Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| introduction_to_digital_systems:calc_logic_example [2021/09/16 23:24] – tfischer | introduction_to_digital_systems:calc_logic_example [2021/09/17 00:08] (aktuell) – tfischer | ||
|---|---|---|---|
| Zeile 1: | Zeile 1: | ||
| ~~REVEAL ~~ | ~~REVEAL ~~ | ||
| - | + | ||
| - | ---> | + | ---->> |
| example for a simplification with the rule for boolean algebra \\ \\ | example for a simplification with the rule for boolean algebra \\ \\ | ||
| Zeile 10: | Zeile 10: | ||
| \end{array} | \end{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | ||
| - | ---> | + | <<---- |
| + | |||
| + | ---->> | ||
| At first we will switch the representation to the following: \\ \\ | At first we will switch the representation to the following: \\ \\ | ||
| Zeile 21: | Zeile 22: | ||
| \end{array} | \end{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |
| - | ---> | + | ---->> |
| At first we will switch the representation to the following: \\ \\ | At first we will switch the representation to the following: \\ \\ | ||
| Zeile 31: | Zeile 32: | ||
| \end{array} | \end{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |
| - | ---> | + | ---->> |
| 1. $\color{blue}{\text{Neutral Element}}$ \\ \\ \\ | 1. $\color{blue}{\text{Neutral Element}}$ \\ \\ \\ | ||
| Zeile 42: | Zeile 43: | ||
| \end{array} | \end{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |
| - | ---> | + | ---->> |
| 1. $\color{blue}{\text{Neutral Element}}$ \\ \\ \\ | 1. $\color{blue}{\text{Neutral Element}}$ \\ \\ \\ | ||
| \begin{align*} | \begin{align*} | ||
| \begin{array}{ll} | \begin{array}{ll} | ||
| - | /(a + (b \cdot (/a + c) \quad ) + a ) & \color{white}{\overline{ab}} | + | /(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{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |
| - | ---> | + | ---->> |
| 2. $\color{blue}{\text{Commutative Law}}$ \\ \\ \\ | 2. $\color{blue}{\text{Commutative Law}}$ \\ \\ \\ | ||
| \begin{align*} | \begin{align*} | ||
| \begin{array}{ll} | \begin{array}{ll} | ||
| - | /(a + \color{blue}{(b \cdot (/a + c) \quad ) + a }) & \color{white}{\overline{ab}} | + | /(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{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |
| - | ---> | + | ---->> |
| 2. $\color{blue}{\text{Commutative Law}}$ \\ \\ \\ | 2. $\color{blue}{\text{Commutative Law}}$ \\ \\ \\ | ||
| \begin{align*} | \begin{align*} | ||
| \begin{array}{ll} | \begin{array}{ll} | ||
| - | /(a + a + (b \cdot (/a + c))) & \color{white}{\overline{ab}} | + | /(a + a + (b \cdot (/a + c) \quad \; )) & \color{white}{\overline{ab}} |
| \quad\quad\quad\quad\quad\quad | \quad\quad\quad\quad\quad\quad | ||
| \end{array} | \end{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |
| - | ---> | + | ---->> |
| 3. $\color{blue}{\text{Idempotence}}$ \\ \\ \\ | 3. $\color{blue}{\text{Idempotence}}$ \\ \\ \\ | ||
| \begin{align*} | \begin{align*} | ||
| \begin{array}{ll} | \begin{array}{ll} | ||
| - | / | + | / |
| \quad\quad\quad\quad\quad\quad | \quad\quad\quad\quad\quad\quad | ||
| \end{array} | \end{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |
| - | ---> | + | ---->> |
| 3. $\color{blue}{\text{Idempotence}}$ \\ \\ \\ | 3. $\color{blue}{\text{Idempotence}}$ \\ \\ \\ | ||
| \begin{align*} | \begin{align*} | ||
| \begin{array}{ll} | \begin{array}{ll} | ||
| - | /(a \quad + (b \cdot (/a + c))) & \color{white}{\overline{ab}} | + | /(a \quad \enspace \: + (b \cdot (/a + c)\quad \;)) & \color{white}{\overline{ab}} |
| \quad\quad\quad\quad\quad\quad | \quad\quad\quad\quad\quad\quad | ||
| \end{array} | \end{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |
| - | ---> | + | ---->> |
| 4. $\color{blue}{\text{Distributive Law}}$ \\ \\ \\ | 4. $\color{blue}{\text{Distributive Law}}$ \\ \\ \\ | ||
| \begin{align*} | \begin{align*} | ||
| \begin{array}{ll} | \begin{array}{ll} | ||
| - | /(a \quad + (\color{blue}{b \cdot (/a + c)})) & \color{white}{\overline{ab}} | + | /(a \quad \enspace \: + (\color{blue}{b \cdot (/a + c)} \quad \;)) & \color{white}{\overline{ab}} |
| \quad\quad\quad\quad\quad\quad | \quad\quad\quad\quad\quad\quad | ||
| \end{array} | \end{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |
| - | ---> | + | ---->> |
| 4. $\color{blue}{\text{Distributive Law}}$ \\ \\ \\ | 4. $\color{blue}{\text{Distributive Law}}$ \\ \\ \\ | ||
| \begin{align*} | \begin{align*} | ||
| \begin{array}{ll} | \begin{array}{ll} | ||
| - | /(a \quad + ((b \cdot /a) + (b \cdot c)))) & \color{white}{\overline{ab}} | + | /(a \quad \, + ((b \cdot /a) + (b \cdot c))) & |
| + | \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 | \quad\quad\quad\quad\quad\quad | ||
| \end{array} | \end{array} | ||
| \end{align*} | \end{align*} | ||
| - | <--- | + | <<---- |