UNIT 06. DIGITAL LOGIC AND DESIGN - Text Book Exercise And Question Answers

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Computer Science For Class X
Unit 06: Digital Logic And Design
Question Answers

By Mrs. Ayesha Arif
Vice Principal
(Jauhar Progressive School)

Q.1: What is data representation ?Also define Truth Table?
Ans: DATA REPRESENTATION:
Data representation refers to the form in which data is stored, processed and transmitted. All digital devices such as smartphones, iPads, and computers store data in digital formats that can be handled by electronic circuitry. These circuits work on two logical binary states i.e, Low and High (1 & 0) pulses.

TRUTH TABLE:
The table that shows the tabular representation of all the possible combinations of the values for inputs and their corresponding outputs is called the truth table.

Truth table is a systematic listing of the values for the dependent variable in terms of all the possible values of independent variable. Truth table representing the condition of input and output circuit which involves two or more variables.
Formula: of truth table is 2ⁿ.
Where 'n' is the number of variables used in operation.

Q.2: What are Logic gates, explain their function in detail.
Ans: GATES:
Boolean functions are implemented in digital computer circuits called gates.
A gate is an electronic device that produces a result based on two or more input values. In reality, gates consist of one to six transistors, but digital designers think of them as a single unit. Integrated circuits contain collections of gates suited to a particular purpose.

LOGIC GATES:
Logic gates are electronic circuits in a digital diagram. They perform logical operations like AND, OR, NOT, NAND, NOR etc.

TYPES OF LOGICAL GATES:
Logical gates are divided into two categories:

Basic Logical Gates
Universal Logical gates

BASIC LOGIC GATES:
The logic gate is the basic unit of digital logic circuits. There are mainly three basic gates:

Logic gate	Operation	Symbol Used
AND	AND	Dot(.), or insertion ( )
OR	OR	Plus (+) or union (U)
NOT	NOT	Prime (`) or Bar ( )

I. AND Gate:

An AND gate is a digital circuit that has two or more inputs and a single output.
It operates on logical multiplication rules.
AND operation using variables A and B is represented “A.B”, here (.) dot is a logical multiplication sign.

Boolean Expression of AND gate: Y = A.B

Truth Table of AND Operation Using Two Input Variables

A	B	Y = A.B
0	0	0
0	1	0
1	0	0
1	1	1

Explanation:

Truth Table of AND gate using two input variables A, B and output is Y.
If any input is 0, then output Y becomes 0.
If all inputs are 1 then output Y becomes 1.

Truth Table of AND Operation Using Three Input Variables:

A	B	C	Y = A.B.C
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	0
1	1	0	0
1	1	1	1

Explanation:

Truth Table of AND gate using three input variables A, B,C and output is Y.
If any input is 0, then output Y becomes 0.
If all inputs are 1 then output Y becomes 1.

II. OR Gate:

An OR gate is a digital circuit that has two or more inputs and a single output.
It operates on logical Addition rules.
Logical OR operation using variables A and B is represented as “A+B”, here (+) is a logical Addition sign.

Boolean Expression of OR gate is Y = A + B.

Truth Table Of OR Gate Operation Using Two Input Variables

A	B	Y = A + B
0	0	0
0	1	1
1	0	1
1	1	1

Explanation:

Truth Table of OR gate using two input variables A, B and output is Y.
If any input is 1, then output Y becomes 1.
If all inputs are 0 then output Y becomes 0.

Truth Table Of OR Operation Using Three Input Variables

A	B	C	Y = A+B+C
0	0	0	0
0	0	1	1
0	1	0	1
0	1	1	1
1	0	0	1
1	0	1	1
1	1	0	1
1	1	1	1

Explanation:

Truth Table of OR gate using three input variables A, B,C and Y is output.
If any input is 1 then output Y becomes 1.
If all inputs are 0 then output Y becomes 0.

III. NOT Gate:

A NOT gate is a digital circuit having single input and a single output.
It is also known as an INVERTER.
The output of NOT gate is the logical inversion of input.
It is symbolically represented by complement sign (‘) Right side on top of the input variable or bar (-) sign on top of the variable.

Boolean expression of NOT gate is Y = A’ or Y = A

Truth Table Of Not Gate Operation Using Two Input Variables

A	Y = A
0	1
1	0

Explanation:

Truth table of NOT gate is A as input and Y = A is output.

2. UNIVERSAL GATES:
The logic gate which can implement any Boolean function without the need to use any other type of basic gate is called Universal gate. There are two universal gates.

NAND Gate
NOR Gate

I. NAND (NOT-AND) Gate:

This Gate is a collection of two logic gates the AND gate and the NOT gate, connected in series.

Boolean expression of NAND gate is Y= (A.B)’or Y = AB.

Truth Table Of NAND Operation Using Two Input Variables

A	B	Y = AB
0	0	1
0	1	1
1	0	1
1	1	0

Explanation:
The Truth table of the NAND gate using two input variables A, B and Y is the output.
When all inputs are “1”, the output, Y is “0”.
If any one of the inputs is “0”, then the output Y is “1”.

II. NOR Gate:

It is the collection of OR Gate and NOT Gate. The output of NOR gate is inverter OR.

The Boolean expression of NOR gate is Y = (A+B)’or Y = A+B.

Truth Table Of NOR Operation Using Two Input Variables

A	B	Y = A+B
0	0	1
0	1	0
1	0	0
1	1	0

Explanation:

The Truth table of the NOR gate using two input variables A, B and Y is the output.
If both inputs are “0”, then the output, Y is “1”.
If any one of the inputs is “1”, then the output Y is “0”.

Q.3: Differentiate between universal and basic gates?
Ans: Difference Between Universal And Basic Gates

S.NO.	BASIC LOGIC GATES	UNIVERSAL LOGIC GATES
1.	AND,OR and NOT are the most basic logic gates.	NAND and NOR are called the universal gates.
2.	By using basic logic gate it is possible to implement all possible Boolean expressions.	Collection of universal gates can be used to achieve any basic AND, OR and NOT operation.
3.	Individual logic gate can be connected to form a variety of different combinational logic circuits.	These gates can implement any Boolean expression without using other type of gate.

Q.4: What is Boolean Algebra? Prove the rules of Boolean Algebra.
Ans: BOOLEAN ALGEBRA:
The Boolean algebra was developed by the English mathematician George Boole in 1954. It deals with statements in mathematical logic, and it can be used to describe the logical operations and processing binary information (i.e. puts them in the form of algebraic equations). It is based on true or false input values to produce a true or false output value.

Logical Algebra:
Boolean algebra is used in designing of logic circuits inside the computer. These circuits perform different types of logical operations. Thus, Boolean algebra is also known as logical algebra or switching algebra.

Boolean Expression:
The mathematical expressions of the Boolean algebra are called Boolean expressions. Boolean algebra describes the Boolean expressions used in the logic circuits. The Boolean expressions are simplified by means of basic theorems.

Boolean Algebra is a convenient and systematic way of expressing and analyzing the operations of logic circuits. It is a mathematical system that contains a set of rules to perform logical operations, whose outcome would be either in the form of true or false.
With regard to digital logic, these rules used to describe circuits state that can be either, 1 (true/on) or 0 (false/off). The Boolean algebra was developed by the English mathematician George Boole in 1847's. Boolean Algebra is applied in the design and analysis of digital system such as computer etc.

RULES OF BOOLEAN ALGEBRA (LAWS OF BOOLEAN ALGEBRA EXPRESSION):
The Boolean arithmetic rules are pre-defined rules that help to simplify the logical expression. There are 12 basic rules which are invented to simplify the gates. To reduce the number of logic gates needed to perform a particular logic operation we can apply a set of rules. These rules are commonly known as the Laws of Boolean Algebra Expressions.

The following table shows the Boolean algebra rules for Boolean Expression Simplification:

These rules are used to simplify and manipulate expression, as:
RULE 1: A + 0 = A
Proof:

L.H.S: 0 + 0 = A
R.H.S: 0+1 = A

A	0	OUTPUT
0	0	0
1	0	1

RULE 2: A + 1 = 1
Proof:

L.H.S: 0 + 1 = 1
R.H.S: 1 + 1 = 1

A	0	OUTPUT
0	1	1
1	1	1

RULE 3: A.0 = 0
Proof:

L.H.S: 0. 0=0
R.H.S: 1.0 =0

A	0	OUTPUT
0	0	0
1	0	0

RULE 4: A.1 = A
Proof:

L.H.S: 0.1 = 0
R.H.S: 1.1 = 1

A	1	OUTPUT
0	1	0
1	1	1

RULE 5: A + A = A
Proof:

L.H.S: 0 + 0 = 0
R.H.S: 1 + 1 = 1

A	A	OUTPUT
0	0	0
1	1	1

RULE 6: A.A = A
Proof:

L.H.S: 0. 0 = A
R.H.S: 1. 1 = A

A	A	OUTPUT
0	0	0
1	1	1

RULE 7: A+A=1
Proof:

L.H.S: 0 + 1 = 1
R.H.S: 1 + 0 = 1

A	1	OUTPUT
0	1	1
1	0	1

RULE 8: A.A = 0
Proof:

L.H.S: 0.1 = 0
R.H.S: 1.0 = 0

A	1	OUTPUT
0	1	0
1	0	0

RULE 9: A.A = A
Proof:

RULE 10: A = A.B = A

A	B	A.B	OUTPUT (A.B)+A
0	0	0	0
0	1	0	0
1.	0	0	1
1	1	1	1

Rule 11: A + A B = A + B

A	B	A+B	AB	A + AB
0	0	0	0	0
0	1	1	1	1
1	0	1	0	1
1	1	1	0	1

Rule 12: (A + B) (A + C) = A + B.C

Q.5: Simplify the following Expressions by using Boolean rules.
(i) AB + AB = A
(ii) (A + B) + (A + B) = A
Ans: Simplification of Expression by using Boolean rules:
(i) AB + AB = A
Solution:

AB + AB	Take L.H.S
A(B + B)	Here take common variable A from both terms
A . (1)	Rule 6^th: A + A = 1 So, B + B = 1
A	Rule 4^th: A . (1) = A
A = A Hence L.H.S = R.H.S A = A (Proved)

(ii) (A + B) + (A + B) = A
Solution:

(A + B) + (A + B)	Take L.H.S
AA + AB + BA + BB	ANDing (Multiplication ) of both expressions
A+ AB + BA + 0	Rule 7^th: A.A = A and Rule 8^th: A.A = 0 or B.B
A + A (B + B) + 0	Take common variable A from A + AB + BA expression
A + A.(1) + 0	Rule 6^th: A + A = 1 or B + B = 1
A + A + 0	Rule 4^th: A.(1) = A
A + 0 = A	Rule 5^th:A + A = A
Hence L.H.S= R.H.S A = A ( proved)

:

Hence L.H.S= R.H.S
A = A proved

Q.6 (a) : Draw the Logic circuit diagram that implements the expression Y = ABC (A + D) using gates having no more than three inputs.
Solution:

Explanation:

Above logic circuit consists of AND, OR and, NOR gates.
The expression A NOT B, C gate connected with AND gate.
A, D is a connected with OR gate and converted in NOR gate.
Finally ABC and A + D connected to AND gate.

Q.6 (b): Draw logic circuit of expression X = AB (C + D).
Solution:

Q.6 (c): Draw logic circuit of expression Q = (A.B) + (A + B).
Solution:

Q.7: Derive the Boolean expression from the given circuit and make a truth table of that Boolean expression.

B. RESPOND THE FOLLOWING:

1. Explain all logic gates and their operations.
Ans: See above "Q.2" - Basic Logic Gates.

2. Differentiate between NAND and NOR gates.
Ans: Difference Between NAND And NOR Gates:

S.NO.	NAND	NOR
1.	The NAND gate represents the complement of the AND operation	The NOR gate represents the complement of the OR operation.
2.	Its name is an abbreviation of NOT AND.	Its name is an abbreviation of NOT OR.
3.	The graphic symbol for the NAND gate consists of an AND symbol with a bubble on the output, denoting that a complement operation is performed on the output of the AND gate.	The graphic symbol for the NOR gate consists of an OR symbol with a bubble on the output, denoting that a complement operation is performed on the output of the OR gate.

3. Why do we use Boolean Algebra?
Ans: Boolean algebra is a branch of mathematics. The two-valued Boolean algebra has important application in the design of modern computing systems. We use it for the following purposes.

It can be used to describe the manipulation and processing of binary information.
It provides a concise way to express the operation of a logic circuit formed by a combination of logic gates so that the output can be determined for various combinations of input values.
We use it to analyse digital gates and circuits.
We use the “Laws of Boolean” to both reduce and simplify a complex Boolean expression in an attempt to reduce the number of logic gates required.

4. Explain the function of Inverter.
Ans: The Not Gate is a digital circuit that has a single input and single output. It is also called inverter because it performs a basic logic function called inversion or complementation. The Inverter has the following functions:

The purpose of the inverter is to change one logic level to the opposite level.
In terms of bits, it changes a 1 to 0 and vice-versa.
NOT gate is called the inverter because output state is always opposite to the input state, so when the input is low signal, output is high signal and vice-versa.
In digital logic, an inverter is a logic gate which implements logical negation.

5. Explain the purpose of Truth table.
Ans: A truth table shows the relationship, in tabular form, between the input values and the result of a specific Boolean operator or function on the input variables. It has following purposes:

It displays the logical operations on input signals in a table format.
Every Boolean expression can be viewed as a truth table.
The truth table identifies all possible input combinations and the output for each.
We can prove that two different logical statements are equivalent or not equivalent to one another.
We can test the validity of the structure of logical arguments.

6. Simplify the following Boolean Expression.
Z= AB + A(B + C) + B(B + C)
Solution:

Step 1: Apply the distributive law to the second and third terms in the expression, as follows:
Z= AB + AB + AC + BB + BC
Step 2: Apply rule 7 (BB = B) to the fourth term.
Z= AB + AB + AC + B + BC
Step 3: Apply rule 5 (AB + AB = AB) to the first two terms.
Z= AB + AC + B + BC
Step 4: Apply rule 10 (B + BC = B) to the last two terms.
Z= AB + AC + B
Step 5: Apply rule 10 (AB + B = B) to the first and third terms.
Z= B+AC

At this point the expression is simplified as much as possible.

Solution (Of Q.6):

AB + A(B + C) + B(B + C)	Take L.H.S
AB + (B + C) (B + A)	Take (B + C) common
According to the Rule 12^th: (A + B) (A + C) = A + BC. so, (B + A) (B + C) = B + AC
AB + B + AC	Rule 12^th: (B + A) (B + C) = B + AC
B(A + 1) + AC	Take B common
B (1) + AC	Rule 2^nd: (A + 1) = 1
B + AC	Rule 4^th: B.(1) = B
Hence L.H.S= R.H.S Z = B + AC ( proved)

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