Digital Unit- II

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UNIT II

Boolean arithmetic

Let us begin our exploration of Boolean algebra by adding numbers together:0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 1

The first three sums mae perfect sense to anyone familiar !ith elementary addition" The

last sum# though# is $uite possibly responsible for more confusion than any other single statement

in digital electronics# because it seems to run contrary to the basic principles of mathematics"

%ell# it does contradict principles of addition for real numbers# but not for Boolean numbers"

&emember that in the !orld of Boolean algebra# there are only t!o possible 'alues for any

$uantity and for any arithmetic operation: 1 or 0" There is no such thing as()( !ithin the scope of

Boolean 'alues" *ince the sum(1 + 1( certainly isnt 0# it must be 1 by process of elimination"

,t does not matter ho! many or fe! terms !e add together# either" -onsider the follo!ing

sums:

0 + 1 + 1 = 1

0 + 1 + 1 + 1 = 1

1 + 0 + 1 + 1 + 1 = 1

1 + 1 + 1 = 1

Tae a close loo at the t!o.term sums in the first set of e$uations" Does that pattern loo

familiar to you/ ,t should ,t is the same pattern of 1s and 0s as seen in the truth table for an

& gate" ,n other !ords# Boolean addition corresponds to the logical function of an(&(

gate# as !ell as to parallel s!itch contacts:

There is no such thing as subtraction in the realm of Boolean mathematics" *ubtraction

implies the existence of negati'e numbers: 2 . 3 is the same thing as 2 + 4.35# and in Boolean

algebra negati'e $uantities are forbidden" There is no such thing as di'ision in Boolean

mathematics# either# since di'ision is really nothing more than compounded subtraction# in the

same !ay that multiplication is compounded addition"

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8ultiplication is 'alid in Boolean algebra# and thanfully it is the same as in real.number

algebra: anything multiplied by 0 is 0# and anything multiplied by 1 remains unchanged:

0 9 0 = 0

0 9 1 = 0

1 9 0 = 0

1 9 1 = 1

This set of e$uations should also loo familiar to you: t is the same pattern found in the

truth table for an ;D gate" ,n other !ords# Boolean multiplication corresponds to the logical

function of an(;D( gate# as !ell as to series s!itch contacts:

Lie(normal( algebra# Boolean algebra uses alphabetical letters to denote 'ariables" 7nlie

(normal( algebra# though# Boolean 'ariables are al!ays -<,TL letters# ne'er lo!ercase"

Because they are allo!ed to possess only one of t!o possible 'alues# either 1 or 0# each and

e'ery 'ariable has a complement : the opposite of its 'alue" or example# if 'ariable(( has a

'alue of 0# and then the complement of has a 'alue of 1" Boolean notation uses a bar abo'e the

'ariable character to denote complementation# lie this:

,f: =0# Then: =1

,f: =1 Then: =0

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,n !ritten form# the complement of(( denoted as(.not( or(.bar(" *ometimes a(

prime( symbol is used to represent complementation" or example# !ould be the complement

of # much the same as using a prime symbol to denote differentiation in calculus rather than the

fractional notation d>dt" 7sually# though# the(bar( symbol finds more !idespread use than the

?prime( symbol# for reasons that !ill become more apparent later in this chapter"

Boolean complementation finds e$ui'alency in the form of the ;T gate# or a normally closed

s!itch or relay contact:

The basic definition of Boolean $uantities has led to the simple rules of addition and

multiplication# and has excluded both subtraction and di'ision as 'alid arithmetic operations" %e

ha'e a symbology for denoting Boolean 'ariables# and their complements"

@ REVIEW:

@ Boolean addition is e$ui'alent to the OR logic function# as !ell as parallel s!itch contacts"

@ Boolean multiplication is e$ui'alent to the AND logic function# as !ell as series s!itch

contacts"

@ Boolean complementation is e$ui'alent to the NOT logic function# as !ell as normally closed

relay contacts"

Basic Laws

,n mathematics# an identity is a statement true for all possible 'alues of its 'ariable or

'ariables" The algebraic identity of x + 0 = x tells us that anything 4x5 added to Aero e$uals the

original (anything#( no matter !hat 'alue that (anything( 4x5 may be" Lie ordinary algebra#

Boolean algebra has its o!n uni$ue identities based on the bi'alent states of Boolean 'ariables"

The first Boolean identity is that the sum of anything and Aero is the same as the

original(anything"( This identity is no different from its real.number algebraic e$ui'alent:




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;o matter !hat the 'alue of # the output !ill al!ays be the same: !hen =1# the output

!ill also be 1 !hen =0# the output !ill also be 0"

The next identity is most definitely different from any seen in normal algebra" Cere !e

disco'er that the sum of anything and one is one:

;o matter !hat the 'alue of # the sum of and 1 !ill al!ays be 1" ,n a sense# the(1(

signal overrides the effect of on the logic circuit# lea'ing the output fixed at a logic le'el of 1"

;ext# !e examine the effect of adding and together# !hich is the same as connecting both

inputs of an & gate to each other and acti'ating them !ith the same signal:

,n real.number algebra# the sum of t!o identical 'ariables is t!ice the original 'ariables

'alue 4x + x = )x5# but remember that there is no concept of()( in the !orld of Boolean math#

only 1 and 0# so !e cannot say that + = )" Thus# !hen !e add a Boolean $uantity to itself#

the sum is e$ual to the original $uantity: 0 + 0 = 0# and 1 + 1 = 1"

,ntroducing the uni$uely Boolean concept of complementation into an additi'e identity#

!e find an interesting effect" *ince there must be one(1( 'alue bet!een any 'ariable and its

complement# and since the sum of any Boolean $uantity and 1 is 1# the sum of a 'ariable and

,ts complement must be 1:

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ust as there are four Boolean additi'e identities 4+0# +1# +# and +5# so there

are also four multiplicati'e identities: x0# x1# x# and x" f these# the first t!o are no

different from their e$ui'alent expressions in regular algebra:

The third multiplicati'e identity expresses the result of a Boolean $uantity multiplied by

itself" ,n normal algebra# the product of a 'ariable and itself is the square of that 'ariable 43 x 3 =

3) = F5" Co!e'er# the concept of(s$uare( implies a $uantity of )# !hich has no meaning in

Boolean algebra# so !e cannot say that x = )" ,nstead# !e find that the product of a

Boolean $uantity and itself is the original $uantity# since 0 x 0 = 0 and 1 x 1 = 1:

The fourth multiplicati'e identity has no e$ui'alent in regular algebra because it uses the

complement of a 'ariable# a concept uni$ue to Boolean mathematics" *ince there must be one

(0( 'alue bet!een any 'ariable and its complement# and since the product of any Boolean

$uantity and 0 is 0# the product of a 'ariable and its complement must be 0:

To summariAe# then# !e ha'e four basic Boolean identities for addition and four for

multiplication:

dditi'e




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+ 0 =

+ 1 = 1

+ =

+ = 1

8ultiplicati'e

0 = 0

1 =

=

= 0

Basic Boolean algebraic identities

nother identity ha'ing to do !ith complementation is that of the double complement : a

'ariable in'erted t!ice" -omplementing a 'ariable t!ice 4or any e'en number of times5 results in

the original Boolean 'alue" This is analogous to negating 4multiplying by .15 in real.number

algebra: an e'en number of negations cancel to lea'e the original 'alue:

Boolean algebraic properties

nother type of mathematical identity# called a (property( or a (la!#( describes ho!

differing 'ariables relate to each other in a system of numbers" ne of these properties is no!n

as the commutative property# and it applies e$ually to addition and multiplication" ,n essence# the

commutati'e property tells us !e can re'erse the order of 'ariables that are either added together

or multiplied together !ithout changing the truth of the expression:

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long !ith the commutati'e properties of addition and multiplication# !e ha'e the

associative property# again applying e$ually !ell to addition and multiplication" This property

tells us !e can associate groups of added or multiplied 'ariables together !ith parentheses

!ithout altering the truth of the e$uations"

Lastly# !e ha'e the distributive property# illustrating ho! to expand a Boolean expression

formed by the product of a sum# and in re'erse sho!s us ho! terms may be factored out of

Boolean sums.of.products:

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Digital Electronics

To summariAe# here are the three basic properties: commutati'e# associati'e# and distributi'e"

Basic Boolean algebraic properties

4B + -5 = B + -

dditi'e

+ 4B + -5 = 4 + B5 + -

+ B = B +

8ultiplicati'e

4B-5 = 4B5-

B = B

DeMorgan’s Theorems

mathematician named De8organ de'eloped a pair of important rules regarding group

complementation in Boolean algebra" By group complementation# ,m referring to the

complement of a group of terms# represented by a long bar o'er more than one 'ariable" Iou

should recall from the chapter on logic gates that in'erting all inputs to a gate re'erses

that gates essential function from ;D to &# or 'ice 'ersa# and also in'erts the output" *o#

an & gate !ith all inputs in'erted 4a ;egati'e.& gate5 beha'es the same as a ;;D gate#

and an ;D gate !ith all inputs in'erted 4a ;egati'e.;D gate5 beha'es the same as a ;&

gate" De8organs theorems state the same e$ui'alence in (bac!ard( form: that in'erting the

output of any gate results in the same function as the opposite type of gate 4;D 's" &5 !ith

in'erted inputs:

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long bar extending o'er the term B acts as a grouping symbol# and as such is entirely

different from the product of and B independently in'erted" ,n other !ords# 4B5 is not e$ual

to B" Because the (prime( symbol 45 cannot be stretched o'er t!o 'ariables lie a bar can#!e are forced to use parentheses to mae it apply to the !hole term B in the pre'ious sentence"

bar# ho!e'er# acts as its o!n grouping symbol !hen stretched o'er more than one 'ariable"

This has profound impact on ho! Boolean expressions are e'aluated and reduced# as !e shall

see"

De8organs theorem may be thought of in terms of breaking a long bar symbol" %hen a long bar

is broen# the operation directly underneath the brea changes from addition to multiplication# or

'ice 'ersa# and the broen bar pieces remain o'er the indi'idual 'ariables"

To illustrate:

Boolean algebra finds its most practical use in the simplification of logic circuits" ,f !e

translate a logic circuitKs function into symbolic 4Boolean5 form# and apply certain algebraic rules

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to the resulting e$uation to reduce the number of terms and>or arithmetic operations# the

simplified e$uation may be translated bac into circuit form for a logic circuit performing the

same function !ith fe!er components" ,f e$ui'alent function may be achie'ed !ith fe!er

components# the result !ill be increased reliability and decreased cost of manufacture"

To this end# there are se'eral rules of Boolean algebra presented in this section for use in

reducing expressions to their simplest forms" The identities and properties already re'ie!ed in

this chapter are 'ery useful in Boolean simplification# and for the most part bear similarity to

many identities and properties of normal algebra" Co!e'er# the rules sho!n in this section are

all uni$ue to Boolean mathematics"

This rule may be pro'en symbolically by factoring an out of the t!o terms# then

applying the rules of + 1 = 1 and 1 = to achie'e the final result:

<lease note ho! the rule + 1 = 1 !as used to reduce the 4B + 15 term to 1" %hen a rule

lie + 1 = 1 is expressed using the letter # it doesnKt mean it only applies to expressions

containing " %hat the stands for in a rule lie + 1 = 1 is any Boolean 'ariable or

collection of 'ariables" This is perhaps the most difficult concept for ne! students to master in

Boolean simplification: applying standardiAed identities# properties# and rules to expressions not

in standard form"




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or instance# the Boolean expression B- + 1 also reduces to 1 by means of the + 1 = 1

identity" ,n this case# !e recogniAe that the term in the identityKs standard form can represent

the entire B- term in the original expression"

The next rule loos similar to the first one sho!n in this section# but is actually $uite

different and re$uires a more cle'er proof:

;ote ho! the last rule 4 + B = 5 is used to un.simplify the first term in the

expression# changing the into an + B" %hile this may seem lie a bac!ard step# it

certainly helped to reduce the expression to something simpler *ometimes in mathematics !e

must tae bac!ard steps to achie'e the most elegant solution" 6no!ing !hen to tae such a

step and !hen not to is part of the art.form of algebra# Must as a 'ictory in a game of chess almost

al!ays re$uires calculated sacrifices"

nother rule in'ol'es the simplification of a product.of.sums expression:




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nd# the corresponding proof:

To summariAe# here are the three ne! rules of Boolean simplification expounded in this section:

arna!gh map

The 6arnaugh map# lie Boolean algebra# is a simplification tool applicable to digital

logic" The 6arnaugh 8ap !ill simplify logic faster and more easily in most cases"

Boolean simplification is actually faster than the 6arnaugh map for a tas in'ol'ing t!o

or fe!er Boolean 'ariables" ,t is still $uite usable at three 'ariables# but a bit slo!er" t four

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Digital Electronics

input 'ariables# Boolean algebra becomes tedious" 6arnaugh maps are both faster and easier"

6arnaugh maps !or !ell for up to six input 'ariables# are usable for up to eight 'ariables" or

more than six to eight 'ariables# simplification should be by CAD 4computer automated design5"

Relationship between a arna!gh Map an" a Tr!th Table

Each ro! in the table 4or minterm5 is e$ui'alent to a a cell on the 6arnaugh 8ap"

Example N1:

Cere is a t!o.input truth table for a digital circuit:

&o! ,nputs utput

The corresponding 6.map is:

Example N):Cere is a three.input truth table for a digital circuit:




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Example N3:

Cere is a four.input truth table for a digital circuit:




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#impli$%ing Boolean E&pressions !sing arna!gh map

To simplify the resulting Boolean expression using a 6arnaugh map adMacent cells

containing one are looped together" This step eliminated any terms of the form AA "

dMacent cells means:

1" -ells that are side by side in the horiAontal and 'ertical directions 4but not diagonal5"

)" or a map ro!: the leftmost cell and the rightmost cell"

3" or a map column: the topmost cell and the bottom most cell"

" or a 'ariable map: cells occupying the four corners of the map"

-ells may only be looped together in t!os# fours# or eights" s fe! groups as possible

must

be formed" Oroups may o'erlap one another and may contain only one cell"

The larger the number of 1s looped together in a group the simpler is the product termthat the group represents"

Example N1:

*implifying the corresponding 6.map of a t!o.input truth table for a digital circuit:

,n Loop 1 the 'ariable has both logic 0 and logic 1 'alues in the same loop" B has a 'alue of

1" Cence minterm e$uation is: F = B




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,n Loop ) Pariable B ha'e both logic 0 and 1 'alues in the same loop" = 1# hence minterm

e$uation is: F = A

The o'erall Boolean expression for is therefore: F = A + B

Example N):

*implifying the corresponding 6.map of a three.input truth table for a digital circuit# ,n

Loop 1 the 'ariable - has both logic 0 and logic 1 'alues in the same loop" has a 'alue of 0 and

B has a logic 'alue of 1" Cence minterm e$uation is: F = AB

,n Loop ) the 'ariable - has both logic 0 and 1 'alues in the same loop" = 1 and B = 0# hence

minterm e$uation is: F = AB "

,n Loop 3 the t!o 'ariables and B both ha'e logic 0 and logic 1 'alues in the same loop" - has

a 'alue of 1" Cence minterm e$uation is: F = C

The o'erall Boolean expression for is therefore: F = AB + AB + C

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Digital Electronics

Example N3:

*implifying the corresponding 6.map of a four.input truth table for a digital circuit# ,n

Loop 1 the t!o 'ariables and D both ha'e logic 0 and logic 1 'alues in the same loop" - has a

'alue of 0 and B has a 'alue of 1" Cence minterm e$uation is: F = BC

,n Loop ) the t!o 'ariables B and - both ha'e logic 0 and logic 1 'alues in the same

loop" has a 'alue of 1 and D has a 'alue of 0" Cence minterm e$uation is: F = AD

,n Loop 3 the 'ariable D has logic 0 and logic 1 'alues in the same loop" and B both

ha'e a 'alue of 0 and - has a 'alue of 1" Cence minterm e$uation is: F = ABC

,n Loop the t!o 'ariables B and - both ha'e logic 0 and logic 1 'alues in the same

loop" has a 'alue of 0 and D has a 'alue of 1" Cence minterm e$uation is: F = AD

,n Loop 2 the 'ariable - has logic 0 and logic 1 'alues in the same loop" and D both ha'e a

'alue of 1 and B has a 'alue of 0" Cence minterm e$uation is: F = ABD"

The o'erall Boolean expression for is therefore: F = BC + AD + ABC + AD + ABD

'!ine(Mc)l!s*e% algorithm

The QuineR8c-lusey algorithm 4or the method of prime implicants5 is a method used

for minimiAation of Boolean functions !hich !as de'eloped by %"P" Quine and Ed!ard "

8c-lusey" ,t is functionally identical to 6arnaugh mapping# but the tabular form maes it more

efficient for use in computer algorithms# and it also gi'es a deterministic !ay to chec that the

minimal form of a Boolean function has been reached" ,t is sometimes referred to as the

tabulation method"

The method in'ol'es t!o steps:

1" inding all prime implicants of the function"

)" 7se those prime implicants in a prime implicant c!art to find the essential prime

implicants of the function# as !ell as other prime implicants that are necessary to co'er

the function"

)omple&it%

lthough more practical than 6arnaugh mapping !hen dealing !ith more than four

'ariables# the QuineR8c-lusey algorithm also has a limited range of use since the problem it

sol'es is ;<.hard: the runtime of the QuineR8c-lusey algorithm gro!s exponentially !ith the

number of 'ariables" ,t can be sho!n that for a function of n 'ariables the upper bound on the

Department of Electronics 6arpagam 7ni'ersity1H

http://en.wikipedia.org/wiki/Boolean_function

http://en.wikipedia.org/wiki/Willard_Van_Orman_Quine

http://en.wikipedia.org/wiki/Edward_J._McCluskey


http://en.wikipedia.org/wiki/Karnaugh_map

http://en.wikipedia.org/wiki/Implicant

http://en.wikipedia.org/wiki/Boolean_satisfiability_problem

http://en.wikipedia.org/wiki/NP-hard

http://en.wikipedia.org/wiki/Run_time_(computing)

http://en.wikipedia.org/wiki/Exponential_growth

http://en.wikipedia.org/wiki/Boolean_function

http://en.wikipedia.org/wiki/Willard_Van_Orman_Quine



http://en.wikipedia.org/wiki/Karnaugh_map

http://en.wikipedia.org/wiki/Implicant

http://en.wikipedia.org/wiki/Boolean_satisfiability_problem

http://en.wikipedia.org/wiki/NP-hard

http://en.wikipedia.org/wiki/Run_time_(computing)

http://en.wikipedia.org/wiki/Exponential_growth



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number of prime implicants is 3n>n" ,f n = 3) there may be o'er G"2 S 1012 prime implicants"

unctions !ith a large number of 'ariables ha'e to be minimiAed !ith potentially non.optimal

heuristic methods# of !hich the Espresso heuristic logic minimiAer is the de.facto standard"

Example

*tep 1: finding prime implicants

8inimiAing an arbitrary function:

B - D f

m0 0 0 0 0 0

m1 0 0 0 1 0

m) 0 0 1 0 0

m3 0 0 1 1 0

m 0 1 0 0 1

m2 0 1 0 1 0

mG 0 1 1 0 0

mH 0 1 1 1 0

mJ 1 0 0 0 1

mF 1 0 0 1 x

m10 1 0 1 0 1

m11 1 0 1 1 1

m1) 1 1 0 0 1

m13 1 1 0 1 0

m1 1 1 1 0 x

m12 1 1 1 1 1

ne can easily form the canonical sum of products expression from this table# simply by

summing the minterms 4lea'ing out donKt.care terms5 !here the function e'aluates to one:

f A# B#C # D = AK BC K DK + ABKC K DK + ABKCDK + ABKCD + ABC K DK + ABCD"

f course# thatKs certainly not minimal" *o to optimiAe# all minterms that e'aluate to one

are first placed in a minterm table" DonKt.care terms are also added into this table# so they can be

combined !ith minterms:

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http://en.wikipedia.org/wiki/Heuristic_algorithm

http://en.wikipedia.org/wiki/Espresso_heuristic_logic_minimizer

http://en.wikipedia.org/wiki/Sum_of_products

http://en.wikipedia.org/wiki/Minterm

http://en.wikipedia.org/wiki/Don't-care_(logic)

http://en.wikipedia.org/wiki/Heuristic_algorithm

http://en.wikipedia.org/wiki/Espresso_heuristic_logic_minimizer

http://en.wikipedia.org/wiki/Sum_of_products

http://en.wikipedia.org/wiki/Minterm

http://en.wikipedia.org/wiki/Don't-care_(logic)



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;umber of 1s 8interm Binary &epresentation

............................................

1 m 0100

mJ 1000

............................................

) mF 1001

m10 1010

m1) 1100

............................................

3 m11 1011

m1 1110

............................................

m12 1111

t this point# one can start combining minterms !ith other minterms" ,f t!o terms 'ary

by only a single digit changing# that digit can be replaced !ith a dash indicating that the digit

doesnKt matter" Terms that canKt be combined any more are mared !ith a S" %hen going from

*iAe ) to *iAe # treat K.K as a third bit 'alue" Ex: .110 and .100 or .11. can be combined# but not

.110 and 011." 4Tric: 8atch up the K.K first"5

;umber of 1s 8interm 0.-ube *iAe ) ,mplicants *iAe ,mplicants

.......................................................................

1 m 0100 m4#1)5 .100S m4J#F#10#115 10..S

mJ 1000 m4J#F5 100. m4J#10#1)#15 1..0S

.............................. m4J#105 10.0 ......................

) mF 1001 m4J#1)5 1.00 m410#11#1#125 1.1.S

m10 1010 ...................

m1) 1100 m4F#115 10.1

.............................. m410#115 101.

3 m11 1011 m410#15 1.10

m1 1110 m41)#15 11.0

.................................................

m12 1111 m411#125 1.11

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Digital Electronics

Boolean functions may be practically implemented by using electronic gates" The

follo!ing points are important to understand"

• Electronic gates re$uire a po!er supply"

• Oate IN,UT# are dri'en by 'oltages ha'ing t!o nominal 'alues# e"g" 0P and 2P

representing logic 0 and logic 1 respecti'ely"

• The -UT,UT of a gate pro'ides t!o nominal 'alues of 'oltage only# e"g" 0P and 2P

representing logic 0 and logic 1 respecti'ely" ,n general# there is only one output to a

logic gate except in some special cases"

• There is al!ays a time delay bet!een an input being applied and the output responding"

Tr!th Tables

Truth tables are used to help sho! the function of a logic gate" ,f you are unsure about

truth tables and need guidance on ho! go about dra!ing them for indi'idual gates or logic

circuits then use the truth table section lin"

Logic gates

Digital systems are said to be constructed by using logic gates" These gates are the ;D#

&# ;T# ;;D# ;&# EU& and EU;& gates" The basic operations are described belo!

!ith the aid of truth tables"

.ND gate

The ;D gate is an electronic circuit that gi'es a high output 415 only if all its inputs are

high" dot 4"5 is used to sho! the ;D operation i"e" "B" Bear in mind that this dot is

sometimes omitted i"e" B

-R gate

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http://www.ee.surrey.ac.uk/Projects/Labview/gatesfunc/TruthFrameSet.htm



http://www.ee.surrey.ac.uk/Projects/Labview/gatesfunc/index.html#truth%23truth







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The & gate is an electronic circuit that gi'es a high output 415 if one or more of its

inputs are high" plus 4+5 is used to sho! the & operation"

N-T gate

The ;T gate is an electronic circuit that produces an in'erted 'ersion of the input at its

output" ,t is also no!n as an inverter " ,f the input 'ariable is # the in'erted output is no!n as

;T " This is also sho!n as K# or !ith a bar o'er the top# as sho!n at the outputs

N.ND gate

This is a ;T.;D gate !hich is e$ual to an ;D gate follo!ed by a ;T gate" The

outputs of all ;;D gates are high if an% of the inputs are lo!" The symbol is an ;D gate !ith

a small circle on the output" The small circle represents in'ersion"

N-R gate

This is a ;T.& gate !hich is e$ual to an & gate follo!ed by a ;T gate" The

outputs of all ;& gates are lo! if an% of the inputs are high" The symbol is an & gate !ith a

small circle on the output" The small circle represents in'ersion"

E/-R gate

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Table ) is a summary truth table of the input>output combinations for the ;T gate together !ith

all possible input>output combinations for the other gate functions" lso note that a truth table

!ith KnK inputs has )n ro!s" Iou can compare the outputs of different gates"

Table 5: Logic gates representation !sing the Tr!th table

The "Universal" NAND Gate

The Logic N.ND +ate is generally classed as a 7ni'ersal gate because it is one of the

most commonly used logic gate types" ;;D gates can also be used to produce any other type

of logic gate function# and in practice the ;;D gate forms the basis of most practical logic

circuits" By connecting them together in 'arious combinations the three basic gate types of ;D#

& and ;T function can be formed using only ;;DKs# for example"

Vario!s Logic +ates !sing onl% N.ND +ates

s !ell as the three common types abo'e# Ex.r# Ex.;or and standard ;& gates can be

formed using Must indi'idual ;;D gates"

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The "Universal" NOR Gate

Lie the ;;D gate seen in the last section# the ;& gate can also be classed as a

7ni'ersal type gate" ;& gates can be used to produce any other type of logic gate function

Must lie the ;;D gate and by connecting them together in 'arious combinations the three basic

gate types of ;D# & and ;T function can be formed using only ;&Ks# for example"

Vario!s Logic +ates !sing onl% N-R +ates

s !ell as the three common types abo'e# Ex.r# Ex.;or and standard ;& gates can also be

formed using Must indi'idual ;& gates"

)2

Digital Unit- II

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Transcript of Digital Unit- II