Unit-2_Boolean Eq. Simplifiaction

8/18/2019 Unit-2_Boolean Eq. Simplifiaction

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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

Prof. Tushar Patel (M.tech/DC)


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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 –

8Prof. Tushar Patel (M.tech/DC)


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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 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 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

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Unit-2_Boolean Eq. Simplifiaction