Dies ist eine alte Version des Dokuments!
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example for a simplification with the rule for boolean algebra
\begin{align*} \begin{array}{ll} \overline{a \lor (b \land (\bar{a} \lor c) \land 1) \lor a} & \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---
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At first we will switch the representation to the following:
\begin{align*}
\begin{array}{ll}
\overline{a \lor (b \land (\bar{a} \lor c) \land 1) \lor a} & \color{white}{\overline{ab}} \\
\quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\
\end{array}
\end{align*}
<---
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At first we will switch the representation to the following:
\begin{align*} \begin{array}{ll} /(a + (b \cdot (/a + c) \cdot 1 ) + a ) & \color{white}{\overline{ab}} \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---
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1. $\color{blue}{\text{Neutral Element}}$
\begin{align*} \begin{array}{ll} /(a + (b \cdot (/a + c) \color{blue}{\cdot 1} ) + a ) & \color{white}{\overline{ab}} \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---
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1. $\color{blue}{\text{Neutral Element}}$
\begin{align*} \begin{array}{ll} /(a + (b \cdot (/a + c) \quad ) + a ) & \color{white}{\overline{ab}} \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---
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2. $\color{blue}{\text{Commutative Law}}$
\begin{align*} \begin{array}{ll} /(a + \color{blue}{(b \cdot (/a + c) \quad ) + a }) & \color{white}{\overline{ab}} \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---
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2. $\color{blue}{\text{Commutative Law}}$
\begin{align*} \begin{array}{ll} /(a + a + (b \cdot (/a + c))) & \color{white}{\overline{ab}} \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---
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3. $\color{blue}{\text{Idempotence}}$
\begin{align*} \begin{array}{ll} /(\color{blue}{a + a} + (b \cdot (/a + c))) & \color{white}{\overline{ab}} \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---
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3. $\color{blue}{\text{Idempotence}}$
\begin{align*} \begin{array}{ll} /(a \quad + (b \cdot (/a + c))) & \color{white}{\overline{ab}} \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---
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4. $\color{blue}{\text{Distributive Law}}$
\begin{align*} \begin{array}{ll} /(a \quad + (\color{blue}{b \cdot (/a + c)})) & \color{white}{\overline{ab}} \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---
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4. $\color{blue}{\text{Distributive Law}}$
\begin{align*} \begin{array}{ll} /(a \quad + ((b \cdot /a) + (b \cdot c)))) & \color{white}{\overline{ab}} \\ \quad\quad\quad\quad\quad\quad & \quad\quad\quad\quad\quad\quad \\ \end{array} \end{align*} <---