It is about the Boolean Algebra according to the syllabus of CSVTU.

1. DIGITAL SYSTEM DESIGN
Course Code:B000313(028)
Mr. Manjeet Singh Sonwani
Assistant Professor
Department of Electronics & Telecomm.
Government Engineering College Raipur

2. Course Contents
UNIT-I BOOLEAN ALGEBRA & MINIMIZATIONTECHNIQUES
UNIT-II COMBINATIONAL CIRCUITS
UNIT-III SEQUENTIAL CIRCUITS
UNIT-IV FINITE STATE MACHINE
UNIT-V DIGITAL LOGIC FAMILIES
Text Books/Reference Books:
1. R.P. Jain, “Modern digital Electronics”, Tata McGraw Hill, 4th edition, 2009.
2. W.H. Gothmann, “Digital Electronics- An introduction to theory and practice”,
PHI, 2nd Edition ,2006.
3. D.V. Hall, “Digital Circuits and Systems”, Tata McGraw Hill, 1989
4. Digital Fundamentals: Floyd & Jain: Pearson Education.
5. Digital Electronics: A. P. Malvino: Tata McGraw Hill.

3. UNIT-I
BOOLEAN ALGEBRA & MINIMIZATIONTECHNIQUES
• Boolean Algebra: Logic Operations; Axioms and Laws of
Boolean Algebra: Complementation Laws, AND Laws,
OR Laws, Commutative Laws, Associative Laws,
Distributive Laws, Absorption Laws, Transposition
Theorem,
• De Morgan’s Theorem;
• Duality;
• Reducing Boolean Expressions; Functionally Complete
Sets of Operations;
• Boolean Functions and their Representation.

4. Objectives
• Understand the relationship between Boolean logic and digital
computer circuits.
• Learn how to design simple logic circuits.
• Understand how digital circuits work together to form complex
computer systems.
4

5. Introduction
• In 1854, George Boole developed an algebraic
system now called Boolean algebra.
• In 1938, C. E. Shannon introduced a two-valued
Boolean algebra called switching algebra that
represented the properties of bistable electrical
switching circuits.
•
5

6. Boolean Algebra
• Boolean algebra is a mathematical system for the
manipulation of variables that can have one of two
values.
• In formal logic, these values are “true” and “false.”
• In digital systems, these values are “on” and “off,” 1 and 0,
or “high” and “low.”
• Boolean expressions are created by performing
operations on Boolean variables.
• Common Boolean operators include AND, OR, and NOT.
6

7. Boolean Algebra
• Boolean Constants
• these are ‘0’ (false) and ‘1’ (true)
• Boolean Variables
• variables that can only take the vales ‘0’ or ‘1’
• Boolean Functions
• each of the logic functions (such as AND, OR and NOT) are represented by
symbols as described above
• Boolean Theorems
• a set of identities and laws

8. Boolean Algebra : Logical Operator
• A Boolean operator can be
completely described using a
truth table.
• The truth table for the Boolean
operators AND and OR are
shown at the right.
• The AND operator is also known
as a Boolean product.
• The OR operator is the Boolean
sum. 8

9. Boolean Algebra: Logical Operator
• The truth table for the
Boolean NOT operator is
shown at the right.
• The NOT operation is most
often designated by an
overbar. It is sometimes
indicated by a prime mark
( ‘ ) or an “elbow” ().
9

10. Boolean Algebra
• A Boolean function has:
• At least one Boolean variable,
• At least one Boolean operator, and
• At least one input from the set {0,1}.
• It produces an output that is also a member of
the set {0,1}.
10
Now you know why the binary numbering
system is so handy in digital systems.

11. Boolean Algebra
• The truth table for the
Boolean function:
is shown at the right.
• To make evaluation of the
Boolean function easier,
the truth table contains
extra (shaded) columns to
hold evaluations of
subparts of the function. 11

12. Boolean Algebra
• As with common
arithmetic, Boolean
operations have rules of
precedence.
• The NOT operator has
highest priority, followed
by AND and then OR.
• This is how we chose the
(shaded) function
subparts in our table.
12

13. Boolean Algebra
• Digital computers contain circuits that implement
Boolean functions.
• The simpler that we can make a Boolean function,
the smaller the circuit that will result.
• Simpler circuits are cheaper to build, consume less
power, and run faster than complex circuits.
• With this in mind, we always want to reduce our
Boolean functions to their simplest form.
• There are a number of Boolean identities that help
us to do this. 13

14. Boolean Algebra
• Boolean algebra is an algebraic structure defined by a set of
elements , together with two binary operators. ‘+’ and ‘-‘, provided
that the postulates are satisfied.
• Postulates OR Axioms: A set of logical operations that accept
without proof and upon which it is possible to deduce rules ,
theorems and properties of the system.
• Most Boolean identities have an AND (product) form as well as an
OR (sum) form. We give our identities using both forms. Our first
group is rather intuitive:
14

15. Boolean Algebra : Law
• Our second group of Boolean identities should be
familiar to you from your study of algebra:
15

16. Boolean Algebra : Identity
• Our last group of Boolean identities are perhaps the
most useful.
• If you have studied set theory or formal logic, these
laws are also familiar to you.
16

17. Boolean Algebra : Identity
• Transposition theorem:
• AB+A’C=(A+C)(A’+B)
17

18. Boolean Algebra
• We can use Boolean identities to simplify the function:
as follows:
18

19. Boolean Algebra : De-Morgan’s Law
• Sometimes it is more economical to build a
circuit using the complement of a function (and
complementing its result) than it is to implement
the function directly.
• De-Morgan’s law provides an easy way of
finding the complement of a Boolean function.
• Recall De-Morgan’s law states:
19

20. Boolean Algebra : De-Morgan’s Law
• De-Morgan’s law can be extended to any number of
variables.
• Replace each variable by its complement and change all
ANDs to ORs and all ORs to ANDs.
• Thus, we find the the complement of:
is:
20

21. Boolean Algebra
• Through our exercises in simplifying Boolean
expressions, we see that there are numerous ways of
stating the same Boolean expression.
• These “synonymous” forms are logically equivalent.
• Logically equivalent expressions have identical truth tables.
• In order to eliminate as much confusion as possible,
designers express Boolean functions in standardized
or canonical form.
21

22. Boolean Algebra
• There are two canonical forms for Boolean
expressions: sum-of-products and product-of-sums.
• Recall the Boolean product is the AND operation and the
Boolean sum is the OR operation.
• In the sum-of-products form, ANDed variables are
ORed together.
• For example:
• In the product-of-sums form, ORed variables are
ANDed together:
• For example:
22

23. Boolean Algebra
• The sum-of-products form for
our function is:
23
We note that this function is not
in simplest terms. Our aim is
only to rewrite our function in
canonical sum-of-products form.

24. Boolean Algebra

25. Boolean Algebra (cont.)

26. Boolean Algebra (cont.)
• Boolean algebra is an algebraic structure defined by a set of
elements B, together with two binary operators. ‘+’ and ‘-‘, provided
that the following (Huntington) postulates are satisfied;
• Principle of Duality
• It states that every algebraic expression is deducible from the postulates
of Boolean algebra, and it remains valid if the operators & identity
elements are interchanged. If the inputs of a NOR gate are inverted we
get a AND equivalent circuit. Similarly when the inputs of a NAND gate
are inverted, we get a OR equivalent circuit.
• Interchanging the OR and AND operations of the expression.
• Interchanging the 0 and 1 elements of the expression.
• Not changing the form of the variables.

27. Boolean Algebra (cont.)
• Proving logical equivalence: Boolean algebra
method
• To prove that two logical functions F1 and F2 are equivalent
• Start with one function and apply Boolean laws to
derive the other function
• Needs intuition as to which laws should be applied
and when Practice helps
• Sometimes it may be convenient to reduce both
functions to the same expression

28. Logic Gates
• We have looked at Boolean functions in abstract
terms.
• In this section, we see that 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.
28

29. Logic Gates
• A logic gate is an electronic circuit/device which makes the logical decisions.
To arrive at this decisions, the most common logic gates used are OR, AND,
NOT, NAND, and NOR gates. The NAND and NOR gates are called universal
gates. The exclusive-OR gate is another logic gate which can be constructed
using AND, OR and NOT gate.
• Logic gates have one or more inputs and only one output. The output is active
only for certain input combinations. Logic gates are the building blocks of any
digital circuit. Logic gates are also called switches. With the advent of
integrated circuits, switches have been replaced by TTL (Transistor -Transistor
Logic) circuits and CMOS circuits. Here I give example circuits on how to
construct simples gates.
• AND ,OR,NOT,NAND,NOR,XOR,XNOR
29

30. Logic Gates
• The three simplest gates are the AND, OR, and NOT
gates.
• They correspond directly to their respective Boolean
operations, as you can see by their truth tables. 30

31. Logic Gates
• Another very useful gate is the exclusive OR
(XOR) gate.
• The output of the XOR operation is true only when
the values of the inputs differ.
31
Note the special symbol 
for the XOR operation.

32. Logic Gates
• AND Gate
• The AND gate performs logical multiplication, commonly known as
AND function. The AND gate has two or more inputs and single
output. The output of AND gate is HIGH only when all its inputs are
HIGH (i.e. even if one input is LOW, Output will be LOW).
• If X and Y are two inputs, then output F can be represented
mathematically as
F = X.Y, Here dot (.) denotes the AND operation. Truth table and
symbol of the AND gate is shown in the figure below.
32

33. Logic Gates
• Two input AND gate using "diode-resistor" logic is shown in figure below, where X, Y are
inputs and F is the output.
• If X = 0 and Y = 0, then both diodes D1 and D2 are forward biased and thus both diodes
conduct and pull F low.
If X = 0 and Y = 0, then both diodes D1 and D2 are forward biased and thus both diodes
conduct and pull F low.
• If X = 0 and Y = 1, D2 is reverse biased, thus does not conduct. But D1 is forward biased,
thus conducts and thus pulls F low.
• If X = 1 and Y = 0, D1 is reverse biased, thus does not conduct. But D2 is forward biased,
thus conducts and thus pulls F low.
• If X = 1 and Y = 1, then both diodes D1 and D2 are reverse biased and thus both the diodes
are in cut-off and thus there is no drop in voltage at F. Thus F is HIGH.
33

34. Logic Gates
• NAND and NOR are two
very important gates.
Their symbols and truth
tables are shown at the
right.
34

35. Universal Logic Gates
• 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 known
universal gates because
they are inexpensive to
manufacture and any
Boolean function can be
constructed using only
NAND or only NOR gates.
• But there are some rules
that need to be followed
when implementing NAND
or NOR based gates.
35

36. Logic Gates
• Gates can have multiple inputs and more than
one output.
• A second output can be provided for the complement
of the operation.
• We’ll see more of this later.
36

37. Digital Components
• The main thing to remember is that combinations
of gates implement Boolean functions.
• The circuit below implements the Boolean
function:
37
We simplify our Boolean expressions so
that we can create simpler circuits.

38. Reducing Boolean expression
• Step1. Multiply all variables necessary to remove parentheses.
• Step2. If there is two or more identical terms then only one term be
detained and rest all terms dropped/ignored.
• Ex:-AB+AB+AB=AB
• Step3. If there will be a variables and its negation present in the same
terms then this term can be dropped/ignored.
• Ex:-A.BB’=A.0=0 ,ACBC’=ABCC’=AB.0=0
38

39. Reducing Boolean expression
• Step4. Pairs of terms are identical except for one
variables then the larger term can be ignored.
• Ex:-ABC’D’+ABC’=ABC’(D’+1)=ABC’
• Step5. The Pairs of terms which have the same
variables with one or more variables completed.
• If a variables in one term of such a pair is completed
• While in the second term is not,then such can be
combined into a single term with that variables
dropped.
Ex:-ABC’D’+ABC’D=ABC’(D+D’)=ABC’.1=ABC’
Que. Reduce the Exp. A[B+C’(AB+AC’)’]
39

40. Converting AND/OR/Invert Logic to
NAND/NOR logic
• Any logic function can be implemented using NAND gates.
To achieve this, first the logic function has to be written in
Sum of Product (SOP) form. Once logic function is
converted to SOP, then is very easy to implement using
NAND gate.
• In other words any logic circuit with AND gates in first level
and OR gates in second level can be converted into a
NAND-NAND gate circuit.
• Consider the following SOP expression
• F = W.X.Y + X.Y.Z + Y.Z.W 40

41. Converting AND/OR/Invert Logic to
NAND/NOR logic
• The procedure is:
• Step1. Draw the circuit / expression in AOI logic.
• Step2a.If NAND logic is chosen, add a circle at the output of AND gate and at the inputs to all
the OR gates.
41

42. Converting AND/OR/Invert Logic to
NAND/NOR logic
Step2b. Realization of logic function using NOR gates
• Any logic function can be implemented using NOR gates. To achieve this, first the logic
function has to be written in Product of Sum (POS) form. Once it is converted to POS, then
it's very easy to implement using NOR gate. In other words any logic circuit with OR gates in
first level and AND gates in second level can be converted into a NOR-NOR gate circuit.
• Consider the following POS expression
• If NOR logic is chosen, add a circle at the output of OR gate and at the inputs to all the AND
gates. F = (X+Y) . (Y+Z)
42

43. Converting AND/OR/Invert Logic to
NAND/NOR logic
• Step3.Add or subtract an inverter on each line that the polarity of the signals on those lines
remains unchanged from that of the original diagram
43