap & Uxy := ∃z (Pxz ∧ Pyz) \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
OX & \seqsplit{Over-Crossing} & OXxy := Oxy ∧ ¬Pxy \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
UX & \seqsplit{Under-Crossing} & UXxy := Uxy ∧ ¬Pyx \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
PO & Proper Overlap & POxy := OXxy ∧ OXyx \tn 
% Row Count 14 (+ 2)
% Row 7
\SetRowColor{white}
PU & Proper Underlap & PUxy := UXxy ∧ UXyx \tn 
% Row Count 16 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{6.24 cm} x{1.76 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Derived Statements}}  \tn
% Row 0
\SetRowColor{LightBackground}
Overlapping is Reflexive & Oxx \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Overlapping is Transitive & Oxy → Oyx \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\{\{nobreak\}\}Proper Parts are not Reflexive & ¬PPxx \tn 
% Row Count 5 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.52 cm} x{4.48 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Extensional Mereology}}  \tn
% Row 0
\SetRowColor{LightBackground}
𝗘𝗠 & Extensional Mereology \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\{\{nobreak\}\}Supplementation Axiom & \{\{nobreak\}\}¬Pxy → ∃z(Pzx ∧ ¬Ozy) \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Weak Supplementation & \{\{nobreak\}\}𝗘𝗠 ⊢ PPxy → ∃z(PPzy ∧ ¬Ozx) \tn 
% Row Count 6 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{If all the proper parts of X are proper parts of Y, X is part of Y} \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{If two objects have the exact same proper parts, they are the same object} \tn 
% Row Count 10 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{1.84 cm} x{6.16 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Closed (Extensional) Mereology}}  \tn
% Row 0
\SetRowColor{LightBackground}
\seqsplit{𝗖𝗘𝗠} & Closed Extensional Mereology \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
℩ & description operator\{\{nl\}\}℩x is "the unique x such that" \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
x+y & sum (or fusion)\{\{nl\}\}Oxy→∃x∀w(Pwz↔(Pwx∧Pwy))\{\{nl\}\}defined as:\{\{nl\}\}℩z∀w(Owz↔(Owx∨Owy)) \tn 
% Row Count 8 (+ 4)
% Row 3
\SetRowColor{white}
x×y & product\{\{nl\}\}Uxy→∃z∀w(Owz↔(Owx∨Owy))\{\{nl\}\}defined as:\{\{nl\}\}℩z∀w(Pwz↔(Pwx∧Pwy)) \tn 
% Row Count 12 (+ 4)
% Row 4
\SetRowColor{LightBackground}
x-y & difference\{\{nl\}\}∃z(Pzx∧¬Ozy)→∃z∀w(Pwz↔(Pwx∧¬Owy))\{\{nl\}\}defined as:\{\{nl\}\}℩z∀w(Pwz↔(Pwx∧¬Owy)) \tn 
% Row Count 16 (+ 4)
% Row 5
\SetRowColor{white}
𝑈 & universe\{\{nl\}\}∃z∀x(Pxz)\{\{nl\}\}defined as:\{\{nl\}\}℩z∀x(Pxz) \tn 
% Row Count 19 (+ 3)
% Row 6
\SetRowColor{LightBackground}
∼x & compliment\{\{nl\}\}U-x \tn 
% Row Count 20 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{1.84 cm} x{6.16 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{General (Extensional) Mereology}}  \tn
% Row 0
\SetRowColor{LightBackground}
\seqsplit{𝗚𝗘𝗠} & General Extensional Mereology \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Fusion Axiom & ∃xΦ → ∃z∀y(Oyz ↔ ∃x(Φ∧Oyx)) \tn 
% Row Count 4 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{1.976 cm} x{2.28 cm} x{3.344 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Ground Topology Axioms}}  \tn
% Row 0
\SetRowColor{LightBackground}
𝗧 & Ground Topology &  \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cxy & x is connect to y &  \tn 
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% Row 2
\SetRowColor{LightBackground}
\seqsplit{Reflexivity} & x is connected to itself & Cxx \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Symmetry &  & Cxy → C yx \tn 
% Row Count 7 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\{\{noshy\}\}Transitivity &  & \{\{nobreak\}\}Pxy → ∀z(Czx → Czy) \tn 
% Row Count 10 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{p{0.912 cm} x{5.928 cm} p{0.76 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Ground Topology Definitions}}  \tn
% Row 0
\SetRowColor{LightBackground}
EC & External Connection &  \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
TP & Tangential Part &  \tn 
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\SetRowColor{LightBackground}
TPP & Tangential Proper Part &  \tn 
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\SetRowColor{white}
IP & Internal Part &  \tn 
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\SetRowColor{LightBackground}
IPP & Internal Proper Part &  \tn 
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\SetRowColor{white}
E & Enclosure &  \tn 
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\SetRowColor{LightBackground}
IE & Internal Enclosure &  \tn 
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\SetRowColor{white}
TE & Tangential Enclosure &  \tn 
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\SetRowColor{LightBackground}
S & Superposition &  \tn 
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\SetRowColor{white}
PS & Proper Superposition &  \tn 
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\SetRowColor{LightBackground}
I & Coincidence &  \tn 
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\SetRowColor{white}
A & Abutting &  \tn 
% Row Count 12 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{p{1.52 cm} x{6.48 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Predicate Logic}}  \tn
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\SetRowColor{LightBackground}
¬ & not \tn 
% Row Count 1 (+ 1)
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\SetRowColor{white}
∧ & and \tn 
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\SetRowColor{LightBackground}
∨ & or \tn 
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\SetRowColor{white}
∀ & for every \tn 
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\SetRowColor{LightBackground}
∃ & there exists \tn 
% Row Count 5 (+ 1)
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\SetRowColor{white}
→ & implies \tn 
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\SetRowColor{LightBackground}
:= & definition \tn 
% Row Count 7 (+ 1)
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\SetRowColor{white}
↔ & iff \tn 
% Row Count 8 (+ 1)
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\SetRowColor{LightBackground}
⊢ & provable \tn 
% Row Count 9 (+ 1)
% Row 9
\SetRowColor{white}
⊨ & entails \tn 
% Row Count 10 (+ 1)
% Row 10
\SetRowColor{LightBackground}
⊤ & tautology \tn 
% Row Count 11 (+ 1)
% Row 11
\SetRowColor{white}
⊥ & contradiction \tn 
% Row Count 12 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Basic Patterns in Mereology}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/apowers313_1477251585_Screen Shot 2016-10-23 at 12.37.40 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{Credit: Varzi 1996, used without permission. The relations \newline in parenthesis hold if there is a larger z including both x and y.}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Basic Patterns in Mereotopology}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/apowers313_1477333731_Screen Shot 2016-10-24 at 11.26.19 AM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{Credit: Varzi 1996, used without permission. Seven basic patterns of the connection relationship.}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.36 cm} x{4.64 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Examples}}  \tn
% Row 0
\SetRowColor{LightBackground}
Part & Your finger is part of your hand \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Reflexivity & Your finger is part of your finger \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\{\{nobreak\}\}Antisymmetry & Your finger is part of your hand, but your hand is not part of your finger \tn 
% Row Count 8 (+ 4)
% Row 3
\SetRowColor{white}
\{\{nobreak\}\}Transitivity & Your finger is part of your hand, and your hand is part of your body, so your finger is part of your body \tn 
% Row Count 13 (+ 5)
% Row 4
\SetRowColor{LightBackground}
Proper Part & A tail is a proper part of a cat \tn 
% Row Count 15 (+ 2)
% Row 5
\SetRowColor{white}
Overlapping & Two roads overlap at their intersection \tn 
% Row Count 17 (+ 2)
% Row 6
\SetRowColor{LightBackground}
\{\{nobreak\}\}Underlapping & Your finger and thumb are underlapping parts of your hand \tn 
% Row Count 20 (+ 3)
% Row 7
\SetRowColor{white}
\{\{nobreak\}\}Supplementation & Road A is not part of Road B, because there is at least some of Road A that doesn't overlap Road B \tn 
% Row Count 25 (+ 5)
% Row 8
\SetRowColor{LightBackground}
\{\{noshy\}\}Weak Supplementation & Road A is not a proper part of Road B, because at least some of Road A is outside Road B \tn 
% Row Count 29 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{2.052 cm} x{3.8 cm} x{1.748 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Alternate Notations}}  \tn
% Row 0
\SetRowColor{LightBackground}
\{\{bb\}\}symbol & \{\{bb\}\}meaning & \{\{bb\}\}from \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
≪ & is a proper part of & Simon 1987 \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
≺ & is an improper part of & Simon 1987 \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
○ & overlaps & Simon 1987 \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
⎱ & is disjoint from & Simon 1987 \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
Pxx & is a part of & Smith \tn 
% Row Count 11 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.116 cm} x{3.268 cm} p{1.216 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Mereological Operations}}  \tn
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\SetRowColor{LightBackground}
⋅ & binary product & x⋅y \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
+ & binary sum & x+y \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
- & difference & x-y \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
σx⌜Fx⌝ & fusion &  \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
𝜋x⌜Fx⌝ & nucleus &  \tn 
% Row Count 5 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{1.44 cm} x{1.944 cm} x{1.296 cm} x{2.52 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{8.4cm}}{\bf\textcolor{white}{Smith (1996) Mereology Definitions}}  \tn
% Row 0
\SetRowColor{LightBackground}
\{\{bb\}\}sym. & \{\{bb\}\}meaning & \{\{bb\}\}ex. & \{\{bb\}\}defn. \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
P & is a part of & xPy &  \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
O & overlaps & xOy & ∃z(zPx ∧ zPy) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
D & discrete & xDy & ¬xOy \tn 
% Row Count 7 (+ 1)
% Row 4
\SetRowColor{LightBackground}
Pt() & is a point & Pt(x) & \seqsplit{∀y(yPx→y=x)} \tn 
% Row Count 9 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}