simplification of boolean expression

8/13/2019 Simplification Of Boolean Expression

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60-265 Computer Architecture I: Digital Design Fall 2010

EXAMPLE ANSWERS

Exercise 2 Simplification of Boolean expressions for efficient design logic

Question 1. [ 2 marks ]

1. Determine by means o a truth table the !ali"ity o De#organ$s theorem or three !ariables:

%A&C'$ ( A$ ) &$ ) C$.

A & C A$ &$ C$ A& A&C A$)&$ %A&C'$ A$)&$)C$

0 0 0 1 1 1 0 0 1 1 1 same

0 0 1 1 1 0 0 0 1 1 1 same

0 1 0 1 0 1 0 0 1 1 1 same

0 1 1 1 0 0 0 0 1 1 1 same

1 0 0 0 1 1 0 0 1 1 1 same

1 0 1 0 1 0 0 0 1 1 1 same

1 1 0 0 0 1 1 0 0 1 1 same

1 1 1 0 0 0 1 1 0 0 0 same

*oting that the !alues in the last t+o columns are the same or all ro+s %ie. all combinations possible

o 0 an" 1', De#organ$s theorem or three !ariables has been sho+n to be !ali". his is an eample

o /erect In"ucti!e proo using truth tables.

2. ist the truth table o a three-!ariable eclusi!e- %sometimes calle" 3o""$' unction:D ( A & C, +here "enotes the 4 operator.

A & C A& A&C

0 0 0 0 0

0 0 1 0 1

0 1 0 1 1

0 1 1 1 0

1 0 0 1 1

1 0 1 1 0

1 1 0 0 01 1 1 0 1

*ote that the !alues in the last %rightmost' column are 1 i the total number o 1$s in the set A,&,C isDD 7 other+ise, i the total 1$s is e!en, the !alue is 0. For this reason, the three !ariable 4 operator

is calle" the DD unction.


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8impliy the ollo+ing epressions using &oolean algebra. In each case, state the Aiom %ie. /ostulate'or heorem being applie" at each step.

For the ans+ers belo+, +e reer to the ollo+ing /ostulates an" heorems %see lecture notes':/1: Closure: here eists ,y in & such that t+o in"epen"ent operations, . %"ot' an" ) %plus' are "eine":

) y . y

/2: I"entity: here eist i"entity elements 0,1 in & relati!e to the operations ) an" . , such that or e!ery in &:

0 ) ( ) 0 ( 1 . ( . 1 (

/9: Commutati!ity: he operations ) an" . are commutati!e or all ,y in &:

) y ( y ) . y ( y . /: Distributi!ity: ;ach operation ) an" . is "istributi!e o!er the other< that is, or all ,y,= in &:

.%y)=' ( .y ) .= )%y.=' ( %)y'.%)='

/5: Complementation: For e!ery element in & there eists an element >, calle" the complement o ,satisying:

) > ( 1 .> ( 0

/6: ;istence: here eist at least t+o elements ,y in & such that ? y.

2: For each 4 in &: 4 ) 1 ( 1 4 . 0 ( 0

a. A ) A&

Deri!ati!e step /ostulate

A.1 ) A.& /2

A%1)&' /

A.1 2

A /2

Final ans+er: A

b. A& ) A&$


A%&)&$' /

A.1 /5

A /2

Final ans+er: A

c. A$&C ) AC


%A$&)A'C /

%A$)A'%&)A'C /

1.%&)A'C /5

%&)A'C /2

Final ans+er: %&)A'C ( %A)&'C ( C%A)&' etc %all similar orms are e@ui!alent'.


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". A$& ) A&C$ ) A&C

Deri!ati!e step /osthm

A$& ) A&C$ ) A&C$ ) A&C I"empotent

&%A$ ) AC$' ) A&%C$ ) C' /9, /

&%A$ ) AC$' ) A&.1 /5

&%A$ ) AC$' ) A& /2&%A$ ) AC$ ) A' /

&%A$ ) A ) AC$' /9

&%1 ) AC$' /5

&.1 2

& /2

Final ans+er: &

e. A& ) A%CD ) CD$'


A& ) AC%D ) D$' /

A& ) AC.1 /5

A& ) AC /2

A%& ) C' /

Final ans+er: A%& ) C' *;: his ans+er is simpler than the secon" last line abo!e "ue

to the act that the number o operations is only 2 %one , one A*D' in the last line, !ersus 9

%one , t+o A*D' in the secon" last line.

. %&C$ ) A$D'%A&$ ) CD$'


&C$%A&$ ) CD$' ) A$D%A&$ ) CD$' /

&C$A&$ ) &C$CD$ ) A$DA&$ ) A$D CD$ /

A&&$C$ ) &CC$D$ ) AA$&$D ) A$CDD$ /9

A.0.C$ ) &.0.D$ ) 0.&$D ) A$C.0 /5

0 ) 0 ) 0 ) 0 2

0 /2

Final ans+er: 0 *;: It is interesting %to say the least' that a complicate" circuit "esignin!ol!ing many inputs may simpliy to a !ery simple circuit, as in this case +here the output is

0 regar"less o the inputs.


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1. Bsing De#organ$s theorem, sho+ that:

a. %A ) &'$%A$ ) &$'$ ( 0

%A ) &'$%A$ ) &$'$ ( %%A ) &' ) %A$ ) &$''$

( %%A ) A$' ) %& ) &$''$ ( % 1 ) 1 '$( 1$( 0 ;D

b. A ) A$& ) A$&$ ( 1

A ) A$& ) A$&$ ( A ) A$%& ) &$'( A ) A$%&$&'$

( A ) A$%0'$

( A ) A$.1

( A ) A$( %A$A'$

( %0'$

( 1 ;D

*;: his eample is notably ineicient in ho+ a higher or"er theorem, such as

"e#organ$s theorem, is use" to get the result. he same result coul" ob!iously ha!e beenachie!e" in e+er steps by applying the postulates "irectly, or simpler theorems.

2. Ei!en the &oolean epression: F ( 4$ ) 4G$

&eore beginning, note that F ( %4$ ) 4G$' ( %4$)G$' %see eamples abo!e'.

a. Deri!e an algebraic epression or the complement F$.

F$ ( %4$ ) 4G$'$ ( %4$'$.%4G$'$( %4$$ ) $'%4$ ) $ ) G$$'

( %4 ) $'%4$ ) $ ) G'

Hhich simpliies to:

( %4)$'4$ ) %4)$'$ ) %4)$'G

( 4$$ ) 4$ ) $ ) 4G ) $G( $ ) 4G ) $G( $ ) 4G

It is possible to sho+ this more "irectly, using:F$ ( %%4$)G$''$ ( %4$)G$'$ ) $ "e#organJ ( 4G ) $ "e#organJ


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b. 8ho+ that FF$ ( 0. %Bse algebra, not truth tables'

I +e start +ith the simpliie" orm o F ( %4$)G$', then:

FF$ ( %%4$)G$''.%$ ) 4G'

( %4G'$%$ ) 4G' "e#organ, Distributi!eJ

( %4G'$%4G' Complement, I"entityJ( 0 ;D ComplementJ

c. 8ho+ that F ) F$ ( 1. %Bse algebra, not truth tables'

I +e start +ith the simpliie" orm o F ( %4$)G$', then:

F)F$ ( %4$)G$' ) %$ ) 4G'

( %4G'$ ) %$ ) 4G'

( $ ) 4G ) %4G'$

( $) %4G'$ ) %%4G' ) %4G'$'%4G ) '( $) %4G'$ ) %4G ) '

( %)$' ) %%4G' ) %4G'$'

( 1 ) 1( 1 ;D

K All inormation on this +ebsite is Copyright K 2010 by obert D. Lent. All rights reser!e".

simplification of boolean expression

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