\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 \\ $\mathcal{M}_2 \not\models \phi$