Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
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introduction_to_digital_systems:calc_logic_example [2021/09/16 23:02] tfischer |
introduction_to_digital_systems:calc_logic_example [2021/09/17 00:08] (aktuell) tfischer |
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| Zeile 1: | Zeile 1: | ||
| ~~REVEAL ~~ | ~~REVEAL ~~ | ||
| - | |||
| - | ----> | ||
| - | example for a simplification with the rule for boolean algebra \\ | ||
| - | $\overline{a \lor (b \land (\bar{a} | + | ---->> |
| - | <---- | + | example for a simplification with the rule for boolean algebra |
| - | ----> | + | \begin{align*} |
| - | At first we will switch the representation to the following: | + | \begin{array}{ll} |
| + | \overline{a \lor (b \land (\bar{a} \lor c) \land 1) \lor a} & \\ | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| - | $/(a + (b \cdot (/a + c) \cdot 1 ) + a )$ | + | << |
| - | <---- | + | ---->> |
| + | At first we will switch the representation to the following: \\ \\ | ||
| - | ----> | + | \begin{align*} |
| - | so lets start\\ | + | \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 | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | <<---- | ||
| + | ---->> | ||
| + | At first we will switch the representation to the following: \\ \\ | ||
| - | $/(a + (b \cdot (/a + c) \cdot 1 ) + a )$ | + | \begin{align*} |
| + | \begin{array}{ll} | ||
| + | /(a + (b \cdot (/a + c) \cdot 1 ) + a ) & | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | ---->> | ||
| + | 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 | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | <<---- | ||
| - | ----> | + | ---->> |
| - | 1. Put space between the digits | + | 1. $\color{blue}{\text{Neutral Element}}$ \\ \\ \\ |
| \begin{align*} | \begin{align*} | ||
| - | \begin{smallmatrix} | + | \begin{array}{ll} |
| - | /(a + (b \cdot (/a + c) \color{blue}{\cdot 1} ) + a ) & \color{blue}{\text{Neutral Element}} | + | /(a + (b \cdot (/a + c) \quad \; ) + a ) & \color{white}{\overline{ab}} |
| - | \quad\quad\quad\quad\quad\quad | + | \quad\quad\quad\quad\quad\quad |
| - | \end{smallmatrix} | + | \end{array} |
| \end{align*} | \end{align*} | ||
| - | <---- | + | <<---- |
| + | ---->> | ||
| + | 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 | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 2. $\color{blue}{\text{Commutative Law}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | /(a + a + (b \cdot (/a + c) \quad \; )) & \color{white}{\overline{ab}} | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 3. $\color{blue}{\text{Idempotence}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | / | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 3. $\color{blue}{\text{Idempotence}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | /(a \quad \enspace \: + (b \cdot (/a + c)\quad \;)) & \color{white}{\overline{ab}} | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 4. $\color{blue}{\text{Distributive Law}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | /(a \quad \enspace \: + (\color{blue}{b \cdot (/a + c)} \quad \;)) & \color{white}{\overline{ab}} | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 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 | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 5. $\color{blue}{\text{Associative Law}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | / | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 5. $\color{blue}{\text{Associative Law}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | /(a \quad \, + \,\,(b \cdot /a) + (b \cdot c)\,\, ) & \color{white}{\overline{ab}} | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 6. $\color{blue}{\text{Absorption Law}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | / | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 6. $\color{blue}{\text{Absorption Law}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | /(a \quad \, + \quad\enspace b \quad\,\, + (b \cdot c) \,\,) & \color{white}{\overline{ab}} | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 7. $\color{blue}{\text{Absorption Law}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | /(a \quad \, + \quad\enspace \color{blue}{b \quad\,\, + (b \cdot c)} \,\,) & \color{white}{\overline{ab}} | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 7. $\color{blue}{\text{Absorption Law}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | /(a \quad \, + \quad\enspace b ) \qquad\qquad\quad\; | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 8. $\color{blue}{\text{DeMorgan}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | \color{blue}{/ | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||
| + | |||
| + | ---->> | ||
| + | 8. $\color{blue}{\text{DeMorgan}}$ \\ \\ \\ | ||
| + | |||
| + | \begin{align*} | ||
| + | \begin{array}{ll} | ||
| + | \;/a \quad \, \cdot \quad\enspace /b \qquad\qquad\quad\; | ||
| + | \quad\quad\quad\quad\quad\quad | ||
| + | \end{array} | ||
| + | \end{align*} | ||
| + | << | ||