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Unit-2Boolean Function and
its Minimization
Prof. Tushar Patel
M.tech(DC)
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Contents
• Simplication of Boolean Function using Booleantheorems
• Canonical & Standard Forms S!P and P!S
•
Minimi"ation Methods – #arnaugh Map Method
– #arnaugh Map Method $ith Don%t care condition
– Taulation Method
•
Concept of Prime 'mplicants• eali"ation of Boolean Functions
sing !nl* +,+D
and +! -ates
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Basic Defnitions
•
Binar* !perators – ,+D
z 0 x 1 y 0 x
y z 02 if x 02 AND
y 02
– !
z 0 x 3 y z 02
if x 02 OR y 02
–
+!T z 0 x 0 x’
z 02 if x 04
• Boolean ,lgera
–
Binar* 5ariales onl* 64% and 62% 7alues –
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Boolean Algebra Postulates
• Commutati7e 9a$
x 1 y 0 y 1 x
x 3 y 0 y 3 x
• 'dentit* :lement x 1 2 0 x
x 3 4 0 x
• Complement
x 1 x’ 0 4
x 3 x’ 0 2
;Prof. Tushar Patel (M.tech/DC)
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Boolean Algebra Theorems
• Dualit* – The dual of a Boolean algeraic
e
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• Theorem 8 Involution –
( x’ )’ 0 x
( x ) 0 x • Theorem
; Associative & Distributive
– ( x 1 y )
1 z 0 x 1
( y 1 z )
( x 3 y )3 z 0 x 3
( y 3 z )
– x 1
( y 3 z ) 0
( x 1 y ) 3
( x 1 z )
x 3
( y 1 z ) 0
( x 3 y )1
( x 3 z )
• Theorem = DeMorgan –
( x 1 y )’ 0 x’
3 y’
( x 3 y )’ 0 x’
1 y’ –
( x 1 y ) 0 x 3 y
( x 3 y ) 0 x
1 y
• Theorem > Absorption >Prof. Tushar Patel
(M.tech/DC)
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DeMorgan’s Theorem
)]([ ed cba ++
)]([ ed cba +++
))(( ed cba ++
))(( ed cba +++
))(( ed cba ++
)( ed cba ++
?Prof. Tushar Patel (M.tech/DC)
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Boolean Functions
•
Boolean :
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Comlement o! a Function
•
DeMorgan%s Theorm
• Dualit* & 9iteral Complement
C B A F ++=
C B A F ++=
C B A F ••=
C B A F ++=
C B A F ••
C B A F ••=
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Canonical Forms
•
Minterm – Product ( AND function)
– Contains all 7ariales
– :7aluates to 62% for aspecic comination
!xa"ple
A 0 4 A $ %
$ 0 4 (4) 1 (4) 1 (4)
% 0 42 1 2 1 2 0 2
A B C Minterm
0 0 0 0 m0
1 0 0 1 m1
2 0 1 0 m2
3 0 1 1 m3
4 1 0 0 m4
5 1 0 1 m5
6 1 1 0 m6
7 1 1 1 m7
C B A
C B A
C B A
C B A
C B A
C B A
C B A
C B A
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•
Ma
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A B C F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 11 1 0 0
1 1 1 1
C B A F =
C B A+ C B A+
ABC +
• Truth Tale to Boolean Function
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• Sum of Minter"s
• Product of Maxter"s
2; @
• Sum of Minter"s 'O()
• Product of Maxter"s (O')
A B C F
0 0 0 0 0
1 0 0 1 1
2 0 1 0 0
3 0 1 1 04 1 0 0 1
5 1 0 1 1
6 1 1 0 0
7 1 1 1 1
ABC C B AC B AC B A F
+++=
7541 mmmm F +++=
∑= )7,5,4,1( F
C AB BC AC B AC B A F
+++=
C AB BC AC B AC B A F
+++=
C AB BC AC B AC B A F
•••=
))()()((
C B AC B AC B AC B A F
++++++++=
6320
M M M M F
=
∏= (0,2,3,6) F
F
1
0
1
10
0
1
0
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"tandard Forms
ABC C B AC B AC B A F
+++=
AC
B B AC
=
+ )(
C B
A AC B
=
+ )(
B A
B A
C C B A
=
=
+
)1()(
)()()(
B B AC C C B A A AC B F
+++++=
AC B AC B F
++=
• Sum of Products (S!P)
2=Prof. Tushar Patel (M.tech/DC)
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)( A AC B +
)( B BC A +
)( C C B A +
)()()(
A AC BC C B A B BC A F
+++++=
C B B AC A F
++=
C AB BC AC B AC B A F
+++=
))()((
C B B AC A F
+++=
• Product of Sums (P!S)
2>Prof. Tushar Patel (M.tech/DC)
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#$amles
2@Prof. Tushar Patel (M.tech/DC)
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#$ercise
2AProf. Tushar Patel (M.tech/DC)
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2-10 Draw the logic diagram !or the !ollowing
Booleanex"reion#
$a% Y & A' B' (
B $ A ( C % $)% Y &
BC ( AC '
$c% Y & A ( CD $d% Y &
$ A ( B% $C ' ( D%
2-12 *im"li!+ the Boolean !,nction T 1 and
T 2 to a minim,mn,m)er o! literal
A B C T 1 T 2
0 0 0 1 0
0 0 1 1 0
0 1 0 1 00 1 1 0 1
1 0 0 0 1
1 0 1 0 1
1 1 0 0 1
1 1 1 0 1
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Minimization Method
– #arnaugh Map Method
– #arnaugh Map Method $ith Don%t carecondition
– Taulation Method
– #%Map , #arnaugh map pro7ides a
s*stematic method for simplif*ing Booleane
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T%o-&ariable '’Ma(
Three-&ariable '’Ma(
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Four-&ariable '’Ma(
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;Prof. Tushar Patel (M.tech/DC)
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Prof. Tushar Patel (M.tech/DC) =
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Don’t Care Condition
Prof. Tushar Patel (M.tech/DC) >
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Fi)e-&ariable '’Ma
Prof. Tushar Patel (M.tech/DC) ?
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#$ercise
Prof. Tushar Patel (M.tech/DC) @
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Tabulation Method(
Prof. Tushar Patel (M.tech/DC) A
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eali"ation of Boolean Fun. sing !nl* +,+D&+! -ates
Prof. Tushar Patel (M.tech/DC) 84
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Using *nl+ ,*
Prof. Tushar Patel (M.tech/DC) 82
