Questionnaire-2024/predicate_logic/1017_sol.tex

\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
\Tree
 [.$\exists x$
    [.$\forall y$
        [.$\lor$
            [.$\imp$
                [.$P$ $x$ $y$ ]
                [.$Q$ $x$ $y$ ]
            ]
            [.$\imp$
                [.$P$ $y$ $x$ ]
                [.$R$ $x$ $y$ ]
            ]
        ]
    ]
 ]
\end{tikzpicture}

\begin{itemize}
  \item $x=a$
\end{itemize}

\begin{multicols}{2}
  $$x=a \land y=a$$
\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
\Tree
        [.$\lor$
            [.$\imp$
                [.$P$ $a$ $a$ ]
                [.$Q$ $a$ $a$ ]
            ]
            [.$\imp$
                [.$P$ $a$ $a$ ]
                [.$R$ $a$ $a$ ]
            ]
        ]
\end{tikzpicture}
 \begin{align*}
   &(P(a,a) \imp Q(a,a)) \lor (P(a,a) \imp R(a,a)) = \\
   &(\false \imp \true) \lor (\false \imp \true) = \true
 \end{align*}
\columnbreak
  $$x=a \land y=b$$
\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
\Tree
        [.$\lor$
            [.$\imp$
                [.$P$ $a$ $b$ ]
                [.$Q$ $a$ $b$ ]
            ]
            [.$\imp$
                [.$P$ $b$ $a$ ]
                [.$R$ $a$ $b$ ]
            ]
        ]
\end{tikzpicture}
 \begin{align*}
   &(P(a,b) \imp Q(a,b)) \lor (P(b,a) \imp R(b,a)) = \\
   &(\true \imp \false) \lor (\true \imp \false) = \false
 \end{align*}
\end{multicols}


\begin{itemize}
  \item $x=b$
\end{itemize}

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