Questionnaire-2024/predicate_logic/0013_sol.tex

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

\begin{multicols}{2}
\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.25cm]
\Tree
[.$\exists y$
    [.$\land$ 
        [.$P$ $a$ $y$ ]
        [.$Q$ $a$ ]
    ]
]
\end{tikzpicture} 
\newline
Subtree: $x=a$ \\

\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.25cm]
\Tree
[.$\land$ 
    [.$P$ $a$ $a$ ]
    [.$Q$ $a$ ]
]
\end{tikzpicture} 
\newline
Subtree: $x=a \land y=a$ \\
Evaluates $\mathcal{M}_1$ to true. \\
Evaluates $\mathcal{M}_2$ to false.

\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.25cm]
\Tree
[.$\land$ 
    [.$P$ $a$ $b$ ]
    [.$Q$ $a$ ]
]
\end{tikzpicture} 
\newline
Subtree: $x=a \land y=b$ \\
Evaluates $\mathcal{M}_1$ to true. \\
Evaluates $\mathcal{M}_2$ to false.

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

\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.25cm]
\Tree
[.$\land$ 
    [.$P$ $b$ $a$ ]
    [.$Q$ $b$ ]
]
\end{tikzpicture} 
\newline
Subtree: $x=b \land y=a$ \\
Evaluates $\mathcal{M}_1$ to true. \\
Evaluates $\mathcal{M}_2$ to false.

\begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.25cm]
\Tree
[.$\land$ 
    [.$P$ $b$ $b$ ]
    [.$Q$ $b$ ]
]
\end{tikzpicture} 
\newline
Subtree: $x=b \land y=b$ \\
Evaluates $\mathcal{M}_1$ to true. \\
Evaluates $\mathcal{M}_2$ to false.
\end{multicols}

$\mathcal{M}_1: \mathcal{A} = \{a, b\}$ \\
$P$ = true \\
$Q$ = true \\ 
$\mathcal{M}_1 \models \phi$ \\ 

$\mathcal{M}_2: \mathcal{A} = \{a, b\}$ \\
$P$ = true \\
$Q$ = false \\ 
$\mathcal{M}_2 \not\models \phi$