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37 lines
1.3 KiB
37 lines
1.3 KiB
\item \self Consider the set $\mathbb{N}_{16} = \{0,1,2,3,\dots,14,15\}$.
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Let $x_0$, $x_1$, $x_2$, and $x_3$ be propositional variables,
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used for symbolic encoding of the elements of $\mathbb{N}_{16}$,
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using standard binary encoding, with $x_0$ being the least
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significant ($2^0$) bit, and $x_3$ being the most significant
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($2^3$) bit.
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Now, consider the following subsets of $\mathbb{N}_{16}$.
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\begin{itemize}
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\item $A = \{0, 1, 2, 3\}$
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\item $B = \{0, 1, 2, 3, 4, 5, 6, 7\}$
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\item $C = \{0, 2, 4, 6, 8, 10, 12, 14\}$
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\item $D = \{8, 10, 12, 14\}$
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\item $E = \{3, 10\}$
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\item $F = \{ \}$
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\end{itemize}
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In the following list of formulas, write the letter of the set that
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the formula encodes into the adjacent box. Note that some sets might
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be encoded by more than one formula. Also note that some formulas
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might not encode any of the above sets; write a ``--'' in the box of
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such formulas.
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\begin{itemize}
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\item[\Huge{$\square$}] $\bot$
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\item[\Huge{$\square$}] $\top$
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\item[\Huge{$\square$}] $(\neg x_3 \wedge \neg x_2 \wedge x_1
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\wedge x_0) \vee (x_3 \wedge \neg x_2 \wedge x_1 \wedge \neg
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x_0)$
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\item[\Huge{$\square$}] $x_0$
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\item[\Huge{$\square$}] $\neg x_0$
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\item[\Huge{$\square$}] $x_3$
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\item[\Huge{$\square$}] $\neg x_3$
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\item[\Huge{$\square$}] $\neg x_3 \vee \neg x_2$
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\item[\Huge{$\square$}] $\neg x_3 \wedge \neg x_2$
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\end{itemize}
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