Entwicklungssatz von Shannon

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Beispiel:

\((b\wedge\bar{c})\vee((c\oplus b)\wedge a\wedge b)\)

Auflösen:

\((b\bar{c})\vee((c\oplus b)ab)=(b\bar{c})\vee((c\bar{b}\vee\bar{c}b)ab)\)

Entwickeln nach a:

\(a[(b\bar{c})\vee((c\bar{b}\vee\bar{c}b)1b)]\vee\bar{a}[(b\bar{c})\vee((c\bar{b}\vee\bar{c}b)0b)]\)

Entwickeln nach b:

\(\mathrm{I:}\) \(a[b[(1\bar{c})\vee((c\bar{1}\vee\bar{c}1)11)]]\vee a[\bar{b}[(0\bar{c})\vee((c\bar{0}\vee\bar{c}0))01]]\vee\)
\(\mathrm{II:}\) \(\bar{a}[b[(1\bar{c})\vee((c\bar{1}\vee\bar{c}1))10]]\vee\bar{a}[\bar{b}[(0\bar{c})\vee((c\bar{0}\vee\bar{c}0)00)]]\)

\(\mathrm{I:}\) \(a[b[(1\bar{c})\vee(c\bar{1}\vee\bar{c}1)]]\vee a[\bar{b}[(0\bar{c})\vee(c\bar{0}\vee\bar{c}0)0]]\vee\)
\(\mathrm{II:}\) \(\bar{a}[b[(1\bar{c})\vee(c\bar{1}\vee\bar{c}1)0]]\vee\bar{a}[\bar{b}[(0\bar{c})\vee(c\bar{0}\vee\bar{c}0)0]]\)

\(\mathrm{I:}\) \(a[b[(1\bar{c})\vee(c\bar{1}\vee\bar{c}1)]]\vee a[\bar{b}[(0\bar{c})\vee 0]]\vee\)
\(\mathrm{II:}\) \(\bar{a}[b[(1\bar{c})\vee 0]]\vee\bar{a}[\bar{b}[(0\bar{c})\vee 0]]\)

\(\mathrm{I:}\) \(a[b[(\bar{c})\vee(0\vee\bar{c})]]\vee a[\bar{b}[0\vee 0]]\vee\)
\(\mathrm{II:}\) \(\bar{a}[b[\bar{c}\vee 0]]\vee\bar{a}[\bar{b}[0\vee 0]]\)

\(\mathrm{I:}\) \(a[b[(\bar{c})\vee(0\vee\bar{c})]]\vee\)
\(\mathrm{II:}\) \(\bar{a}[b[\bar{c}\vee 0]]\)

Entwickeln nach c:

\(\mathrm{I:}\) \(a[b[c[(\bar{1})\vee(0\vee\bar{1})]]]\vee\Rightarrow abc\wedge 0\Rightarrow 0\)
\(\mathrm{II:}\) \(a[b[\bar{c}[(\bar{0})\vee(0\vee\bar{0})]]]\vee\Rightarrow ab\bar{c}\wedge 1\Rightarrow ab\bar{c}\)
\(\mathrm{III:}\) \(\bar{a}[b[c[\bar{1}\vee 0]]]\Rightarrow\bar{a}bc\wedge 0\Rightarrow 0\)
\(\mathrm{IV:}\) \(\bar{a}[b[\bar{c}[\bar{0}\vee 0]]]\Rightarrow\bar{a}b\bar{c}\wedge 1\Rightarrow\bar{a}b\bar{c}\)

DNF:

\((ab\bar{c})\vee(\bar{a}b\bar{c})\)
\(\Rightarrow b\bar{c}\)