csc3011 – discrete structures

7/26/2019 CSC3011 Discrete Structures

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CSC3011 Discrete

StructuresBoolean Algebra


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OutlineO Introduction

O Boolean Functions and expressions

O Logic Netor!s and "arnaug# $apsO Structure o% Boolean Algebra

O Application& Se'en seg$ent displa(


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IntroductionO Boolean algebra as %or$ulated b(

)eorge Boole

O It is an essential tool %or designingnetor!s to sol'e input and outputproble$s

O It is t#e basic $at#e$atics re*uired

%or t#e stud( o% t#e logic design o%digital s(ste$s

O Digital co$puters use to+statede'ices t#at produce to distinctoutput signals


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IntroductionO De,nition

O A Boolean Algebra -B./...0.12 consists o%operations / and on B. a unar( operation on

B. and speci,ed ele$ents 0 and 1 in B suc#t#at t#e %olloing axio$s %or Boolean algebra#old %or all x. (. and 4 in B

O Boolean Axio$sO x/( 5 (/x x( 5 (x

O -x/(24 5 -x4/(42 -x(24 5 x-(42

O x/-(42 5 -x/(2-x/42 x-(/42 5 x( / x4

O x/0 5 x x0 5 0

O x/x 5 1 xx 5 0


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Boolean Las %or Set 6#eor(O 78 5 87 7 8 5 8 7

O -7829 5 7-892 -782 9 5 7 -8 92

O 7-892 5 -7 82-7 92 7 -8 92 5 -7 82 -7 92

O 75 7 7: 5 7

O 7 7 5 : 77 5 O6#e s(ste$ -;-:2. . .. .:2 is an exa$ple o% a

Boolean Algebra


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Boolean AlgebraO6#eore$

O x( / x( 5 x

O -x/(2-x/(25x

O ;roo%

O x( / x( 5 x-(/(2 5 x1 5 x

O6#eore$& :ni*ue co$ple$ent la

O For eac# ele$ent x B. x is t#e uni*ue

ele$ent in B satis%(ing x/x51 and xx50O ;roo%

O S#o t#at x/(51 and x(50 i$plies (5x


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Boolean AlgebraO ;roo% ctd

O Let x / ( 5 1 and x(50 t#en

( 5 (1

5(-x / x2

5(x / (x

5x( / x(

50 / x(

5xx / x(

5x-x / (2

5x1

5x


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Boolean AlgebraO6#eore$& In'olution Las

O -x2 5 x

O 150 and 051O ;roo%

O


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Boolean AlgebraO ;roo% ctd

x( /-x / (2 5 -x( / x2 / (

5 -x / x(2 / ( 5-x / x2-x / (2 / (

51-x / (2 / (

5x / -( / (2

5x / 1 5 1 51

So -x / (2 5 -x(2


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Boolean FunctionsO De,nition& Boolean Function

O Let =0. 1>ndenote t#e set o% all n+tuples o% 0sand 1s A Boolean Function is a %unction o% n

'ariables %&=0.1>n=0.1> #ere n1O ie

O %-x1.x..xn2 5 0 or

O %-x1.x..xn2 5 1 %or an( Boolean %unction %

O =0.1> de,ned

b(O %-02 5 1 and %-12 5 0 % can be represented as


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Boolean FunctionO Boolean %unctions are usuall(

represented b( tables called trut#

tablesO A trut# table speci,es t#e 'alues o% a

Boolean %unction %or e'er( possibleco$bination o% 'alues in =0.1> o% t#e'ariables

O 3=0.1> de,ned b( g-1.0.125 g-1.1.12 5 1 and g is 0 ot#erise


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Boolean Functions

I Find Card-B2 and Card-B32II Find t#e nu$ber o% Boolean %unctions %ro$ B3

to B


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Boolean


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Boolean


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Boolean


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Boolean


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Boolean nto =0. 1> b( %or$ula%-x1..xn2 5 0 %or all -x1..xn2 =0.1>n

O 1 de,nes t#e constant Boolean %unction%ro$ =0. 1>nto =0. 1> b( %or$ula%-x1..xn2 5 1 %or all -x1..xn2 =0.1>n

O Boolean expression in $ 'ariables cande,ne a Boolean %unction %& =0. 1>n=0.1> %or n$


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Boolean


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Boolean


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Boolean


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Boolean


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Booleans


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Boolean


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Boolean


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Logic netor!sO De,nition

O A logic gate is a binar( electronicde'ice it# one or $ore input signalsto produce one output signal

O6#ere are t#ree basic logic gatescalled t#e OH gate. t#e AND gate.and t#e NO6 gate -in'erter2


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Logic Netor!sO6#e AND

gate

O6#e OH gate

O6#e NO6

gate


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Logic Netor!sO6#e OH gate represents t#e Boolean

addition in =0.1>O

6#e output o% t#e OH gate is 1 i% oneor bot# input 'alues are 1 it is 0ot#erise

O6#e AND gate represents Boolean$ultiplication in =0.1>O6#e output o% t#e AND gate is 1 i% and

onl( #en bot# input 'alues are 1 itis 0 ot#erise


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Logic Netor!sO A logic netor! -also called

sitc#ing. gating. co$binatorialnetor! or logic circuit etc2 is so$eco$bination o% t#e OH. AND and NO6gates

O E#en e use single 'ariables %orinputs. t#e output o% t#e logic

netor! is represented b( a Booleanexpression o% t#e input 'ariables

O


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Logic Netor!sO6#e logic netor!s in t#e exa$ple

de,ne t#e sa$e Boolean %unction %&=0.1>=0.1>

O E#en to output Booleanexpressions are e*ui'alent. e sa(t#at t#e to corresponding logicnetor!s are e*ui'alent. and 'ice

'ersa

O )i'en a logic netor!. it is possibleto deri'e t#e Boolean expression itde,nes


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Logic Netor!sO


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Logic Netor!sO Since e'er( Boolean expression

de,nes a Boolean %unction. e'er(Logic netor! also de,nes a Boolean%unction

O Alt#oug# $an( e*ui'alent Booleanexpressions -#ence. logic netor!s2can correspond to t#e sa$e Boolean

%unction g. t#eres onl( one standardsu$ o% products %or$ t#atcorresponds to g


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Logic Netor!sO


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Logic Netor!sO Soln

O De,ne 'aribles x and ( de,ned

Ox %or $aster not ansering t#etelep#one a%ter rings

O ( %or t#e sitc# being on

O Ee ant t#e robot to anser i% andonl( i% ( 5 1 or -(50 and x512

O %-x.(2 5 ( / (x


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Logic Netor!sO6#e #al%+adder netor! #as to

output. s and c as described in t#etableO s is t#e binar( su$ o% t#e to inputs

O c is t#e carr( resulting #en to inputdigits are added

O s 5 x(/x(

O c 5 x(


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Logic Netor!sO6#e designing o% logic netor!s can

be 'ieed as a t#ree+step process&O

Speci%( t#e input and output tas! it#a Boolean %unction %

O Find t#e Boolean expression


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Logic Netor!sO De,nition

O A $ini$al su$ o% products F %or a Booleanexpression < #as t#e %olloing properties

O F 5

csc3011 – discrete structures

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