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100 lines
3.9 KiB
100 lines
3.9 KiB
\begin{enumerate}
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\item using \textit{variable order} $c < a < d < b$:\\
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\begin{minipage}{0.5\textwidth}
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\begin{tabbing}
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$f$ \= $= (a \lor b \lor c) \land \lnot d$\\
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\>$ f_c $ \= $= \lnot d$ \\
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\>\>$ f_{ca} $ \= $= \lnot d$ \\
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\>\>$ f_{c \lnot a} $ \= $= \lnot d$ \\
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\>\>\Rightarrow a does not have an influence\\
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\>\>on the formula.\\
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\>\>These cofactors can be skipped.\\
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\>\>$ f_{cd} $ \= $= \bot$ \\
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\>\>$ f_{c \lnot d} $ \= $= \top $ \\
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\>$ f_{\lnot c} $ \= $= (a \lor b) \land \lnot d$ \\
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\>\>$ f_{\lnot ca} $ \= $= \lnot d = f_{c}$ \\
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\>\>$ f_{\lnot c \lnot a} $ \= $= b \land \lnot d$ \\
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\>\>\>$ f_{\lnot c \lnot ad} $ \= $= \bot$ \\
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\>\>\>$ f_{\lnot c \lnot a \lnot d} $ \= $= b$ \\
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\>\>\>\>$ f_{\lnot c \lnot a \lnot db} $ \= $= \top$ \\
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\>\>\>\>$ f_{\lnot c \lnot a \lnot db} $ \= $= \bot$ \\
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\end{tabbing}
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\end{minipage}
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\begin{minipage}{0.5\textwidth}
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\begin{center}
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\begin{forest}
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for tree={circle, draw, no edge,
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minimum size=2em,
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inner sep=0pt,
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s sep=6mm,
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l sep=6mm}
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[$f$, rectangle, draw, tikz={\draw [fulldot=.5] () to (c.north);}
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[,phantom]
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[$c$, name=c, tikz={\draw [line] () to (d1.north); \draw [dot=.5] () to (a.north);}
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[$f_{c}$, rectangle, draw, tikz={\draw [fulldot=.5] () to (d1.north west);}]
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[,phantom]
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[$a$, name=a, tikz={\draw [line,bend left=30] () to (d1.north); \draw [dot=.5] () to (d2.north);}
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[$d$, name=d1, tikz={\draw [line] () to (-2,-5); \draw [fulldot=.5] () to (-1,-5);}]
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[,phantom]
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[$d$, name=d2, tikz={\draw [line] () to (0.6,-5); \draw [fulldot=.5] () to (b.north);}
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[,phantom]
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[$b$, name=b, tikz={\draw [line] () to (1.3,-6.3); \draw [fulldot=.5] () to (2.3,-6.3);}]
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]
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[$f_{\lnot c \lnot a \lnot d}$, rectangle, draw, tikz={\draw [line] () to (b.north east);}]
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]
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[$f_{\lnot c \lnot a}$, rectangle, draw, tikz={\draw [fulldot=.5] () to (d2.north east);}]
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]
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[$f_{\lnot c}$, rectangle, draw, tikz={\draw [fulldot=.5] () to (a.north east);}]
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]
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\end{forest}
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\end{center}
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\end{minipage}
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\item
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using \textit{variable order} $d < a < b < c$:\\
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\begin{minipage}{0.5\textwidth}
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\begin{tabbing}
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$f$ \= $= (a \lor b \lor c) \land \lnot d$\\
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\>$ f_d $ \= $= \bot$ \\
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\>$ f_{\lnot d} $ \= $= a \lor b \lor c$ \\
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\>\>$ f_{\lnot da} $ \= $= \top$ \\
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\>\>$ f_{\lnot d \lnot a} $ \= $= b \lor c$ \\
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\>\>\>$ f_{\lnot d \lnot ab} $ \= $= \top$ \\
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\>\>\>$ f_{\lnot d \lnot a \lnot b} $ \= $= c$ \\
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\>\>\>\>$ f_{\lnot d \lnot a \lnot bc} $ \= $= \top$ \\
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\>\>\>\>$ f_{\lnot d \lnot a \lnot b \lnot c} $ \= $= \bot$ \\
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\end{tabbing}
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\end{minipage}
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\begin{minipage}{0.5\textwidth}
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\begin{center}
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\begin{forest}
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for tree={circle, draw, no edge,
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minimum size=2em,
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inner sep=0pt,
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s sep=6mm,
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l sep=6mm}
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[$f$, rectangle, draw, tikz={\draw [fulldot=.5] () to (d.north);}
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[,phantom]
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[$d$, name=d, tikz={\draw [line] () to (-0.5,-2.3); \draw [fulldot=.5] () to (a.north);}
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[,phantom]
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[,phantom]
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[$a$, name=a, tikz={\draw [line] () to (0,-3.6); \draw [dot=.5] () to (b.north);}
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[,phantom]
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[,phantom]
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[$b$, name=b, tikz={\draw [line] () to (0.5,-4.9); \draw [dot=.5] () to (c.north);}
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[,phantom]
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[,phantom]
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[$c$, name=c, tikz={\draw [line] () to (1.3,-6.2); \draw [fulldot=.5] () to (2.3,-6.2);}
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]
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[,phantom]
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]
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[$f_{\lnot d \lnot a \lnot b}$, rectangle, draw, tikz={\draw [line] () to (c.north east);}]
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]
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[$f_{\lnot d \lnot a}$, rectangle, draw, tikz={\draw [line] () to (b.north east);}]
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]
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[$f_{\lnot d}$, rectangle, draw, tikz={\draw [line] () to (a.north east);}]
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]
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\end{forest}
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\end{center}
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\end{minipage}
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\end{enumerate}
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