Questionnaire-2024/predicate_logic/0014_sol.tex

\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.25cm]
\Tree
 [.$\exists x$
    [.$\forall y$
        [.$\imp$ 
            [.$P$ $x$ $y$ ]
            [.$\lor$
                [.$Q$ $x$ $y$ ]
                [.$R$ $x$ $y$ ]
            ]
        ]
    ]
 ]
\end{tikzpicture}
\newline

\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.25cm]
\Tree
[.$\forall y$
    [.$\imp$ 
        [.$P$ $b$ $y$ ]
        [.$\lor$
            [.$Q$ $b$ $y$ ]
            [.$R$ $b$ $y$ ]
        ]
    ]
]
\end{tikzpicture}
\newline 
Subtree: $x=b$ \\

\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.25cm]
\Tree
[.$\imp$ 
    [.$P$ $b$ $a$ ]
    [.$\lor$
        [.$Q$ $b$ $a$ ]
        [.$R$ $b$ $a$ ]
    ]
]
\end{tikzpicture}
\newline 
Subtree: $x=b \land y=a$

\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.25cm]
\Tree
[.$\imp$ 
    [.$P$ $b$ $b$ ]
    [.$\lor$
        [.$Q$ $b$ $b$ ]
        [.$R$ $b$ $b$ ]
    ]
]
\end{tikzpicture}
\newline 
Subtree: $x=b \land y=b$

The model $\mathcal{M}$ satisfies the formula.