Technische Informatik
Boolesche Algebra
Thorsten Thormählen
01. November 2022
Teil 3, Kapitel 1
Thorsten Thormählen
01. November 2022
Teil 3, Kapitel 1
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Typ | Schriftart | Beispiele |
---|---|---|
Variablen (Skalare) | kursiv | $a, b, x, y$ |
Funktionen | aufrecht | $\mathrm{f}, \mathrm{g}(x), \mathrm{max}(x)$ |
Vektoren | fett, Elemente zeilenweise | $\mathbf{a}, \mathbf{b}= \begin{pmatrix}x\\y\end{pmatrix} = (x, y)^\top,$ $\mathbf{B}=(x, y, z)^\top$ |
Matrizen | Schreibmaschine | $\mathtt{A}, \mathtt{B}= \begin{bmatrix}a & b\\c & d\end{bmatrix}$ |
Mengen | kalligrafisch | $\mathcal{A}, B=\{a, b\}, b \in \mathcal{B}$ |
Zahlenbereiche, Koordinatenräume | doppelt gestrichen | $\mathbb{N}, \mathbb{Z}, \mathbb{R}^2, \mathbb{R}^3$ |
$s$ | $l$ |
0 | 0 |
1 | 1 |
$s_1$ | $s_2$ | $l$ |
0 | 0 | 0 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
$s_1$ | $s_2$ | $l$ |
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 1 |
$s$ | $l$ |
0 | 1 |
1 | 0 |
$s_1$ | $s_2$ | $l$ |
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
$s_1$ | $s_2$ | 16 mögliche Funktionen $l_0$ bis $l_{15}$ | |||||||||||||||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Definition mittels Huntington'scher Axiome:
Operator $\bullet$ | Operator $\circ$ | |
kommutativ | $a \bullet b = b \bullet a$ | $a \circ b = b \circ a$ |
distributiv | $a \bullet (b \circ c) = (a \bullet b) \circ (a \bullet c)$ | $a \circ (b \bullet c) = (a \circ b) \bullet (a \circ c)$ |
Identität | $a \bullet 1 = a$ | $a \circ 0 = a$ |
Inversion | $a \bullet a^{-1} = 0 $ | $a \circ a^{-1} = 1$ |
Operator $\cdot$ | Operator $+$ | |
kommutativ | $a \cdot b = b \cdot a$ | $a + b = b + a$ |
distributiv | $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ | $a + (b \cdot c) = (a + b) \cdot (a + c)$ |
Identität | $a \cdot 1 = a$ | $a + 0 = a$ |
Inversion | $a \cdot a^{-1} = 0$ | $a + a^{-1} = 1$ |
Operator $\land$ | Operator $\lor$ | |
kommutativ | $a \land b = b \land a$ | $a \lor b = b \lor a$ |
distributiv | $a \land (b \lor c) = (a \land b) \lor (a \land c)$ | $a \lor (b \land c) = (a \lor b) \land (a \lor c)$ |
Identität | $a \land 1 = a$ | $a \lor 0 = a$ |
Inversion | $a \land \lnot a = 0 $ | $a \lor \lnot a = 1$ |
$a$ | $b$ | $y = a \land b$ |
0 | 0 | 0 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
$a$ | $b$ | $y = a \lor b$ |
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 1 |
$a$ | $y = \lnot a$ |
0 | 1 |
1 | 0 |
$x_1$ | $x_2$ | $y$ |
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
$x_1$ | $x_2$ | $x_3$ | $y$ |
0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 1 |
0 | 1 | 1 | 0 |
1 | 0 | 0 | 1 |
1 | 0 | 1 | 0 |
1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 |
$x_1$ | $x_2$ | $y$ |
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
# | Name | AND-Operator $\land$ | OR-Operator $\lor$ |
1 | Identität | $a \land 1 = a$ | $a \lor 0 = a$ |
2 | Elimination | $a \land 0 = 0$ | $a \lor 1 = 1$ |
3 | Idempotenz | $a \land a = a$ | $a \lor a = a$ |
4 | Involution | $\lnot(\lnot a) = a$ | |
5 | Komplement | $a \land \lnot a = 0 $ | $a \lor \lnot a = 1$ |
6 | Kommutativität | $a \land b = b \land a$ | $a \lor b = b \lor a$ |
7 | Assoziativität | $(a \land b) \land c = a \land (b \land c)$ | $(a \lor b) \lor c = a \lor (b \lor c)$ |
8 | Distributivität | $a \land (b \lor c) = (a \land b) \lor (a \land c)$ | $a \lor (b \land c) = (a \lor b) \land (a \lor c)$ |
9 | Vereinigung | $(a \lor b) \land (a \lor \lnot b) = a$ | $(a \land b) \lor (a \land \lnot b) = a$ |
10 | Absorption | $a \land (a \lor b)= a$ | $a \lor (a \land b)= a$ |
11 | Absorption (2) | $(a \land \lnot b) \lor b= a \lor b$ | $(a \lor\lnot b) \land b= a \land b$ |
12 | Faktorisierung | $(a \lor b) \land (\lnot a \lor c) = (a \land c) \lor (\lnot a \land b)$ | $(a \land b) \lor (\lnot a \land c) = (a \lor c) \land (\lnot a \lor b) $ |
13 | Konsens | $(a \lor b) \land (b \lor c) \land (\lnot a \lor c) = (a \lor b) \land (\lnot a \lor c)$ | $(a \land b) \lor (b \land c) \lor (\lnot a \land c) = (a \land b) \lor (\lnot a \land c)$ |
14 | de Morgans Gesetz | $\lnot (a \land b \land \dots) = \lnot a \lor \lnot b \lor \dots$ | $\lnot (a \lor b \lor \dots) = \lnot a \land \lnot b \land \dots$ |
$a$ | $b$ | $a \land b $ | $\lnot(a \land b)$ | $\lnot a$ | $\lnot b$ | $\lnot a \lor \lnot b$ |
0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 0 | 0 | 0 |
$a$ | $b$ | $a \lor b $ | $\lnot(a \lor b)$ | $\lnot a$ | $\lnot b$ | $\lnot a \land \lnot b$ |
0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 0 | 1 | 0 | 0 |
1 | 0 | 1 | 0 | 0 | 1 | 0 |
1 | 1 | 1 | 0 | 0 | 0 | 0 |
$a$ | $b$ | $y = a \rightarrow b$ |
0 | 0 | 1 |
0 | 1 | 1 |
1 | 0 | 0 |
1 | 1 | 1 |
$a$ | $b$ | $y = a \leftarrow b$ |
0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 1 |
1 | 1 | 1 |
$a$ | $b$ | $y = a \leftrightarrow b$ |
0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
$a$ | $b$ | $y = a \nleftrightarrow b$ |
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
$a$ | $b$ | $y = a \mathbin{\bar{\land}} b$ |
0 | 0 | 1 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
$a$ | $b$ | $y = a \mathbin{\bar{\lor}} b$ |
0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 0 |
Operator | Benennung | Beispiel | Sprechweise |
---|---|---|---|
$\lnot$ $\overline{\ \ }$ | Negation, NOT | $y = \lnot a = \overline{a}$ | nicht $a$ |
$\land$ | Konjunktion, AND | $y = a \land b$ | $a$ und $b$ |
$\lor$ | Disjunktion, OR | $y = a \lor b$ | $a$ oder $b$ |
$\rightarrow$ | Implikation | $y = a \rightarrow b$ | $a$ impliziert $b$ |
$\leftrightarrow$ | Äquivalenz | $y = a \leftrightarrow b$ | $a$ äquivalent $b$ |
$\nleftrightarrow$ | Antivalenz, XOR | $y = a \nleftrightarrow b$ | $a$ xor $b$ |
$\mathbin{\bar{\land}}$ | Sheffer-Funktion, NAND | $y = a \mathbin{\bar{\land}} b$ | $a$ nand $b$ |
$\mathbin{\bar{\lor}}$ | Peirce-Funktion, NOR | $y = a \mathbin{\bar{\lor}} b$ | $a$ nor $b$ |
Operator | Benennung | Beispiel | Sprechweise |
---|---|---|---|
$\overline{\ \ }$ | negation, NOT | $y = \overline{a}$ | not $a$ |
$\cdot$ | conjunction, AND | $y = a \cdot b = ab$ | $a$ and $b$ |
$+$ | disjunction, OR | $y = a + b$ | $a$ or $b$ |
$\implies$ | implication | $y = a \implies b$ | $a$ implies $b$ |
$\equiv$ $\overline{\ \oplus \ }$ | equivalence, (E)XNOR | $y = a \equiv b = \overline{a \oplus b}$ | $a$ exnor $b$ |
$\oplus$ | exclusive disjunction, (E)XOR | $y = a \oplus b$ | $a$ exor $b$ |
$\overline{\ \cdot \ }$ | NAND | $y = \overline{a \cdot b} = \overline{ab}$ | $a$ nand $b$ |
$\overline{\ + \ }$ | NOR | $y = \overline{a + b}$ | $a$ nor $b$ |
$a$ | 0011 | |
$b$ | 0101 | |
$y=0$ | 0000 | Nullfunktion |
$y=a \land b$ | 0001 | Konjunktion |
$y=a \land \lnot b$ | 0010 | |
$y=a$ | 0011 | |
$y=\lnot a \land b$ | 0100 | |
$y=b$ | 0101 | |
$y=a \nleftrightarrow b$ | 0110 | Antivalenz |
$y=a \lor b$ | 0111 | Disjunktion |
$y=a \mathbin{\bar{\lor}} b$ | 1000 | Peirce-Funktion, NOR |
$y=a \leftrightarrow b$ | 1001 | Äquivalenz |
$y=\lnot b$ | 1010 | |
$y=a \leftarrow b$ | 1011 | Inverse Implikation |
$y=\lnot a$ | 1100 | |
$y= a \rightarrow b$ | 1101 | Implikation |
$y=a \mathbin{\bar{\land}} b$ | 1110 | Sheffer-Funktion, NAND |
$y=1$ | 1111 | Einsfunktion |
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