Questionnaire-2024/symbolic_encoding/practical_questions/2_7_symbrep_self.tex

\item \self Consider the set $\mathbb{N}_{16} = \{0,1,2,3,\dots,14,15\}$.
    Let $x_0$, $x_1$, $x_2$, and $x_3$ be propositional variables,
    used for symbolic encoding of the elements of $\mathbb{N}_{16}$,
    using standard binary encoding, with $x_0$ being the least
    significant ($2^0$) bit, and $x_3$ being the most significant
    ($2^3$) bit.

    Now, consider the following subsets of $\mathbb{N}_{16}$.

\begin{itemize}
\item $A = \{0, 1, 2, 3\}$
\item $B = \{0, 1, 2, 3, 4, 5, 6, 7\}$
\item $C = \{0, 2, 4, 6, 8, 10, 12, 14\}$
\item $D = \{8, 10, 12, 14\}$
\item $E = \{3, 10\}$
\item $F = \{ \}$
\end{itemize}

In the following list of formulas, write the letter of the set that
the formula encodes into the adjacent box. Note that some sets might
be encoded by more than one formula. Also note that some formulas
might not encode any of the above sets; write a ``--'' in the box of
such formulas.

\begin{itemize}
\item[\Huge{$\square$}] $\bot$
\item[\Huge{$\square$}] $\top$
\item[\Huge{$\square$}] $(\neg x_3 \wedge \neg x_2 \wedge x_1
    \wedge x_0) \vee (x_3 \wedge \neg x_2 \wedge x_1 \wedge \neg
    x_0)$
\item[\Huge{$\square$}] $x_0$
\item[\Huge{$\square$}] $\neg x_0$
\item[\Huge{$\square$}] $x_3$
\item[\Huge{$\square$}] $\neg x_3$
\item[\Huge{$\square$}] $\neg x_3 \vee \neg x_2$
\item[\Huge{$\square$}] $\neg x_3 \wedge \neg x_2$
\end{itemize}