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88 lines
2.0 KiB
88 lines
2.0 KiB
\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
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\Tree
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[.$\exists x$
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[.$\forall y$
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[.$\lor$
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[.$\imp$
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[.$P$ $x$ $y$ ]
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[.$Q$ $x$ $y$ ]
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]
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[.$\imp$
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[.$P$ $y$ $x$ ]
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[.$R$ $x$ $y$ ]
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]
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]
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]
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]
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\end{tikzpicture}
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\begin{itemize}
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\item $x=a$
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\end{itemize}
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\begin{multicols}{2}
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$$x=a \land y=a$$
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\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
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\Tree
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[.$\lor$
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[.$\imp$
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[.$P$ $a$ $a$ ]
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[.$Q$ $a$ $a$ ]
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]
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[.$\imp$
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[.$P$ $a$ $a$ ]
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[.$R$ $a$ $a$ ]
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]
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]
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\end{tikzpicture}
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\begin{align*}
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&(P(a,a) \imp Q(a,a)) \lor (P(a,a) \imp R(a,a)) = \\
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&(\false \imp \true) \lor (\false \imp \true) = \true
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\end{align*}
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\columnbreak
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$$x=a \land y=b$$
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\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
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\Tree
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[.$\lor$
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[.$\imp$
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[.$P$ $a$ $b$ ]
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[.$Q$ $a$ $b$ ]
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]
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[.$\imp$
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[.$P$ $b$ $a$ ]
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[.$R$ $a$ $b$ ]
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]
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]
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\end{tikzpicture}
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\begin{align*}
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&(P(a,b) \imp Q(a,b)) \lor (P(b,a) \imp R(b,a)) = \\
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&(\true \imp \false) \lor (\true \imp \false) = \false
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\end{align*}
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\end{multicols}
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\begin{itemize}
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\item $x=b$
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\end{itemize}
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\begin{centering}
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$$x=b \land y=a$$
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\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
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\Tree
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[.$\lor$
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[.$\imp$
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[.$P$ $b$ $a$ ]
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[.$Q$ $b$ $a$ ]
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]
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[.$\imp$
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[.$P$ $a$ $b$ ]
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[.$R$ $b$ $a$ ]
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]
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]
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\end{tikzpicture}
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\begin{align*}
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&(P(b,a) \imp Q(b,a)) \lor (P(a,b) \imp R(b,a)) = \\
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&(\true \imp \false) \lor (\true \imp \false) = \false
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\end{align*}
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\end{centering}
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We do not need to evaluate $x=b \land y=b$. $M\nmodels\varphi$.
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