6th Boolean Algebra

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6Boolean Algebra ~

CHAPTER

B o o lean A lg eb ra)Q.1: Complete the following statements.

(i) In Boolean algebra plus sign stands for _ operation.(ii) AND operation is used for logical _(iii) Double complementation has effect.(iv) In Boolean Algebra 1+1+1 is equal to _(v) According to the distributive law {A+(BC)} =

Answers:(i)OR(iv) I

. (ii) multiplication(v) (A+B).(A+C)

(iii) cancellation

Q.2: Tick the following statements either True or False.i) AND operation is also called as logical addition-ii). In a NOT gate, the output is negative of the inputiii) Two series switches can represent AND operation.iv) NOT of(A+B) is equal to NOT of A OR NOT ofBv) A variable that is same within two adjacent squares of a

Karnaugh map is dropped.

True/FalseTrue/FalseTrue/FalseTrue/FalseTrue/False

Answers:(i) False(iv) True

(ii) False(v) False

(iii) True

Q.3: Encircle one choice A, B,C or 0 in each-case.(i) , Boolean Algebra isalso known as

a. Logical algebrab. Code Algebrac. Switching algebrad. Digital Algebra'

(ii) An OR operation ha&A variables. The possible number of combinationsin its truth table are:-a. 4b. 8c. 16d. 32

(iii) The- output will be one if all inputs except one are zero in case of:-


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5 Star Key to Computer Science 9-10.(Federal Board)a. NOT operationb. AND operationc. OR operationd. NOT or OR

(iv) According to the Demogron's law A + B is equal to:-a . A+Bb. A Bc. ABd .. A+B

(v) According to absorption law A . (A+B) is equal to:-a. 1 +Bb. ABc.. Ad. A +1

,~

Answers:(i) C(iv) C

(ii) C(v) C (iii) C -

Q.4: Match the items given in Column Iwith those given in Column II.Column I Column II

i) AND a) 1 ,ii) OR b) A+B riii) A+A c) Union riv) A+AB , d) Av) A + (A B) e) Intersection -;:-

Answers:(i) e(iv) b

(ii) c(v) d (iii) a

.Q.5: What is Boolean algebra?Answer: .

-The Boolean algebra was developed by the English mathematician GeorgeBoole. It deals with statements in mathematical logic, and puts them in the form ofalgebraic equations. The Boolean algebra was further developed by the modernAmerican mathematician Claude Shannon, in order to apply it to computers. Thebasic techniques described by Shannon were adopted almost universally for thedesign and analysis of switching circuits. Because of the analogous relationshipbetween the actions of relays and of modern electronic circuits, the same techniques


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Boolean Algebra ~which were developed for the design of relay circuits are still being used in the design.of mode~n high speed computers. Thus the Boolean algebra founds its applications inmodem computers after almost one hundred years of its discovery.

Boolean algebra provides an economical and straightforward approach to thedesign of relay and other types of switching circuits. J4


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\Star Key to Computer Science 9-10 (Federal Board), .Q.7: Find the values of theBoolean expressions.

Answer:i). 'xv +XY W hen X = I a nd Y = 0

. = = (1 . 0 ) +'( I. I )= 0 +1I = 1(X + Y) . ( X Y) W hen X =I and Y =0 '

(1 +0) . ( I . 0 )= 1.0= 0(X + Y) . (X + Y) W hen X = I and Y = I

(1+0).(0+1)= . 1 . I

1

ii)

iii)J

, , ,Q.S: State and prove the .two basic Demogran's theorem. Find out thecomplement of the following Boolean'expressions.

Answer: '.DEMORGAN'S LAW:-- --(a) AB = AB (b) AB =A+Ba) AB = = AB'.Proof: (a) LHS = A+B\.. = A+B

= AB=R.H.S.L.H.S. = AS

=A. S ,

( by d ua lit y p ri nc ip le )

(b)) (b y d ua li ty p rin ci ple )= A+B'=R.H.S.

Complements of BooJeanExpressionsif XY +XY=(X+Y).(x+Y)

ii) ( X +Y ) . ( X Y )= (X ,. Y )+(X+Y)


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..

iii) (X + Y) . ex +Y). =(X . 'Y) +(' x . Y),

Q.9: What is a Truth Table? Construct a truth table for AND or NOT of .AND operation for the three variables X, Yand Z.Answer:TRUTH TABLE

A truth table is table that shows the result of a Boolean expression for all thepossible combinations of the values given to the variables relater by some operator inthe expression. For example, the truth "tables for operations AND, OR and NOT givenbelow clearly define these operators.

AND OPERATION OR OPERATION NOT OPERATION

~ I X ~ A IA B X=AB0 0 00 1 01 0 01 1 1 .

A B X=A+B0 0 00 1 11 0 11 1 1

Truth tabl f OR for opera IonX Y Z F=X+Y+Z0 O . 0 00 0 - 1 10 . 1 0 10 1 1 .11 " 0 0 11 0 I 11 1 - 0 11 1 1 1

Truth table r AND tlr opera IonX Y I Z F=X. Y.Z0 0 0 00 0 1 00 1 0 00 1 1 0

~ 1 0 0 01 0 , 1 . I 01 1 0 01 1 1 1


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6 Star Key to Computer Science 9-10 (Federal Board)Truth tabl ti NOT f ANDor 0 operation

, NOTX Y Z F=X. 'Y . 'Z/ (j 0 0 I0 o . I' I0 I 0 . I0 I I 0-I 0 0, 0 f 0 , I . . 0-I 1 O Ir I I 0

Q.10: State and prove the following laws: .'(i) Idempotent Law' (ii) Involution Law (iii) Absorption LawAnswer:(I) IDEMPOTENT LAW

(a)A + A = A (b) A . A =AProof: (a) L.H.S. =A+A= (A +A) . 1 [by axiom 1(b)]= (A + A) . (A + A) [by axiom 5 (aj]= A + (A A) [by axiom 4 (b)]= A +0 [by axiom 5 (b)]= A' [by axiom I (a)] .=R.H.S.

(b) L.H.S. =A A=AA= A . A +0 [by axiom I (aj]=AA+A.A [by axiom 5 (b)]= A . (A + ,A) [by axiom 4 (a)]= A . I [by axiom 'S (a)]= A [by axiom I (b)]=R.H.S. .

(II) INVOLUTION LAWIt states that double complementation has cancellation effect. This. can beproved by the method of perfect induction as shown in truth table below. In thismethod a relation is checked by a truth table.


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Boolean Algebra I!!l. .Truth Table

(III) ABSORPTION LAW(a)A+(AB)=A (b)A (A+B)=A

Proof: (a) L.H.S. = A +(A . B)= (A 1) +(A B) [by axiom 1 (b)]= A ) (l +B) [by axiom 4 (a)]= A . 1 [by theorem 2 (a)].= A [by axiom 1 (b)]=R.RS.

(b) L.H.S. = A . (A+B)= (A +0) . (A +B) . [by axiom 1 on= A o f: (0 B) [by axiom 4 (b)]= A +0 [by theorem 2 (b)]= A [by axiom 1 (a)]-=A=R.H.S.

Q.11: Construct a truth table for the followinsj ~oolean expre!sion.(i)XY+ XZ+YZ (ii)(X+ vr. (XY) . (iii)XY +XZ+YZ

Answer:(i) XY + X Z + YZx Y Z 'X XY 'XZ YZ F=XY + 'XZ + YZ0 O 0 1 0 0 0 00 0 1 1 O 1 0 1 I0 1 0 1 0 0 0 - 0 ~0 1 1 1 0 1 1 11 0 0 0 0 0 ,0 01 '0 1 0 0 0 0 01 1 0 0 1 0 0 11 . 1 1 . 0 1 1 1 .. 1


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64 Star Key to Computer Science 9-10 (Federal Board)

(ii) (X + Y) . (XY)X Y X+Y' XY F=( X + Y ) . ( XY )- -0 0 O . 0 00 1 1 0 01 0 1 0 01 1 1 1 1 .-(iii) X Y +XZ +YZ

X Y 'Y Z 'z X'Y XZ Y'Z X'Y + XZ + Y"Z0 0 1 0 1 0 0 0 0-0 0 . 1 1 0 ,0 0 0 00 1 0 0 1 0 0 1 . 10 1 0 1 0 0 0 0 0~1 0 1 0 1 1 0 0 11 0 1 1 0 1 1 0 11 1 0 0 1 0 0 1 11 1 0 1 0 0 1 0 1.

Q.12~Simpli~the fol~~ng Boolea~expressi~ _(i) A C + A.B + ABC +BC (ii) X Y Z +X Y Z + X Y Z +, X Y Z- - - -(iii)(A + B + C) . (A + B + C) . (A + B + C) . (A + B + C)

. Answer:(l) A C + A.B + ABC + BC .- - -= A C + A B+C (A B + B)- - -= AC+ AB+C {(B+A). (B+B)}

= A C + A-B + C (B + A) : 1- -= A C + A B + C (B + A)- -= A C + A B + BC + AC.- -=C(A +B+A)+ AB- -=C(A +A+B)+ AB.=C(l +B)+ AB=C.l+AB=C+'AB


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66 Star Key to Computer Science 9-10 (Federal Board)" , I=AB+AC

(iii) ABC + ABC +ABC

I I I=B\C+ABC

(iv) A B + A C + B C

I I I\

IAB+AC+B'

"

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6th Boolean Algebra