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NPP Number System, Boolean Algebra and Logic Circuits 205

The output F is, F = M . M . M . M 7
0
5
4
or F = (X + Y+ Z ) (X. + Y + Z ) (X. + Y + Z ) (X. + Y + ) Z This is POS form of F..
3.23 Karnaugh Map 3.23 H$maZm°\$ _on
Karnaugh Map is a very powerful tool to BgH$s ghm`Vm go EH$ ~hþV ~‹S>o ~y{b`Z ì`O§H$ H$mo
simplify a complex Boolean expression. The gab {H$`m Om gH$Vm h¡Ÿ& Bg _on H$mo h_ gË`-Vm{bH$m
Karnaugh map can be derived from a truth `m ~y{b`Z ì`§OH$ H$s ghm`Vm go ~Zm gH$Vo h¢Ÿ& AV…
table or Boolean expression. Thus, it provides
the same information as provided by the truth H$moB© ^r H$m°aZm\$ _on dhr OmZH$mar àXmZ H$aVm h¡ Omo
table or Boolean function. but in a specific way. gË`Vm{bH$m Ûmam àXmZ H$s OmVr h¡Ÿ& bo{H$Z Bg
What is that specific way? To answer this OmZH$mar H$mo _on _| ^aZo Ho$ {bE EH$ {deof ~mV H$m Ü`mZ
question consider the following Boolean aIZm Mm{hEŸ& `h {deof ~mV Š`m h¡ ? BgH$mo OmZZo
function: Ho$ {bE Xmo nXm| Ho$ \$bZ H$mo XoImo…
C . B . A + C . B . A
If B.C. is taken common: B.C. H$mo C^`{Zð> boZo na
(A + A ) .B. C

But A+ A = 1, Therefore we can write bo{H$Z A+ A = 1, AV: h_ {cI gH$Vo h¢§ {H$:

C . B . A + C . B . A = C . B
In the original expression the two terms CnamoŠV \$bZ _| Ho$db EH$ hr {~Q> _| A§Va Wm
differ by one bit position only. (That is, A (EH$ go Xygao na OmZo na A go A hmo J`m WmŸ& ~mH$s
becomes A , while moving from one term to XmoZm| {Z`V WoŸ&) Bgr{bE A§{V_ ì`§OH$ _| A hQ> J`mŸ&
another). Therefore A is eliminated. Thus, the
Karnaugh map uses the principle: AV… H$maZm°\$ _on Bg {gÕm§V H$m Cn`moJ H$aVm h¡ …
“While making groups, when we move “O~ h_ EH$ g§»`m go Xygar g§»`m na OmE Vmo
from one term to another term, there must be {g\©$ EH$ {~Q> H$s pñW{V _| n[adV©Z AmZm Mm{hEŸ&”
a change in single bit position.”
Eliminate those variables which are {OZ Ma am{e`m| _| n[adV©Z Am aho h¢, CÝh| hQ>m
changing their form. XoZm Mm{hEŸ&
Two-variable K-map Xmo-Ma K-_on
Consider the following map: {ZåZ _on H$mo XoImo:

B
A 0 1

0

1