Digital Electronics (18EC32) Notes
UNIT-I : Fundamentals of Digital Electronics
1. 1. Introduction
Digital electronics is a type of electronics that deals with the digital systems which
processes the data/information in the form of binary(0s and 1s) numbers, whereas analog
electronics deals with the analog systems which processes the data/information in the form of
continuous signals.
Continuous signals
A Continuous signal is function f(t), whose value is defined for all time 't'.
in other words
Continuous signal a varying quantity with respect to independent variable time.
Example: Figure 1.1(a) shows the continuous signal.
Figure 1.1(a): Continuous signals.
Digital signals
A digital signal is a quantized discrete time signals.
Example: Figure 1.1(b) shows the discrete and digital signals.
Figure 1.1(b1): Discrete signal.
Figure 1.1(b2): Digital signal.
Prepared by Mohankumar V., Assistant Professor, Dept. of ECE, Dr.AIT, B-56. Page 1
,Digital Electronics (18EC32) Notes
1.2. Boolean Algebra
Boolean algebra is a branch of Algebra (Mathematics) that deals with operations on logical
values with Boolean variables, Boolean variables are represented as binary numbers which takes
logic 1 and logic 0 values. Hence, the Boolean algebra is also called two-valued logic, Binary
Algebra or Logical Algebra. The Boolean algebra was introduced by great mathematician George
Boole in 1847. The Boolean algebra is a fundamental for the development of digital electronic
systems, and is provided for in all programming languages. Set theory and statistics fields also
uses Boolean algebra for the representation, simplification and analysis of mathematical
quantities.
Logic levels are classified into two types
1. Positive logic
Logic 0 = False, 0V, Open Switch, OFF
Logic 1= True, +5V, Closed Switch, ON
2. Negative logic
Logic 0 = True, +5V, Closed Switch, ON
Logic 1= False, 0V, Open Switch, OFF
Boolean algebra differs from normal or elementary algebra. Latter deals with numerical
operations such as, addition, subtraction, multiplication and division on decimal numbers. And
former deals with the logical operations such as conjunction (OR), disjunction(AND) and
negation(NOT).
In present context, positive logic has been used for the entire discussion, representation
and simplification of Boolean variables.
1.2.1. Rules and properties of Boolean Algebra
1. Boolean variables takes only two values, logic 1 and logic 0, called binary numbers.
2. Basic operations of Boolean algebra are complement of a variable, ORing and ANDing of two
or more variables.
3. Mathematical description of Boolean operations using variables is called Boolean expression.
4. Complement of variable is represented by an over-bar (-).
Example: 𝑌 = 𝐴̅, Y is the output variable
5. ORing of variables is represented by a plus symbol (+)
Example:𝑌 = 𝐴 + 𝐵, Y is the output variable
6. ANDing of variables is represented by a dot symbol (.)
Example:𝑌 = 𝐴. 𝐵, Y is the output variable
7. Boolean operations are different from binary operations.
Example : 1+1=10 in Binary Addition
1+1=1 in Boolean algebra.
Table 1.1, shows the complement operation of a variable, table 1.2 summarized the OR operation
and table 1.3, summarized the AND operation of two variables.
A ̅
𝒀=𝑨
0 1
1 0
Table 1.1: Complement of variable A
Prepared by Mohankumar V., Assistant Professor, Dept. of ECE, Dr.AIT, B-56. Page 2
,Digital Electronics (18EC32) Notes
A B Y=A+B
0 0 0
0 1 1
1 0 1
1 1 1
Table 1.2: OR operation on A and B
A B Y=A.B
0 0 0
0 1 0
1 0 0
1 1 1
Table 1.3: AND operation on A and B
The present chapter deals with the simplification of Boolean expressions and
representation using sum of product form and product of sum forms.
1.2.2. Boolean Laws:
Law-1: Commutative law
The sequence of changing the variables does not effect on the result even after changing
their sequence while performing OR/AND operations of Boolean expression.
i. e. , A. B = B. A and A + B = B + A
Law-2: Associative law
The order of operations on variables is independent.
A. (B. C) = (A. B). C and A + (B + C) = (A + B) + C
Law-3: Distributive Laws
𝐴. (𝐵 + 𝐶) = 𝐴. 𝐵 + 𝐴. 𝐶
𝐴 + 𝐵𝐶 = (𝐴 + 𝐵)(𝐴 + 𝐶)
Law-4: AND Laws
𝐴. 0 = 0
𝐴. 1 = 𝐴
𝐴. 𝐴 = 𝐴
𝐴. 𝐴̅ = 0
Law-5: OR Laws
𝐴+0=𝐴
𝐴+1=1
𝐴+𝐴 = 𝐴
𝐴 + 𝐴̅ = 1
Law-6: Inversion/Complement/NOT Laws
0̅ = 1
Prepared by Mohankumar V., Assistant Professor, Dept. of ECE, Dr.AIT, B-56. Page 3
, Digital Electronics (18EC32) Notes
1̅ = 0
𝐴̅ = 𝐴
Law-7: Absorption Law
𝑨(𝑨 + 𝑩) = 𝑨
𝑨 + 𝑨𝑩 = 𝑨
𝑨+𝑨 ̅𝑩 = 𝑨 + 𝑩
Law-8: Demargon's Laws
De-Morgan's First Law
Statement: Sum of complement of two or more variables is equal to the product of the
complement of their variables.
i. e. , ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
A + B + C + ⋯..= A ̅. B
̅. ̅̅̅̅̅̅̅
C…..
Proof:
consider three variables for the proof shown in figure 1.4
A B C ̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑨 +𝑩+𝑪 ̅. 𝑩
𝑨 ̅
̅. 𝑪
0 0 0 1 1
0 0 1 0 0
0 1 0 0 0
0 1 1 0 0
1 0 0 0 0
1 0 1 0 0
1 1 0 0 0
1 1 1 0 0
Table 1.4: De-Morgan's First Law
De-Morgan's Second Law
Statement: Product of complement of two or more variables is equal to the sum of the
complement of their variables.
i. e. , ̅̅̅̅̅̅̅̅̅̅̅̅̅̅
ABC + ⋯ . . = ⋯ 𝐴̅ + 𝐵̅ + 𝐶̅ ….
Proof:
consider three variables for the proof shown in figure 1.5
A B C ̅̅̅̅̅̅̅̅
𝑨. 𝑩. 𝑪 ̅+B
A ̅ + C̅
0 0 0 1 1
0 0 1 1 1
0 1 0 1 1
0 1 1 1 1
1 0 0 1 1
1 0 1 1 1
1 1 0 1 1
1 1 1 0 0
Table 1.5: De-Morgan's Second Law
Boolean expressions must be simplified and evaluated using the order of operator precedence
shown in table 1.6
Prepared by Mohankumar V., Assistant Professor, Dept. of ECE, Dr.AIT, B-56. Page 4
UNIT-I : Fundamentals of Digital Electronics
1. 1. Introduction
Digital electronics is a type of electronics that deals with the digital systems which
processes the data/information in the form of binary(0s and 1s) numbers, whereas analog
electronics deals with the analog systems which processes the data/information in the form of
continuous signals.
Continuous signals
A Continuous signal is function f(t), whose value is defined for all time 't'.
in other words
Continuous signal a varying quantity with respect to independent variable time.
Example: Figure 1.1(a) shows the continuous signal.
Figure 1.1(a): Continuous signals.
Digital signals
A digital signal is a quantized discrete time signals.
Example: Figure 1.1(b) shows the discrete and digital signals.
Figure 1.1(b1): Discrete signal.
Figure 1.1(b2): Digital signal.
Prepared by Mohankumar V., Assistant Professor, Dept. of ECE, Dr.AIT, B-56. Page 1
,Digital Electronics (18EC32) Notes
1.2. Boolean Algebra
Boolean algebra is a branch of Algebra (Mathematics) that deals with operations on logical
values with Boolean variables, Boolean variables are represented as binary numbers which takes
logic 1 and logic 0 values. Hence, the Boolean algebra is also called two-valued logic, Binary
Algebra or Logical Algebra. The Boolean algebra was introduced by great mathematician George
Boole in 1847. The Boolean algebra is a fundamental for the development of digital electronic
systems, and is provided for in all programming languages. Set theory and statistics fields also
uses Boolean algebra for the representation, simplification and analysis of mathematical
quantities.
Logic levels are classified into two types
1. Positive logic
Logic 0 = False, 0V, Open Switch, OFF
Logic 1= True, +5V, Closed Switch, ON
2. Negative logic
Logic 0 = True, +5V, Closed Switch, ON
Logic 1= False, 0V, Open Switch, OFF
Boolean algebra differs from normal or elementary algebra. Latter deals with numerical
operations such as, addition, subtraction, multiplication and division on decimal numbers. And
former deals with the logical operations such as conjunction (OR), disjunction(AND) and
negation(NOT).
In present context, positive logic has been used for the entire discussion, representation
and simplification of Boolean variables.
1.2.1. Rules and properties of Boolean Algebra
1. Boolean variables takes only two values, logic 1 and logic 0, called binary numbers.
2. Basic operations of Boolean algebra are complement of a variable, ORing and ANDing of two
or more variables.
3. Mathematical description of Boolean operations using variables is called Boolean expression.
4. Complement of variable is represented by an over-bar (-).
Example: 𝑌 = 𝐴̅, Y is the output variable
5. ORing of variables is represented by a plus symbol (+)
Example:𝑌 = 𝐴 + 𝐵, Y is the output variable
6. ANDing of variables is represented by a dot symbol (.)
Example:𝑌 = 𝐴. 𝐵, Y is the output variable
7. Boolean operations are different from binary operations.
Example : 1+1=10 in Binary Addition
1+1=1 in Boolean algebra.
Table 1.1, shows the complement operation of a variable, table 1.2 summarized the OR operation
and table 1.3, summarized the AND operation of two variables.
A ̅
𝒀=𝑨
0 1
1 0
Table 1.1: Complement of variable A
Prepared by Mohankumar V., Assistant Professor, Dept. of ECE, Dr.AIT, B-56. Page 2
,Digital Electronics (18EC32) Notes
A B Y=A+B
0 0 0
0 1 1
1 0 1
1 1 1
Table 1.2: OR operation on A and B
A B Y=A.B
0 0 0
0 1 0
1 0 0
1 1 1
Table 1.3: AND operation on A and B
The present chapter deals with the simplification of Boolean expressions and
representation using sum of product form and product of sum forms.
1.2.2. Boolean Laws:
Law-1: Commutative law
The sequence of changing the variables does not effect on the result even after changing
their sequence while performing OR/AND operations of Boolean expression.
i. e. , A. B = B. A and A + B = B + A
Law-2: Associative law
The order of operations on variables is independent.
A. (B. C) = (A. B). C and A + (B + C) = (A + B) + C
Law-3: Distributive Laws
𝐴. (𝐵 + 𝐶) = 𝐴. 𝐵 + 𝐴. 𝐶
𝐴 + 𝐵𝐶 = (𝐴 + 𝐵)(𝐴 + 𝐶)
Law-4: AND Laws
𝐴. 0 = 0
𝐴. 1 = 𝐴
𝐴. 𝐴 = 𝐴
𝐴. 𝐴̅ = 0
Law-5: OR Laws
𝐴+0=𝐴
𝐴+1=1
𝐴+𝐴 = 𝐴
𝐴 + 𝐴̅ = 1
Law-6: Inversion/Complement/NOT Laws
0̅ = 1
Prepared by Mohankumar V., Assistant Professor, Dept. of ECE, Dr.AIT, B-56. Page 3
, Digital Electronics (18EC32) Notes
1̅ = 0
𝐴̅ = 𝐴
Law-7: Absorption Law
𝑨(𝑨 + 𝑩) = 𝑨
𝑨 + 𝑨𝑩 = 𝑨
𝑨+𝑨 ̅𝑩 = 𝑨 + 𝑩
Law-8: Demargon's Laws
De-Morgan's First Law
Statement: Sum of complement of two or more variables is equal to the product of the
complement of their variables.
i. e. , ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
A + B + C + ⋯..= A ̅. B
̅. ̅̅̅̅̅̅̅
C…..
Proof:
consider three variables for the proof shown in figure 1.4
A B C ̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑨 +𝑩+𝑪 ̅. 𝑩
𝑨 ̅
̅. 𝑪
0 0 0 1 1
0 0 1 0 0
0 1 0 0 0
0 1 1 0 0
1 0 0 0 0
1 0 1 0 0
1 1 0 0 0
1 1 1 0 0
Table 1.4: De-Morgan's First Law
De-Morgan's Second Law
Statement: Product of complement of two or more variables is equal to the sum of the
complement of their variables.
i. e. , ̅̅̅̅̅̅̅̅̅̅̅̅̅̅
ABC + ⋯ . . = ⋯ 𝐴̅ + 𝐵̅ + 𝐶̅ ….
Proof:
consider three variables for the proof shown in figure 1.5
A B C ̅̅̅̅̅̅̅̅
𝑨. 𝑩. 𝑪 ̅+B
A ̅ + C̅
0 0 0 1 1
0 0 1 1 1
0 1 0 1 1
0 1 1 1 1
1 0 0 1 1
1 0 1 1 1
1 1 0 1 1
1 1 1 0 0
Table 1.5: De-Morgan's Second Law
Boolean expressions must be simplified and evaluated using the order of operator precedence
shown in table 1.6
Prepared by Mohankumar V., Assistant Professor, Dept. of ECE, Dr.AIT, B-56. Page 4
