Page 168 - FUNDAMENTALS OF COMPUTER

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168 Fundamentals of Computers NPP

Algebraic Simplification of Bool- ~y{b`Z \$bZ H$m gabrH$aU
ean Expression

Boolean: These are the variables which ~y{b`Z Ma am{e… `o Bg àH$ma H$s Ma am{e`m± h¢
have only two values. For example the input {OZHo$ {g\©$ Xmo hr _mZ hmoVo h¢& O¡go JoQ> Ho$ BZnwQ> VWm
and output variables used with gates are AmCQ>nwQ> _| Omo am{e`m± Cn`moJ _| AmVr h¢ do ~y{b`Z Ma
boolean variables. am{e`m± h¡ Š`m|{H$ BZHo$ Ho$db Xmo _mZ hmo gH$Vo h¢Ÿ&
Boolean Algebra: The algebra which deals ~y{b`Z ~rOJ{UV… Omo ~rOJ{UV ~y{b`Z Ma
with boolean variable is called Boolean algebra. am{e`m| na H$m`© H$aVm h¡, ~y{b`Z ~rOJ{UV H$hbmVm h¡&
The operations performed are called logic gånÝZ {H$`o OmZo dmbo Am°naoeÝg bm°{OH$ Am°naoeÝg
operations. H$hbmVo h¢Ÿ&
Boolean Function: This is a boolean ~y{b`Z \$bZ… `h EH$ Eogr ~y{b`Z Ma am{e hmoVr
variable which may depend on other Boolean h¡ {OgH$m _mZ Xygar ~y{b`Z Ma am{e`m| na {Z^©a H$aVm
Variables. For example consider the following
equation: F = A.B + C h¡Ÿ& {ZåZ{b{IV g_rH$aU H$mo XoImo … F = A.B + C
In this equation F is a Boolean function `hm± na F EH$ ~y{b`Z \$bZ h¡ Omo ~y{b`Z Ma
which depends upon the values of A, B and C. am{e`m| A, B Am¡a C na {Z^©a H$aVm h¡ Ÿ& Bgr{bE
Thus, we can write: F = f(A, B, C). h_ Eogm ^r {bI gH$Vo h¢… F = f(A, B, C)
Any Boolean function may also have only {H$gr ^r ~y{b`Z \$bZ Ho$ Xmo hr _mZ hmo gH$Vo
two values. For example in the above expres-
sion F may be ‘1’ or ‘0’ for all possible combi- h¢ 0 `m 1; O¡go Cnamoº$ g_rH$aU _| A, B, C Ho$ g^r
nations of A, B and C. _mZm| Ho$ {bE F Ho$ {g\©$ Xmo _mZ hmo gH$Vo h¢Ÿ&
3.11 Logic Diagram 3.11 cm°{OH$ S>m`J«m_
When Boolean function is transformed into O~ {H$gr ~y{b`Z \$bZ H$mo EH$ Eogo Vm{H©$H$
a circuit containing logic gates, it is called a n[anW _| ~Xb {X`m OmE Omo CgHo$ Ma am{e`m| _|
logic diagram. For example the Boolean gå~ÝY ñWm{nV H$a| Vmo Bgo bm°{OH$ S>m`J«m_ H$hVo h¢Ÿ&
function F shown below can be transformed to CXmhaUV… {ZåZ ~y{b`Z \$bZ F Ho$ {bE bm°{OH$
the following logic diagram: S>m`J«m_ Bg àH$ma go ~Zm`m Om gH$Vm h¡…

A
B

C F = A.B+C

Boolean algebra is used to show the relation ~y{b`Z ~rOJ{UV H$m Cn`moJ H$aHo$ BZnwQ>m| d
between inputs and outputs in the form of an AmCQ>nwQ> _| gå~ÝY H$mo ì`§OH$, gË` Vm{bH$m `m bm°{OH$
expression. This relationship may be known to us
from a truth table, or logic diagram. Boolean S>m`J«m_ go g_P gH$Vo h¢Ÿ& gmW hr EH$ gab ì`§OH$
Algebra shows how to simplify the expression so àmßV H$aHo$ n[anW H$mo N>moQ>m VWm gñVm ~Zm`m Om
as to obtain a simple and economical logic design. gH$Vm h¡Ÿ&