Questionnaire-2024/predicate_logic/0016_sol.tex

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

$\mathcal{M}_1: \mathcal{A} = \{a, b\}$ \\
$P^{\mathcal{M}_1}$ = true \\
$Q^{\mathcal{M}_1}$ = true \\
$\mathcal{M}_1 \models \phi$ \\
$f$ can be defined arbitrarily in this case.

$\mathcal{M}_2: \mathcal{A} = \{a, b\}$ \\
$P^{\mathcal{M}_2}$ = true \\
$Q^{\mathcal{M}_2}$ = false \\
$\mathcal{M}_2 \not\models \phi$
$f$ can be defined arbitrarily in this case.