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CHAPTER2

BOOLEANALGEBRALOGICANDLOGICGATES

Basic Theorems and Properties of Booean A!e"ra#

Duality- We simply interchange OR and AND operators and replace 1s by 0s and 0s by 1s.

Also, the duality principle states that if to !oolean e"pressions are e#ual, then theirduals are also e#ual. $"%X& 0 'X(X)1 'X

!asic *heorem- !oolean Addition%

o " & 0 ' 0 & " ' " +0 is called the additie identity-

o " & 1 ' 1 +X'AB& CAB& C& 1 ' 1-

- !oolean ultiplicationo " $0 ' 0

o " $1 ' 1$" ' " +1 is called the multiplicatie identity-- /dempotent as

o " & " ' "

o " " ' "

- /nolution ao +"- ' "

- as of 2omplemento " & " ' 1

o " $ " ' 0

- 2ommutatie aso " & y ' y & "

o " y ' y "

- Associatie aso +" & y- & 3 ' " & +y & 3- ' " & y & 3

o +" $ y- $ 3 ' " +y $ 3- ' " $ y $ 3

- Distributie ao " $+y & 3- ' +" $ y- & +" $ 3-

o " & +y $3- ' +" & y- $+" & 3-

- Deorgans aso +" & y- ' " y

o +" $ y- ' " & y

-2onsensus ao " $y & " $ 3 & y $ 3 ' " $ y & " $ 3

*he term y3 is referred to as the 4consensus term4. A consensus term is a redundantterm and it can be eliminated. 5ien a pair of terms for hich a ariable appears in oneterm and the complement of that ariable in another term, the consensus term is formedby multiplying the to original terms together, leaing out the selected ariables and itscomplement.

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- A field is a set of elements, together ith to binary operators.- *he set of real numbers together ith the binary operators & and $ form the field of

real numbers.- *he field of real numbers is the basis for arithmetic and ordinary algebra. *he

operators and postulates hae the folloing meanings%o

*he binary operator & defines addition.o *he additie identity is 0.

o *he additie inerse defines subtraction.

o *he binary operator $ defines multiplication.

o *he multiplicatie identity is 1.

o *he multiplicatie inerse of a' 1/adefines subtraction, i.e. a $1/a = 1.

o *he only distributie la applicable is that of $ oer &%

a (b + c) = (a $b)& (a $c)

- A to6alued !oolean Algebra is defined on a set of 7 elements, B' 80, 19, ithrules for the 7 binary operators & and $

x y x $ y " y " & y " "

0 0 0 0 0 0 0 10 1 0 0 1 1 1 01 0 0 1 0 11 1 1 1 1 1

" y 3 y & 3 " $ +y & 3- " $ y " $3+" $ y- & +" $ 3-

0 0 0 0 0 0 00

0 0 1 1 0 0 00

0 1 0 1 0 0 00

0 1 1 1 0 0 0 01 0 0 0 0 0 0 01 0 1 1 1 0 1 1

1 1 0 1 1 1 0 11 1 1 1 1 1 1 1

*ruth table to erify the Distributie a.

A) X Y X + Y X + Y B) X Y X Y X Y

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0 0 0 1 0 0 1 1 10 1 1 0 0 1 1 0 01 0 1 0 1 0 0 1 01 1 1 0 1 1 0 0 0

*ruth table to erify Deorgans a.+" & y- ' " y +" ) y- ' " & y

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Lo!ic Gates

Operator Precedence- *he operator precedence for ealuating !oolean e"pressions is +1- +-, +7- NO*, +:-

AND, and +;- OR. $"% +x& y-.

%enn Dia!ram

- 1, 7 > 7 ?age ;;.

Booean &'nctions

- A !oolean function is an e"pression formed ith binary ariables, the 7 to binaryoperators ORand AND, and unary operator NOT( parentheses, and an e#ual sign)

=or a gien alue of the ariables, the function can be either *or +.- $"% F1= xyz. *he function is e#ual to 1 ifx' 1 and y' 1 z' 1@ otherise F1 ' 0.

- *o represent a function in a truth table, e a list of 7 ncombinations of +s and *s ofnbinary ariables and a column to sho the combination for hich the function ' *,+.

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*able 767Tr'th ta"es for F1= xyz( F2= x + yz( F3= xyz + xyz + xy( and F4= xy + xz

x y z F1 F2 F3 F4

0 0 0 0 0 0 00 0 1 0 1 1 10 1 0 0 0 0 00 1 1 0 0 1 11 0 0 0 1 1 11 0 1 0 1 1 11 1 0 1 1 0 01 1 1 0 1 0 0

- *o functions of n binary alues are said to be e#ual if they hae the same

alue for all possible 7ncombination of the nariables. =:and =;- A !oolean function may be transformed from an algebraic e"pression into a logic

diagram composed of AND, OR, and NO* gates. =ig. 7 > ;.

-

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(x + y) (x + z) (y + z) = (x $ x + x $ z + x $ z + y $ z) (y + z)

0= (x $ y$ z + x $ z $ z + x $ y $ y + x $ y $ z + y $ z $ z + y $ z $ z)

z y z z

= (x $ y$ z + x $ z + x $ y + x $ y $ z + y $ z)= x $ z (1 + y) + x $ y (1 + z) + y $ z= 1 1= x $ z + x $ y + y $ z= x $ x + x $ z + x% $ y + y $ z= 0= (x + y) (x + z)

- =unctions 1 and 7 are the duals of each other and use dual e"pressions incorresponding steps.

Complement of a function

- *he complement of a function Fis Fand is obtained from an interchange of 0s for1s and 1s for 0s in the alues of F.

- *he complement of a function may be deried algebraically through Deorgans

- (A + B + C + & + ' + F) = A B C & ' F

- (A B C & ' F) = A + B + C + & + ' + F

- *he generali3ed form of Deorgans theorem states that the complement of afunction is obtained by interchanging AND and OR operates and complementing

each literal.- F1= x y z + x y z

F1= (x + y + z) (x + y + z)F2= x (y z + y z)F2= x + (y + z) (y + z)

Representations of A &'nction- A function can be specified or represented in any of the folloing ays%

o A truth table

o A circuit

o A !oolean e"pression

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Canonica And Standard &orms,interms and ,a-terms- A binary ariable may appear in normal form +x- or in complement form +x-- $"% *o binary ariablesx andy combined ith an AND gate. *here are four

combinations%xy xy xy xy.-

$ach of these four AND terms represents one of the distinct areas in the Benndiagram and is called a #inter#or a *tandard r"dct.-

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=unctions of *hree Bariables

x y zFncti"n !1 Fncti"n !2

0 0 0 0 00 0 1 1 00 1 0 0 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

- /f e taCe the complement of f1, e obtain the function f1%!1' +x + y + z) (x + y + z)+x + y + z)+x + y + z) (x + y + z) = -0$ -2$ -3$ -$ -

-

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?roduct of a"terms*ruth *able for F = xy + xz

x y z F

0 0 0 00 0 1 10 1 0 00 1 1 11 0 0 01 0 1 01 1 0 11 1 1 1

56e *# "! #inter#* i*F (x y z) ' F +1, :, G, H-

*he sum of ma"terms isF (x y z) ' I+0, 7, ;, -

$"% $"press the !oolean function F = xy + xz in a product of ma"terms. 2onert the function into ORterms using the distributie la%F = xy + xy = (xy + x) (xy + z) = (x + x) (y + x) (x + z) (y + z) = (x + y) (x + z) (y + z)

*he function has : ariables% ", y, ( 3. $ach OR term is missing one ariable%x + y = x + y + zz = (x + y +z) (x + y + z)x + z = x + z + yy = (x + y + z) (x + y + z)y + z = y + z + xx = (x + y + z) (x + y + z)

2ombine the terms and remoe the terms that appear more than once%F = (x + y + z) (x + y + z) (x + y +z) (x + y + z) = -0+ -2+ -4+ -*he function is e"pressed as follos% F (x y z) = I(0 2 4 )

2onersion beteen 2anonical =orms- $"% *he complement of

F (A B C) = 7 (1 4 ) i*F (A B C) = 7 (0 2 3) = # 0+ #2+ #3

- /f e apply the complement of Fusing Deorgans theorem, e obtain Fin a different form%

= ' +#0+ #2+ #3) = #0$ #2$#3 ' -0+ -2+ -3 ' I(0 2 3)

- *he last conersion follos from the definition of minterms and ma"terms. /t is clear that the folloing

relation holds true% #8 ' -8the ma"term ith subscript8 is a complement of the minterm ith thesame subscript8 and ice ersa.

- *o conert from one canonical form to another, interchange the symbol F and I and list thosenumbers missing from the original form.

Inte!rated Circ'its .IC/

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- /ts a small silicon semiconductor, called a chip, containing the electroniccomponents for the digital gates. *he gates are interconnected inside the chip toform the re#uired circuit.

eels of /ntegration-

9#all:*cale ;nterati"n+