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changeset 58:8946d732625f default tip

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Use less dangerous command names
author Markus Kaiser <markus.kaiser@in.tum.de>
date Thu, 10 Apr 2014 00:10:34 +0200
parents a78ea627829e
children
files notes/tex/logic.tex
diffstat 1 files changed, 87 insertions(+), 87 deletions(-) [+]
line wrap: on
line diff
--- a/notes/tex/logic.tex	Thu Apr 10 00:10:18 2014 +0200
+++ b/notes/tex/logic.tex	Thu Apr 10 00:10:34 2014 +0200
@@ -208,32 +208,32 @@
 \defineUnit{aussagenlogikaequivalenzen}{%
 {
     \newcommand{\spc}{\hfill}
-    \newcommand{\F}{F}
-    \newcommand{\G}{G}
-    \newcommand{\K}{H}
+    \newcommand{\VarF}{F}
+    \newcommand{\VarG}{G}
+    \newcommand{\VarK}{H}
 
     \begin{frame}
         \frametitle{Äquivalenzregeln}
 
         \begin{description}[Triviale Kontradiktion\quad]
-            \item[Identität] $\F \wedge \true \equiv \F \spc \F \vee \false \equiv \F$
-            \item[Dominanz] $\F \vee \true \equiv \true \spc \F \wedge \false \equiv \false$
-            \item[Idempotenz] $\F \vee \F \equiv \F \spc \F \wedge \F \equiv \F$
-            \item[Doppelte Negation] $\neg \neg \F \equiv \F$
-            \item[Triviale Tautologie] $\F \vee \neg \F \equiv \true$
-            \item[Triviale Kontradiktion] $\F \wedge \neg \F \equiv \false$
+            \item[Identität] $\VarF \wedge \true \equiv \VarF \spc \VarF \vee \false \equiv \VarF$
+            \item[Dominanz] $\VarF \vee \true \equiv \true \spc \VarF \wedge \false \equiv \false$
+            \item[Idempotenz] $\VarF \vee \VarF \equiv \VarF \spc \VarF \wedge \VarF \equiv \VarF$
+            \item[Doppelte Negation] $\neg \neg \VarF \equiv \VarF$
+            \item[Triviale Tautologie] $\VarF \vee \neg \VarF \equiv \true$
+            \item[Triviale Kontradiktion] $\VarF \wedge \neg \VarF \equiv \false$
                 \bigskip
-            \item[Kommutativität] $\F \vee \G \equiv \G \vee \F$\\
-                $\F \wedge \G \equiv \G \wedge \F$
-            \item[Assoziativität] $(\F \vee \G) \vee \K \equiv \F \vee (\G \vee \K)$\\
-                $(\F \wedge \G) \wedge \K \equiv \F \wedge (\G \wedge \K)$
-            \item[Distributivität] $\F \vee (\G \wedge \K) \equiv (\F \vee \G) \wedge (\F \vee \K)$\\
-                $\F \wedge (\G \vee \K) \equiv (\F \wedge \G) \vee (\F \wedge \K)$
-            \item[De Morgan] $\neg(\F \wedge \G) \equiv \neg \F \vee \neg \G$\\
-                $\neg(\F \vee \G) \equiv \neg\F \wedge \neg\G$
+            \item[Kommutativität] $\VarF \vee \VarG \equiv \VarG \vee \VarF$\\
+                $\VarF \wedge \VarG \equiv \VarG \wedge \VarF$
+            \item[Assoziativität] $(\VarF \vee \VarG) \vee \VarK \equiv \VarF \vee (\VarG \vee \VarK)$\\
+                $(\VarF \wedge \VarG) \wedge \VarK \equiv \VarF \wedge (\VarG \wedge \VarK)$
+            \item[Distributivität] $\VarF \vee (\VarG \wedge \VarK) \equiv (\VarF \vee \VarG) \wedge (\VarF \vee \VarK)$\\
+                $\VarF \wedge (\VarG \vee \VarK) \equiv (\VarF \wedge \VarG) \vee (\VarF \wedge \VarK)$
+            \item[De Morgan] $\neg(\VarF \wedge \VarG) \equiv \neg \VarF \vee \neg \VarG$\\
+                $\neg(\VarF \vee \VarG) \equiv \neg\VarF \wedge \neg\VarG$
                 \bigskip
-            \item[Implikation] $\F \rightarrow \G \equiv \neg \F \vee \G$
-            \item[Bikonditional] $\F \leftrightarrow \G \equiv \neg(\F \otimes \G) \left[ \equiv (\F \rightarrow \G) \wedge (\G \rightarrow \F) \right]$
+            \item[Implikation] $\VarF \rightarrow \VarG \equiv \neg \VarF \vee \VarG$
+            \item[Bikonditional] $\VarF \leftrightarrow \VarG \equiv \neg(\VarF \otimes \VarG) \left[ \equiv (\VarF \rightarrow \VarG) \wedge (\VarG \rightarrow \VarF) \right]$
         \end{description}
     \end{frame}
 
@@ -242,24 +242,24 @@
 
         %\vspace{-2em}
         %\begin{align}
-            %\F \wedge \true &\equiv \F \spc \F \vee \false \equiv \F \tag{\structure{Identität}}\\
-            %\F \vee \true &\equiv \true \spc \F \wedge \false \equiv \false \tag{\structure{Dominanz}}\\
-            %\F \vee \F &\equiv \F \spc \F \wedge \F \equiv \F \tag{\structure{Idempotenz}}\\
-            %\neg \neg \F &\equiv \F \tag{\structure{Doppelte Negation}}\\
-            %\F \vee \neg \F &\equiv \true \tag{\structure{Triviale Tautologie}}\\
-            %\F \wedge \neg \F &\equiv \false \tag{\structure{Triviale Kontradiktion}}\\
+            %\VarF \wedge \true &\equiv \VarF \spc \VarF \vee \false \equiv \VarF \tag{\structure{Identität}}\\
+            %\VarF \vee \true &\equiv \true \spc \VarF \wedge \false \equiv \false \tag{\structure{Dominanz}}\\
+            %\VarF \vee \VarF &\equiv \VarF \spc \VarF \wedge \VarF \equiv \VarF \tag{\structure{Idempotenz}}\\
+            %\neg \neg \VarF &\equiv \VarF \tag{\structure{Doppelte Negation}}\\
+            %\VarF \vee \neg \VarF &\equiv \true \tag{\structure{Triviale Tautologie}}\\
+            %\VarF \wedge \neg \VarF &\equiv \false \tag{\structure{Triviale Kontradiktion}}\\
                 %\bigskip
-            %\F \vee \G &\equiv \G \vee \F \tag{\structure{Kommutativität}}\\
-                %\F \wedge \G &\equiv \G \wedge \F\\
-            %(\F \vee \G) \vee \K &\equiv \F \vee (\G \vee \K) \tag{\structure{Assoziativität}}\\
-                %(\F \wedge \G) \wedge \K &\equiv \F \wedge (\G \wedge \K)\\
-            %\F \vee (\G \wedge \K) &\equiv (\F \vee \G) \wedge (\F \vee \K) \tag{\structure{Distributivität}}\\
-                %\F \wedge (\G \vee \K) &\equiv (\F \wedge \G) \vee (\F \wedge \K)\\
-            %\neg(\F \wedge \G) &\equiv \neg \F \vee \neg \G \tag{\structure{De Morgan}}\\
-                %\neg(\F \vee \G) &\equiv \neg\F \wedge \neg\G\\
+            %\VarF \vee \VarG &\equiv \VarG \vee \VarF \tag{\structure{Kommutativität}}\\
+                %\VarF \wedge \VarG &\equiv \VarG \wedge \VarF\\
+            %(\VarF \vee \VarG) \vee \VarK &\equiv \VarF \vee (\VarG \vee \VarK) \tag{\structure{Assoziativität}}\\
+                %(\VarF \wedge \VarG) \wedge \VarK &\equiv \VarF \wedge (\VarG \wedge \VarK)\\
+            %\VarF \vee (\VarG \wedge \VarK) &\equiv (\VarF \vee \VarG) \wedge (\VarF \vee \VarK) \tag{\structure{Distributivität}}\\
+                %\VarF \wedge (\VarG \vee \VarK) &\equiv (\VarF \wedge \VarG) \vee (\VarF \wedge \VarK)\\
+            %\neg(\VarF \wedge \VarG) &\equiv \neg \VarF \vee \neg \VarG \tag{\structure{De Morgan}}\\
+                %\neg(\VarF \vee \VarG) &\equiv \neg\VarF \wedge \neg\VarG\\
                 %\bigskip
-            %\F \rightarrow \G &\equiv \neg \F \vee \G \tag{\structure{Implikation}}\\
-            %\F \leftrightarrow \G &\equiv (\F \rightarrow \G) \wedge (\G \rightarrow \F) \tag{\structure{Bikonditional}}\\
+            %\VarF \rightarrow \VarG &\equiv \neg \VarF \vee \VarG \tag{\structure{Implikation}}\\
+            %\VarF \leftrightarrow \VarG &\equiv (\VarF \rightarrow \VarG) \wedge (\VarG \rightarrow \VarF) \tag{\structure{Bikonditional}}\\
         %\end{align}
     %\end{frame}
 }
@@ -568,9 +568,9 @@
 
 \defineUnit{natuerlichesschliessen}{%
 {
-    \newcommand{\F}{\tau}
-    \newcommand{\G}{\varphi}
-    \newcommand{\K}{\chi}
+    \newcommand{\VarF}{\tau}
+    \newcommand{\VarG}{\varphi}
+    \newcommand{\VarK}{\chi}
     \newcommand{\subproof}[2]{%
         \begin{tikzpicture}[y=.9em]
             \path
@@ -595,69 +595,69 @@
             & \structure{Introduktion} & \structure{Elimination}\\\tabucline[1pt]{-}
             \capt{$\wedge$} &
                 \topproof{
-                    \AxiomC{$\F$}
-                    \AxiomC{$\G$}
+                    \AxiomC{$\VarF$}
+                    \AxiomC{$\VarG$}
                     \RightLabel{\scriptsize $+\wedge$}
-                    \BinaryInfC{$\F \wedge \G$}
+                    \BinaryInfC{$\VarF \wedge \VarG$}
                 }
             &
                 \topproof{
-                    \AxiomC{$\F \wedge \G$}
+                    \AxiomC{$\VarF \wedge \VarG$}
                     \RightLabel{\scriptsize $-\wedge_1$}
-                    \UnaryInfC{$\F$}
+                    \UnaryInfC{$\VarF$}
                 }
                 \quad
                 \topproof{
-                    \AxiomC{$\F \wedge \G$}
+                    \AxiomC{$\VarF \wedge \VarG$}
                     \RightLabel{\scriptsize $-\wedge_2$}
-                    \UnaryInfC{$\G$}
+                    \UnaryInfC{$\VarG$}
                 }
             \\
             \capt{$\vee$} &
                 \topproof{
-                    \AxiomC{$\F$}
+                    \AxiomC{$\VarF$}
                     \RightLabel{\scriptsize $+\vee_1$}
-                    \UnaryInfC{$\F \vee \G$}
+                    \UnaryInfC{$\VarF \vee \VarG$}
                 }
                 \quad
                 \topproof{
-                    \AxiomC{$\G$}
+                    \AxiomC{$\VarG$}
                     \RightLabel{\scriptsize $+\vee_2$}
-                    \UnaryInfC{$\F \vee \G$}
+                    \UnaryInfC{$\VarF \vee \VarG$}
                 }
             &
                 \topproof{
-                    \AxiomC{$\F \vee \G$}
-                    \AxiomC{\subproof{$\F$}{$\K$}}
-                    \AxiomC{\subproof{$\G$}{$\K$}}
+                    \AxiomC{$\VarF \vee \VarG$}
+                    \AxiomC{\subproof{$\VarF$}{$\VarK$}}
+                    \AxiomC{\subproof{$\VarG$}{$\VarK$}}
                     \RightLabel{\scriptsize $-\vee$}
-                    \TrinaryInfC{$\K$\vphantom{$\F\G$}}
+                    \TrinaryInfC{$\VarK$\vphantom{$\VarF\VarG$}}
                 }
             \\
             \capt{$\rightarrow$} &
                 \topproof{
-                    \AxiomC{\subproof{$\F$}{$\G$}}
+                    \AxiomC{\subproof{$\VarF$}{$\VarG$}}
                     \RightLabel{\scriptsize $+\rightarrow$}
-                    \UnaryInfC{$\F \rightarrow \G$}
+                    \UnaryInfC{$\VarF \rightarrow \VarG$}
                 }
             &
                 \topproof{
-                    \AxiomC{$\F$}
-                    \AxiomC{$\F \rightarrow \G$}
+                    \AxiomC{$\VarF$}
+                    \AxiomC{$\VarF \rightarrow \VarG$}
                     \RightLabel{\scriptsize $-\rightarrow$, MP}
-                    \BinaryInfC{$\G$}
+                    \BinaryInfC{$\VarG$}
                 }
             \\
             \capt{$\neg$} &
                 \topproof{
-                    \AxiomC{\subproof{$\F$}{$\bot$}}
+                    \AxiomC{\subproof{$\VarF$}{$\bot$}}
                     \RightLabel{\scriptsize $+\neg$}
-                    \UnaryInfC{$\neg \F$}
+                    \UnaryInfC{$\neg \VarF$}
                 }
             &
                 \topproof{
-                    \AxiomC{$\F$}
-                    \AxiomC{$\neg\F$}
+                    \AxiomC{$\VarF$}
+                    \AxiomC{$\neg\VarF$}
                     \RightLabel{\scriptsize $-\neg$}
                     \BinaryInfC{$\bot$}
                 }
@@ -675,20 +675,20 @@
                 \topproof{
                     \AxiomC{$\bot$}
                     \RightLabel{\scriptsize $-\bot$}
-                    \UnaryInfC{$\F$}
+                    \UnaryInfC{$\VarF$}
                 }
             \\
             \capt{$\neg\neg$} &
                 \topproof{
-                    \AxiomC{$\F$}
+                    \AxiomC{$\VarF$}
                     \RightLabel{\scriptsize $+\neg\neg$}
-                    \UnaryInfC{$\neg\neg\F$}
+                    \UnaryInfC{$\neg\neg\VarF$}
                 }
             &
                 \topproof{
-                    \AxiomC{$\neg\neg\F$}
+                    \AxiomC{$\neg\neg\VarF$}
                     \RightLabel{\scriptsize $-\neg\neg$}
-                    \UnaryInfC{$\F$}
+                    \UnaryInfC{$\VarF$}
                 }
         \end{tabu}
         \begin{itemize}
@@ -701,22 +701,22 @@
                 \topproof{
                     \AxiomC{ }
                     \RightLabel{\scriptsize LEM}
-                    \UnaryInfC{$\F \vee \neg\F$}
+                    \UnaryInfC{$\VarF \vee \neg\VarF$}
                 }
             &
                 \topproof{
-                    \AxiomC{\subproof{$\neg\F$}{$\bot$}}
+                    \AxiomC{\subproof{$\neg\VarF$}{$\bot$}}
                     \RightLabel{\scriptsize $-\neg$, PBC}
-                    \UnaryInfC{$\F$}
+                    \UnaryInfC{$\VarF$}
                 }
             \\
             &
             &
                 \topproof{
-                    \AxiomC{$\neg\G$}
-                    \AxiomC{$\F \rightarrow \G$}
+                    \AxiomC{$\neg\VarG$}
+                    \AxiomC{$\VarF \rightarrow \VarG$}
                     \RightLabel{\scriptsize MT}
-                    \BinaryInfC{$\neg\F$}
+                    \BinaryInfC{$\neg\VarF$}
                 }
         \end{tabu}
     \end{frame}
@@ -833,9 +833,9 @@
 
 \defineUnit{natuerlichesschliessenquantoren}{%
 {
-    \newcommand{\F}{\tau}
-    \newcommand{\G}{\varphi}
-    \newcommand{\K}{\chi}
+    \newcommand{\VarF}{\tau}
+    \newcommand{\VarG}{\varphi}
+    \newcommand{\VarK}{\chi}
     \newcommand{\subproof}[2]{%
         \begin{tikzpicture}[y=.9em]
             \path
@@ -856,8 +856,8 @@
         \frametitle{Natürliches Schließen}
 
         \begin{definition}[Ersetzung]
-            Sei $\G$ eine Formel und $a$ eine Konstante.\\
-            Mit \structure{$\G[x/a]$} bezeichnen wir die Formel die man erhält, wenn man alle \alert{freien} Vorkommnisse von \structure{$x$} in \structure{$\G$} durch \structure{$a$} \structure{ersetzt}.
+            Sei $\VarG$ eine Formel und $a$ eine Konstante.\\
+            Mit \structure{$\VarG[x/a]$} bezeichnen wir die Formel die man erhält, wenn man alle \alert{freien} Vorkommnisse von \structure{$x$} in \structure{$\VarG$} durch \structure{$a$} \structure{ersetzt}.
         \end{definition}
         \vfill
         \tabulinesep=4pt
@@ -865,29 +865,29 @@
             & \structure{Introduktion} & \structure{Elimination}\\\tabucline[1pt]{-}
             \capt{$\exists$} &
                 \topproof{
-                    \AxiomC{$\F[x/a]$}
+                    \AxiomC{$\VarF[x/a]$}
                     \RightLabel{\scriptsize $+\exists$}
-                    \UnaryInfC{$\exists x. \F$}
+                    \UnaryInfC{$\exists x. \VarF$}
                 }
             &
                 \topproof{
-                    \AxiomC{$\exists x. \F$}
-                    \AxiomC{\subproof{$a. \F[x/a]$}{$\K$}}
+                    \AxiomC{$\exists x. \VarF$}
+                    \AxiomC{\subproof{$a. \VarF[x/a]$}{$\VarK$}}
                     \RightLabel{\scriptsize $-\exists$}
-                    \BinaryInfC{$\K$}
+                    \BinaryInfC{$\VarK$}
                 }
             \\
             \capt{$\forall$} &
                 \topproof{
-                    \AxiomC{\subproof{$a$}{$\F[x/a]$}}
+                    \AxiomC{\subproof{$a$}{$\VarF[x/a]$}}
                     \RightLabel{\scriptsize $+\forall$}
-                    \UnaryInfC{$\forall x. \F$}
+                    \UnaryInfC{$\forall x. \VarF$}
                 }
             &
                 \topproof{
-                    \AxiomC{$\forall x. \F$}
+                    \AxiomC{$\forall x. \VarF$}
                     \RightLabel{\scriptsize $-\forall$}
-                    \UnaryInfC{$\F[x/a]$}
+                    \UnaryInfC{$\VarF[x/a]$}
                 }
         \end{tabu}
         \begin{itemize}