# Digitale Schaltungstechnik/ Gray-Code nach Binär

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## Tabelle

Wert D C B A Z Y X W
0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 1
2 0 0 1 1 0 0 1 0
3 0 0 1 0 0 0 1 1
4 0 1 1 0 0 1 0 0
5 0 1 1 1 0 1 0 1
6 0 1 0 1 0 1 1 0
7 0 1 0 0 0 1 1 1
8 1 1 0 0 1 0 0 0
9 1 1 0 1 1 0 0 1
10 1 1 1 1 1 0 1 0
11 1 1 1 0 1 0 1 1
12 1 0 1 0 1 1 0 0
13 1 0 1 1 1 1 0 1
14 1 0 0 1 1 1 1 0
15 1 0 0 0 1 1 1 1

## KV-Diagramme

Wert B A B A B A B A
D C 0 1 2 3
D C 7 6 5 4
D C 8 9 10 11
D C 15 14 13 12

W B A B A B A B A
D C 0 1 0 1
D C 1 0 1 0
D C 0 1 0 1
D C 1 0 1 0
${\displaystyle W=}$

X B A B A B A B A
D C 0 0 1 1
D C 1 1 0 0
D C 0 0 1 1
D C 1 1 0 0
${\displaystyle X={\overline {D}}\ {\overline {C}}B\lor {\overline {D}}C{\overline {B}}\lor DCB\lor D{\overline {C}}\ {\overline {B}}}$

Y B A B A B A B A
D C 0 0 1 1
D C 1 1 0 0
D C 1 1 0 0
D C 0 0 1 1
${\displaystyle Y={\overline {C}}B\lor C{\overline {B}}}$

Z B A B A B A B A
D C 0 0 0 0
D C 0 0 0 0
D C 1 1 1 1
D C 1 1 1 1
${\displaystyle Z=D}$


## Vereinfachnung

${\displaystyle Y={\overline {C}}B\lor C{\overline {B}}}$
${\displaystyle Y=C\oplus B}$

 ${\displaystyle X={\overline {D}}\ {\overline {C}}B\lor {\overline {D}}C{\overline {B}}\lor DCB\lor D{\overline {C}}\ {\overline {B}}}$ ${\displaystyle {\overline {D}}}$ ausklammern ${\displaystyle X={\overline {D}}({\overline {C}}B\lor C{\overline {B}})\lor DCB\lor D{\overline {C}}\ {\overline {B}})}$ ${\displaystyle D}$ ausklammern ${\displaystyle X={\overline {D}}({\overline {C}}B\lor C{\overline {B}})\lor D(CB\lor {\overline {C}}\ {\overline {B}})}$ ${\displaystyle {\overline {C}}B\lor C{\overline {B}}}$ ist C xor B ${\displaystyle X={\overline {D}}(C\oplus B)\lor D(CB\lor {\overline {C}}\ {\overline {B}})}$ ${\displaystyle CB\lor {\overline {C}}\ {\overline {B}}}$ ist C equ B, C equ B ist C xnor B ${\displaystyle X={\overline {D}}(C\oplus B)\lor D({\overline {C\oplus B}})}$ da ${\displaystyle Y=C\oplus B}$ können wir die Ausdrücke mit Y substituieren ${\displaystyle X={\overline {D}}Y\lor D{\overline {Y}}}$ ${\displaystyle X=D\oplus Y}$