BOOLEAN LOGIC || Computer Science || Class 11

 

BOOLEAN LOGIC

Very Short answer Type Questions Q.1 Prove X. (X+Y) = X using truth table.

Ans: Proof:

X	Y	X+Y	X.(X+Y)
0	0	0	0
0	1	1	1
1	0	1	1
1	1	1	1

Q.2     Give duals for the following – 

 (a) X+X‟Y  (b) XY+XY‟+X‟Y  (c) AB+A‟B  (d) ABC+AB‟C+A‟BC‟ Ans: (a) X.X‘ + Y    (b) (X+Y).(X+Y‘).(X‘+Y)

(c) (A+B).(A‘+B)   (d) (A+B+C).(A+B‘+C).(A‘+B+C‘)  Q.3     What is tautology?

Ans: If the result of any logical statement or expression is always TRUE or 1 for all input

combinations, it is called tautology.  

Q.4     What is Fallacy?      

Ans: If the result of any logical statement or expression is always FALSE or 0 for all input

combinations, it is called fallacy.  

Q.5            What is a truth table? What is its significance?

Ans: A truth table is a table which represents all the possible values of logical variables/statements

along with all the possible results of the given combinations of values.

Short Answer Type Questions  Q.1 Prove X.(X+Y)=X by algebraic method.

Ans: L.H.S.              X. (X+Y) = X.X + X.Y

                                                = X+X.Y                 since X.X = X

                                                = X.(1+Y)

                                                = X (R.H.S.)             since 1+Y = 1

Q.2          Draw logic circuit diagram for the following expression – 

           (a)    Y=AB+B‟C+C‟A‟                (b) R=XYZ‟ + Y.(X+Z‟)

Ans: 

 

 

 

 

 

Q.3   Obtain the Boolean Expression for the logic circuit shown bellow – 

                                         Ans: (X.Y‘)‘+(Z‘+W)

 

 

 

Q.4    Find the complement of the following Boolean function : F1=AB‟ + C‟D‟

Ans: Complement of F1 will be (A‘+B).(C+D) Q.5 State and verify Involution law.

Ans: Involution law says that (X‘)‘ = X.  

X	X‘	(X‘)‘
0	1	0
1	0	1

Q.6   Prove algebraically X.Y + X‟.Z + Y.Z = X.Y + X‟.Z

Ans: L.H.S.      X.Y + X‘.Z + Y.Z = X.Y + X‘.Z + 1.Y.Z

                                               = X.Y + X‘Z + (X+X‘).Y.Z

                                               = X.Y + X‘Z + X.Y.Z + X‘.Y.Z

                                             = X.Y + X.Y.Z + X‘Z + X‘.Y.Z

                                               = X.Y.(1+Z) + X‘.Z.(1+Y)

                                                  =X.Y + X‘.Z              Since 1+Z= 1 and 1+Y = 1

Q.7          State and verify Absorption law in Boolean Algebra.

Ans: (a) X+XY = X                                                                                (b) X(X+Y) = X

          LHS = X+XY = X(1+Y) = X                                               LHS = X(X+Y) = X.X + X.Y 

    = X + XY = X(1+Y) = X Q.8  State DeMorgan‟s law of Boolean Algebra and verify them using truth table.  Ans: (a) (X + Y)‘ = X‘.Y‘     (b) (X.Y)‘ = X‘ + Y‘

Q.9       What do you understand by “logical function”? Give examples for logical functions.

Ans: Boolean algebra is the algebra of logic that deals with the study of binary variables and logical operations. It was founded by the mathematician George Boole.Boolean variables are the variables which have only two states i.e. true/ false or right/ wrong or on/off or 0/1. Boolean function or more commonly known as a logic function is an expression expressed algebraically.

There are 3 logical operators: AND, OR and NOT.

Q.10 What are basic postulates of Boolean Algebra? Ans: Postulates

(1)	A + 0 = A	A · 1 = A	identity
(2)	A + NOT[A] = 1	A · NOT[A] = 0	complement
(3)	A + B = B + A	A · B = B · A	commutative law
(4)	A + (B + C) = (A + B) + C	A · (B · C) = (A · B) · C	associative law
(5)	A + (B · C) = (A + B) · (A + C)	A · (B + C) = (A · B) + (A · C)	distributive law

Q.11 What does duality principal state? What is its usage in Boolean algebra?

Ans: The duality principle ensures that "if we exchange every symbol by its dual in a formula, we get

the dual result".

•        Everywhere we see 1, change to 0.

•        Everywhere we see 0, change to 1.

•        Similarly, (+) to (.) , and (.)to (+).

For example if A+B=1 then its dual will be A.B=0

Q.12    Prove the complementarity law of Boolean algebra using truth table?

Ans: (a) X+X‘=1                                                                                                                       (b) X.X‘=0

 

Q.13    Why are NAND and NOR Gates more popular?

Ans: The NOR Gate has two or more input signals but only one output signal. If all the inputs are 0 (i.e. low) then output signal is 1(high).  

            

 

The NAND Gate has two or more input signals but only one output signal. If all the inputs are 1 (i.e. High) then output signal is 0(low).

 

           NAND and NOR Gates are known as universal gates because fundamental gates can be made using them. And By using NAND and NOR Gates the cost and size of circuit gets reduced.

 

 

 

Q.14    Prove algebraically that (X+Y).(X+Z)=X+Y.Z

Ans: L.H.S. (X+Y).(X+Z) = X.X + X.Z + X.Y + Y.Z

                                        =X + X.Z + X.Y + Y.Z                       Since X.X=X

                                        =X + X.Y + Y.Z                                         Since X+XZ=X

                                        = X + Y.Z    R.H.S.                        Since X+X.Y=X

Q.15    Draw the AND OR Circuit for : y=AB‟C‟D‟ + ABC‟D‟  + ABCD Ans: 

 

 

 

 

 

Q.16    Make AND Gate using NAND Gate.

Ans: 

 

Q.17    Make OR Gate using NAND Gate.

Ans: 

 

 

Q.18    Make NOT Gate using NAND Gate.

Ans: 

 

 

Q.19    Make AND Gate using NOR Gate.

Ans: 

 

 

 

 

 

Q.17    Make OR Gate using NOR Gate.

Ans: 

 

 

Q.18    Make NOT Gate using NOR Gate.

Ans:   

Q.19    Which gate is the following circuit equivalent to ?

            

             (a) AND        (b) OR          (c) NAND                                (d) NOR       (e) None of these

Ans: (b) OR