Page 173 - FUNDAMENTALS OF COMPUTER

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NPP Number System, Boolean Algebra and Logic Circuits 173

3.14 Simplification of Boolean Functions 3.14 ~y{b`Z \$bZ H$m gabrH$aU

Important Formulae Used _hËdnyU© gyÌ
1. A + A = A 4. A A . = 0
2. A.A = A
5. A+ A B . = A+ B
3. A + A = 1
6. A = A
Note : 1. The above laws and formulae ZmoQ>:1. Cnamoº$ {Z`_ Xmo go A{YH$ Ma am{e`m| Ho$
used are also applicable to more than two {bE bmJy hmoVo h¢Ÿ& O¡go,
variables. e.g.

A + A + A + ................... = A
.
.
A + B + C + ................... = A BC ............
2. The variables may be replaced by 2. Ma am{e`m| H$mo ì`§OH$m| go {dñWm{nV {H$`m Om
expressions. e.g.
gH$Vm h¡…
A.C + A.C = A.C
XY + P = XY . P

A B. = AB.
Problem 3.23 àíZ 3.23

Simplify the following: {ZåZ H$mo gab H$s{OE…
(a) F =A + A.B + A +AB +AB (b) Y = C . B . A + C . B . A + B . A + B . A

(c) Q = XY+ X Y + Y . X + XYZ+ Z . Y . X
Solution: hc:
(a) F = A + A B . + A + A B + AB (b) Y = A.B.C + A.B.C + A.B + AB

F = (A + A ) + A ( B + ) B + AB Y = AB (C + C + ) 1 + A B
F = ( ) + A A + .1 AB = (A + A ) + AB Y = A B. .1 + AB = AB + AB
(
F = +1 AB = 1 Y = AB + ) B = A
(c) Q = Y . X + X Y + Y . X + XYZ + Z . Y . X

Q = XY + X + X Z . Y
Q = XY (1+ Z ) X+ (Y+ Y ) X+ Z . Y .
Q= X (Y+ Y Z ) X+ ........ Since Y + = Y.Z Y Z
+
Q= X (Y+ Z ) X+ = XY + (XZ + X ) ........ Since X + = X.Z X + Z
Q= XY+ X+ Z ........ Since X + = X.Y X Y = Y + X + Z
+