**TOPIC- 1**

**Basics of Boolean Algebra**

**Very Short Answer Type Questions (1 Mark each)**

**Question 1:**

Express the OR operator in terms of AND and NOT operator.

**Answer:**

**Question 2:**

**Answer: **(a) x+x’=1-Complementary law

(b) xy+x=x & absorption law

**Question 3:**

Which gates are known as universal gates? Why?

**Answer: **Universal gates are the ones which can be used for implementing any gate like AND, OR and NOT or any combination of these basic gates. NAND and NOR gates are universal gates.

**Question 4:**

State and verify Associative law using Truth Table.

**Answer: **Associative Law: This law states that:

(A+B)+C=A+(B+C)

(A.B).C=A.(B.C)

**Proof:**

From the above tables,

(A+B)+C=A+(B+C)

Similarly, we can prove,

A.(B.C)=(A.B).C

**Question 5:**

Draw the equivalent logic circuit for the following Boolean experession :

**Answer:**

**Short Answer Type Questions**

**Question 1:**

State and define principle of duality. Why is it so important in Boolean Algebra?

**Answer: **

**Principle of duality:**Duality principle states that from every boolean relation another boolean relation can be derived by:

- Changing each OR sign (+) to an AND sign (•)
- Changing each AND sign (.) to an OR sign (+)

ex : Dual of A + A’B = A . (A^{1} + B)

**Importance in Boolean Algebra:** The principle of duality is an important concept in Boolean algebra, particularly in proving various theorems. The principle of duality is used extensively in proving Boolean algebra theorem. Once we prove that an expression is valid, by the principle of duality, its dual is also valid. Hence, our effort in proving various theorems is reduced to half.

**Question 2:**

Write the equilent boolean expression for the following logic circuit.

**Answer: **((X’.Y’)’+(X’.Y’)’)’

**Question 3: **Write the equilent boolean expression for the following logic circuit.

**Answer:**

**Question 4:**

Write the equivalent boolean expression for the following logic circuit:

**Answer:**

**Question 5:**

Name the law shown below & verify it using a truth table.

**Answer:**

This law is called “Absorption Law” also reffered as redundance law.

**Question 6:**

Obtain the Boolean Expression for the logic circuit shown below.

**Answer:**

**Question 7: **Verify the following using truth table:

(i) X+0=X

(ii) X+x’=1

**Answer:**

**Question 8: **Obtain the Boolean Expression for the logic circuit shown below.

**Answer:**

**Question 9:**

State Demorgan’s law. Verify one of them using truth table

**Answer:**

**Question 10:**

Draw a logic circuit for the Boolean expression

**Answer:**

**Question 11:**

Obtain the boolean expression for the logic circuit shown below:

**Answer:**

**Question 12:**

Verify the following using Boolean Lawas:

X+Z=X+X’Z+Y.Z

**Answer:**

**Question 13: **Verify the following using Boolean Lawas:

A+C=A+A’.C+B.C

**Answer:**

**Question 14:**

Obtain the Boolean Expression for the logic circuit shown below:

**Answer:**

**Question 15: **Verify the following using truth table:

(i) X.X’=0

(ii) X+1=1

**Answer:**

**Question 16:**

Write the equivalent boolean expression for the following logic circuit:

**Answer:**

**Question 17:**

**Answer:**

Distribution law: This law states that

(i) x(y+z)=x.y+x.z

(ii) x+yz=(x+y)(x+z)

Noe let us prove using truth table.

**Question 18: **Name the law shown below & verify it using a truth table.

A+B.C=(A+B).(A+C).

**Answer:**

This law is called “Distribute Law”.

Prove ny Truth table

**Question 19:**

Write the POS form of a Boolean Function F, which is represented in a truth table as follows:

**Answer:**

(P+Q+R).(P’+Q+R).(P’+Q’+R)

**Question 20:**

Write the Boolean Expression for the result of the Logic Circuit as shown below:

**Answer: **The boolean expression is :

I=[(A+b).(A’+C’)].(B+C)

**Question 21:**

Derive a Cannonical POS expression for a Boolean function G, represented by the following truth table:

**Answer:**

**Question 22:**

Correct the following boolen statements:

**Answer:**

**Question 23:**

Derive a Cannonical SOP expression for a Boolean function G, represented by the following truth table:

**Answer:**

**Question 24:**

Derive a Canonical POS expression for a Boolen funtion F, represented by the following truth table:

**Answer:**

**Question 25:**

Write the Boolen Expression for the result of the Logic Circuit shown below:

**Answer:**

C=P.Q’+P.R+Q.R’

**Question 26: **Verify the following using Boolen Laws

P’+Q’.R=P’.Q’.R’+P’.Q.R’+P’.Q.R+P’.Q’.R+P.Q’.R

**Answer:**

**Question 27: **Verify the following using Boolen Laws x’+y’z = x’y’z’+x’y’z’+x’yz+x’y’z+xy’z

**Answer:**

**Question 28: **State and prove De-Morgan’s laws in boolean algebra.

DeMorgan’s theorem states that:

(i) (X+Y)’=X’.Y’

(ii) (X.Y)’=X’+Y’

**Answer:**

Now ,let us proove using truth table

**Question 29:**

Write the equivalent boolean expression for the following logic circuit:

**Answer:**

**Question 30: **Verify the following using truth table:

**Answer:**

**Question 31:**

Write the equivalent boolean expression for the following logic circuit:

**Answer: **F=(X+Y’).(X+Z’)

**Question 32:**

Verify the following using truth table:

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

**Answer:**

**Question 33:**

Write the equivalent boolean expression for the following logic circuit:

**Answer: **PQ’+PR’

=P(Q’+R’)

**Question 34: **State abd cerify De-Morgan’s laws algebraically

**Answer:**

**Question 35:**

Write the equivalent boolean expression for the following logic circuit:

**Answer: **F(P,Q)=(P’+Q).(P+Q’)

**Question 36: **State and algebraically verify Absorption law

**Answer:**

**Question 37: **Write the equivalent boolean expression for the following logic circuit:

**Answer:**

F(U,V)=(U’+V).(U+V’)

**Question 38: **Represent the expression X’.Y+Y’.Z+Z’X using NAND gates

**Answer:**

**Question 39:**

Write the equivalent expression for the following logical circuit:

**Answer: **F=A’.B’+B’C’

**Question 40: **Verify the following algebraically.

(A’+B’).(A+B)=A’.B+A.B’

**Answer:**

**Question 41:**

Write the equivalent boolean expression for the following logic circuit:

**Answer: **Y=(P+Q’).(Q+R’)

**Question 42: **Verify the following algebrically:

X’.Y+X.Y’=(X’+Y’).(X=Y)

**Answer:**

**Question 43: **Write the equivalent boolean expression for the following logic circuit:

**Answer:**

(U’+V).(V’+W)

**Question 44:**

**Answer:**

**TOPIC-2**

** Karnaugh Map Minimization and Applications of Boolean Algebra**

** Very Short Answer Type Questions**

**Question 1: **Convert the following Boolean expression into its equivalent Canonical Sum of Products form (SOP):

(U + V + W)(U+V + W’)(U’+V + W) (U’ + V’ + W’ )

**Answer:**

**Question 2:**

Write Product of Sum expression of the function F (a, b, c, d) from the given truth table.

**Answer:**

**Question 3:**

Write the Sum of Product form of the function F(P, Q, R) for the following truth table represen¬tation of F :

**Answer:**

**Question 4:**

Write the Product of Sum form of the function F(X, Y, Z) for the following truth table representation of F :

**Answer:**

**Question 5:**

Write the Product of Sum form of the function G(U, V, W) for the following truth table representation of G :

**Answer:**

**Question 6:**

Write the Product of Sum form of function G(U, V, W) for the following truth table representation of G :

**Answer:**

**Question 7:**

Write the Sum of Product form of the function F(A, B, C) for the following truth table reprsen- tation of F.

**Answer:**

**Question 8:**

Write the SOP form of a boolean function F, which is represented in a truth table as follows:

**Answer: **F(X,Y,Z)=X’.Y’Z’+X’.Y’.Z’+X.Y’.Z’+X.Y.Z

**Question 9:**

Write the POS form of boolean function G, which is represented in a truth table as follows :

**Answer: **G(A,B,C)=(A+B+C).(A+B’+C’).(A’+B+C).(A’+B+C’)

**Question 10: **Write the SOP form of a boolean function F, which is represented in a truth table as follows :

**Answer:**

SOP form is given by

X’Y’Z’ + X’Y’Z + X’YZ + XY’Z’ + XYZ

**Question 11:**

Write the POS form of a boolean function F, which is represented in a truth table as follows :

**Answer: **(A + B + C). (A+B’ + C’).(A’ + B + C’). (A’ + B’ + C)

**Question 12:**

Write the SOP form of a boolean function F, which is represented in a truth table as follows :

**Answer: **F=(0,3,6,7)

So, SOP form of F is

F = (U’.V’W’) + (U’.VW) + (U.VW’) + (U.VW)

**Question 13:**

Write the POS form of a boolean function F, which is represented in a truth table as follows :

**Answer: **F (U, V, W) = (U + V + W’).(U+V’+W’). (U’+V+W’)

**Question 14:**

Write the SOP form of a Boolean function F, which is represented in a truth table as follows :

**Answer: **G (P, Q, R) = P’QR’ + PQ’R’ + PQR’ + PQR

**Question 15: **Write the POS form of a boolean function H, which is represented in a truth table as follows :

**Answer:**

H (x, y, z) = (x + y + z’). (x’ + y + z’) . (x’ + y’ + z)

**Question 16: **Write the SOP form of a boolean function G, which is represented in a truth table as follows :

**Answer: **G (R Q, R) = P’QR’ + P’QR + PQ’R’ + PQR’ + PQR

**Question 17: **Write the POS form of a boolean function G, which is represented in a truth table as follows :

**Answer:**

G(U. V W) = (U + V + W). (U’ + V + W’). (U’ + V’ + W)

**Question 18: **Write the POS form of a boolean function F, which is represented in a truth table as follows :

**Answer: **F(U, V, W) = (U + V + W’) . (U + V’ + W’) . (U’ + V + W’)

**Question 19: **Reduse the following Boolean Expression to its simplest form using K-Map :

**Answer:**

**Question 20: **Write the SOP form of a Boolean function G, which is represented in a truth table as follows :

**Answer:**

G(P, Q, R) = P’.Q.R’ + PQ’.R’ + EQ.R’ + EQ.R

**Question 21: **Write the POS equivalent of ABC + AB’C’ + ABC’+ A’B’C

**Answer:**

**Short Answer Type questions-II**

**Question 1: **Obtain the minimal SOP form for the following Boolean expression using K-Map

**Answer:**

**Question 2:**

Reduce the following Boolean expression using K-Map:

F(A,B,C,D) = π(0,1,2,4,5,6,8,10)

**Answer:**

**Question 3:**

Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 4:**

Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 5:**

Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 6:**

Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 7:**

Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 8:**

Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 9:**

Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 10: **Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 11: **Reduce the following Boolean Expression using K-Map:

F(A,B,C,D)=S(0,1,3,5,6,7,11,13,14,15)

**Answer:**

**Question 12:**

Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 13: **Simplify the following boolean expression using K-Map:

**Answer:**

**Question 14:**

Reduce the following boolean expression using K-Map:

**Answer:**

**Question 15:**

**Answer:**

**Question 16: **Reduce the following Boolean Expression using K-Map:

**Answer:**

**Question 17: **Reduce the follwing Boolean Expression using K-Map:

**Answer:**