Unit3_DIS.ppt

Noida Institute of Engineering and Technology, Greater Noida
Algebraic Structures
RAHUL KUMAR
Assistant Professor
11/11/2022
1
Rahul Kumar Discrete Structures (ACSE0306) Unit 3
B.Tech (CSE)
IIIrd Sem
UNIT-3
Discrete Structures

2
Sl.
No.
Subject Codes Subject Name
Periods Evaluation Schemes
End
Semester Total Credit
L T P CT TA TOTAL PS TE PE
WEEKS COMPULSORY INDUCTION PROGRAM
1 AAS0303 Statistics and Probability 3 1 0 30 20 50 100 150 4
2 ACSE0306 Discrete Structures 3 0 0 30 20 50 100 150 3
3 ACSE0305
Computer Organization &
Architecture
3 0 0 30 20 50 100 150 3
4 ACSE0302
Object Oriented Techniques
using Java
3 0 0 30 20 50 100 150 3
5 ACSE0301 Data Structures 3 1 0 30 20 50 100 150 4
6 ACSDS0301
Foundations of Data
Science
3 0 0 30 20 50 100 150 3
7 ACSE0352
Object Oriented Techniques
using Java Lab
0 0 2 25 25 50 1
8 ACSE0351 Data Structures Lab 0 0 2 25 25 50 1
9 ACSDS0351 Data Analysis Lab 0 0 2 25 25 50 1
10 ACSE0359 Internship Assessment-I 0 0 2 50 50 1
11 ANC0301 /
ANC0302
Cyber Security* /
Environmental
Science*(Non Credit)
2 0 0 30 20 50 50 1000
12
MOOCs (For B.Tech.
Hons. Degree)
GRAND TOTAL 1100 24
NOIDA INSTITUTE OF ENGINEERING & TECHNOLOGY, GREATER NOIDA(An
Autonomous Institute EVALUATION SCHEME SEMESTER-III
11/11/2022
Rahul Kumar Discrete Structures (ACSE0306) Unit
3

11/11/2022
3
3
Subject Syllabus

11/11/2022
3
4
Subject Syllabus

11/11/2022
3
5
Subject Syllabus

11/11/2022
3
6
Subject Syllabus

11/11/2022 7
Application in CSE
1. Discrete Structures are useful in studying and describing objects and problems in branches
of computer science such as computer algorithms, programming languages.
2. Computer implementations are significant in applying ideas from discrete mathematics to real-
world problems, such as in operations research.
3. It is a very good tool for improving reasoning and problem-solving capabilities.
4. Discrete mathematics is used to include theoretical computer science, which is relevant to
computing.
5. Discrete structures in computer science with the help of process algebras.

• The subject enhances one’s ability to develop logical thinking and ability to
problem solving.
• The objective of discrete structure is to enables students to formulate
problems precisely, solve the problems, apply formal proofs techniques and
explain their reasoning clearly.
11/11/2022 8
Course Objective

Course
Outcome
(CO)
At the end of course , the student will be able to Bloom’s
Knowledge
Level (KL)
CO1 Apply the basic principles of sets, relations & functions and
mathematical induction in computer science & engineering related
problems
K3
CO2 Understand the algebraic structures and its properties to solve
complex problems
K2
CO3 Describe lattices and its types and apply Boolean algebra to simplify
digital circuit.
K2,K3
CO4 Infer the validity of statements and construct proofs using predicate
logic formulas.
K3,K5
CO5 Design and use the non-linear data structure like tree and graphs to
solve real world problems.
K3,K6
Course Outcome
11/11/2022 9
3

Program Outcomes
Engineering Graduates will be able to Understand:
1. Engineering knowledge
2. Problem analysis
3. Design/development of solutions
4. Conduct investigations of complex
5. Modern tool usage
6. The engineer and society
7. Environment and sustainability
8. Ethics
9. Individual and team work
10. Communication
11. Project management and finance
12. Life-long learning
11/11/2022
3
10
Program Outcome

PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12
ACSE0306.1
2 2 3 3 2 2 - - 2 1 - 3
ACSE0306.2
1 3 2 3 2 2 - 1 1 1 2 2
ACSE0306.3
2 2 3 2 2 2 - 2 2 1 2 3
ACSE0306.4
2 2 2 3 2 2 - 2 2 1 1 3
ACSE306.5
3 2 2 2 2 2 - 2 1 1 1 2
Average 2 2.2 2.4 2.6 2 2 - 1.4 1.6 1 1.2 2.6
11/11/2022
11
CO-PO Mapping
CO-PO correlation matrix Discrete Structures (ACSE0306)

11/11/2022
3
12
End Semester Question Paper Templates (Offline Pattern/Online
Pattern

11/11/2022
3
13
Pattern

11/11/2022
3
14
Pattern

11/11/2022
3
15
Pattern

11/11/2022
3
16
Pattern

11/11/2022
3
17
Pattern

11/11/2022
3
18
Pattern

• Brief review of Relations and Equivalence of Relations.
11/11/2022
19
Prerequisite and Recap

11/11/2022
3
20
Discrete mathematics is the study of mathematical structures that are
fundamentally discrete rather than continuous. In contrast to real numbers that
have the property of varying "smoothly", the objects studied in discrete
mathematics – such as integers, graphs, and statements in logic.
• https://www.youtube.com/watch?v=C-C2mtqMVfo
•https://www.youtube.com/watch?v=35YXM8PcloY
Brief Introduction about the subject with video

• Bounded and complemented lattice
• Distributive Lattice
• Introduction of Boolean algebra
• Axioms and Theorems of Boolean algebra
• Algebraic manipulation of Boolean expressions
• Truth table & Logic gates
• Principle of duality
• Digital circuits and Boolean algebra
• Canonical form
• Simplification of Boolean Functions
• Karnaugh maps
11/11/2022
3
21
Unit Content

11/11/2022
22
Content
• Video links
• Daily Quiz
• Weekly Assignment
• MCQ
• Old Question papers
• Expected Question for University Exam
• Summary
• References.

11/11/2022 23
Unit Objective
• Student will able to understand represent a finite partially
ordered set, in the form of a drawing of its transitive reduction.
• Define Join and Meet.
• Student will able to draw Hasse Diagram.

The student will be able to:
– Represent a Hasse Diagram.
– Give examples of finite and infinite sets.
– Build new sets from existing sets by applying various combinations of
the set operations for example intersection union, difference, and
complement.
– Determine Posets.
– Sets are used to define the concepts of relations and functions. The
study of geometry, sequences, probability, etc. requires the knowledge
of sets.
11/11/2022 24
Topic Objectives: Lattice and Boolean Algebra (CO3)

11/11/2022 25
Ordered Sets & Posets
• A Set is a collection of elements. The elements that make up a set can be any kind
of mathematical objects: numbers, symbols, points in space, lines, other
geometrical shapes, variables, or even other sets.
• Example: A collection of natural numbers, as in, N = {1,2,3,4,5...}.
• A relation can also be used to define an order on a set.
• An Ordered set is a relational structure (S, ≤) such that the realtion ≤ is an
ordering.
• An ordered pair is a pair of numbers (x,y) written in a particular order

11/11/2022 26
Terminologies
• Cartesian Product: the product of two sets A and B such that every element of set A relates to
every other element of set B to form ordered pairs.
A*B = {(a,1),(a,2),(b,1),(b,2)}
• Reflexive Relation:
A relation R, over a set A, is reflexive if every element of the set is 'related' to itself.
Let's consider set A as follows:
A = {p,q,r}
Therefore, A*A = {(p,p),(p,q),(p,r),(q,p),(q,q),(q,r),(r,p),(r,q),(r,r)}
Then the reflexive pairs in A*A would be all the diagonal elements of the
matrix
i.e. {(p,p),(q,q),(r,r)} as every element relates to itself.

11/11/2022 27
Terminologies
• Symmetric: A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,
b) ∈ R, for all a, b ∈ A.
• Anti-symmetric Relation: : A relation R on a set A such that for all a, b ∈ A, if (a,
b) ∈ R and (b, a) ∈ R, then a = b is called antisymmetric.
a b
a b
Note:
• A relation can be both symmetric and
antisymmetric.
• A relation can be neither symmetric nor
antisymmetric.

11/11/2022 28
Terminologies
• Transitive: A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b,
c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A.
Relation R={(1,2),(2,3),(1,3)} on set A={1,2,3} is transitive.
Asymmetric relation: Asymmetric relation is opposite of symmetric relation. A
relation R on a set A is called asymmetric if no (b,a) € R when (a,b) € R.
a b
c

11/11/2022 29
POSET(Partially Ordered Set).
A relation R, over a set A, is a partial order relation if it is reflexive, anti-
symmetric and transitive.
For this, we will
1. Check if it is reflexive
2. Check if it is anti-symmetric
3. Check if it is transitive
Step 1: The subset is reflexive as it contains the pairs, (p,p), (q,q) and (r,r).
Step 2: the relation R is said to be antisymmetric on a set A, if xRy and yRx hold when x = y. Or it can
be defined as, relation R is antisymmetric if either (x,y)∉R or (y,x)∉R whenever x ≠ y.
Step 3: The subset contains (p,p) and (p,r). Therefore according to the definition of a
transitive relation, it must contain (p,r), which, you can see, is already present in it. Hence it
is transitive.
Since all the three conditions are satisfied, we could now call the subset as a partially
ordered set.

11/11/2022 30
Elements of POSET:
1. Maximal Element: An element a ∈ A is called a maximal element of A if there is no
element in c in A such that a ≤ c.
2. Minimal Element: An element b ∈ A is called a minimal element of A if there is no
element in c in A such that c ≤ b.
Example: Determine all the maximal and minimal elements of the poset whose Hasse
diagram is shown in fig:
The maximal elements are b and f.
The minimal elements are d and
e.

11/11/2022 31
Comparable Elements: Consider an ordered set A. Two elements a and b of set A are called
comparable if a≤b or b ≤ a
Non-Comparable Elements:
Consider an ordered set A. Two elements a and b of set A are called non-comparable if neither a
≤ b nor b ≤ a.
Example: Consider A = {1, 2, 3, 5, 6, 10, 15, 30} is ordered by divisibility. Determine all the
comparable and non-comparable pairs of elements of A.
Solution: The comparable pairs of elements of A are:
{1, 2}, {1, 3}, {1, 5}, {1, 6}, {1, 10}, {1, 15}, {1, 30}
{2, 6}, {2, 10}, {2, 30}
{3, 6}, {3, 15}, {3, 30}
{5, 10}, {5, 15}, {5, 30}
{6, 30}, {10, 30}, {15, 30}
The non-comparable pair of elements of A are:
{2, 3}, {2, 5}, {2, 15}
{3, 5}, {3, 10}, {5, 6}, {6, 10}, {6, 15}, {10, 15}

11/11/2022 32
Hasse Diagram
A POSET can be represented in the form of a simple diagram called the Hasse diagram.
Let us consider a set B = {3,4,5,6}
B*B = {(3,3), (3,4), (3,5), (3,6), (4,3), (4,4), (4,5), (4,6), (5,3), (5,4), (5,5), (5,6), (6,3), (6,4),
(6,5), (6,6)}
From this Cartesian product, let us create another set R having the following pairs of elements:
R = {(3,3), (3,4), (3,5), (3,6), (4,4), (4,5), (4,6), (5,5), (5,6), (6,6)}
Now represent each of the pair in the graph
To further simplify the graph omit the self-loops and transitive edges in order to avoid repetition.
The simplified graph would then look like this:

11/11/2022 33
HASSE Diagram
Example: Consider the set A = {4, 5, 6, 7}. Let R be the relation ≤ on A. Draw the directed graph and the
Hasse diagram of R.
Solution: The relation ≤ on the set A is given by
R = {{4, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}, {4, 4}, {5, 5}, {6, 6}, {7, 7}}
The directed graph of the relation R

11/11/2022 34
HASSE Diagram
To draw the Hasse diagram of partial order, apply the following points:
1. Delete all edges implied by reflexive property i.e.
(4, 4), (5, 5), (6, 6), (7, 7)
2. Delete all edges implied by transitive property i.e.
(4, 7), (5, 7), (4, 6)
3. Replace the circles representing the vertices by dots.
4. Omit the arrows.
• Upper Bound: Consider B be a subset of a partially ordered set A. An element x ∈ A is called an upper
bound of B if y ≤ x for every y ∈ B.
• Lower Bound: Consider B be a subset of a partially ordered set A. An element z ∈ A is called a lower
bound of B if z ≤ x for every x ∈ B.
The Hasse diagram

11/11/2022 35
HASSE Diagram
• Upper Bound: Consider B be a subset of a partially ordered set A. An element x ∈ A is called an upper
bound of B if y ≤ x for every y ∈ B.
• Lower Bound: Consider B be a subset of a partially ordered set A. An element z ∈ A is called a lower
bound of B if z ≤ x for every x ∈ B.
Example: Consider the poset A = {a, b, c, d, e, f, g} be ordered shown in fig. Also let B = {c, d, e}. Determine the upper
and lower bound of B.
Solution:
The upper bound of B is e, f, and g because every element of B is '≤'
e, f, and g.
The lower bounds of B are a and b because a and b are '≤' every
elements of B.

11/11/2022 36
HASSE Diagram
• Least Upper Bound(SUPREMUM) : Let A be a subset of a partially ordered set S. An element M in S
is called an upper bound of A if M succeeds every element of A, i.e. if, for every x in A, we have x
<=M
If an upper bound of A precedes every other upper bound of A, then it is called the supremum of A and
is denoted by Sup (A)
• Greatest Lower Bound (INFIMUM) : An element m in a poset S is called a lower bound of a subset A
of S if m precedes every element of A, i.e. if, for every y in A, we have m <=y
If a lower bound of A succeeds every other lower bound of A, then it is called the infimum of A and is
denoted by Inf (A)
Example: Determine the least upper bound and greatest lower bound of B = {a, b, c} if they exist, of the
poset whose Hasse diagram is
Solution: The least upper bound is c.
The greatest lower bound is k.

11/11/2022 37
Lattice
• A lattice is a poset in (L,≤) in which every subset {a,b} consisiting of two elements
has a least upper bound and a greatest lower bound.
LUB({a,b}) is denotead by a v b and is called the join of a and b.
GLB({a,b}) is denoted by a Λ b and is called the meet of a and b.
a) Is a lattice b) Is not a lattice because
f V g does not exist

• Bounded Lattices:
• A lattice L is called a bounded lattice if it has greatest element 1 and a least element 0.
• Example:
• The power set P(S) of the set S under the operations of intersection and union is a bounded
lattice since ∅ is the least element of P(S) and the set S is the greatest element of P(S).
• The set of +ve integer I+ under the usual order of ≤ is not a bounded lattice since it has a least
element 1 but the greatest element does not exist.
• Sub-Lattices:
• Consider a non-empty subset L1 of a lattice L. Then L1 is called a sub-lattice of L if L1 itself is a
lattice i.e., the operation of L i.e., a ∨ b ∈ L1 and a ∧ b ∈ L1 whenever a ∈ L1 and b ∈ L1.
11/11/2022
3
38
Types of Lattice

11/11/2022
3
39
Types of Lattice
Ex: Determine all the sub-lattices of D30 that contain at least four elements, D30={1,2,3,5,6,10,15,30}.
• Solution: The sub-lattices of D30 that contain at least four elements are as follows:
• 1. {1, 2, 6, 30} 2. {1, 2, 3, 30}
3. {1, 5, 15, 30} 4. {1, 3, 6, 30}
5. {1, 5, 10, 30} 6. {1, 3, 15, 30}
7. {2, 6, 10, 30}
• Isomorphic Lattices:
• Two lattices L1 and L2 are called isomorphic lattices if there is a bijection from L1 to L2 i.e., f: L1⟶ L2, such
that f (a ∧ b) =f(a)∧ f(b) and f (a ∨ b) = f (a) ∨ f (b)
• Example: Determine whether the lattices shown in fig are isomorphic.
• Solution: The lattices shown in fig are isomorphic. Consider the mapping f = {(a, 1), (b, 2), (c, 3), (d, 4)}.For
example f (b ∧ c) = f (a) = 1. Also, we have f (b) ∧ f(c) = 2 ∧ 3 = 1

11/11/2022
3
40
Types of Lattice
• Distributive Lattice:
• A lattice L is called distributive lattice if for any elements a, b and c of L ,it satisfies following distributive
properties:
• a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
• a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
• If the lattice L does not satisfies the above properties, it is called a non-distributive lattice.
• Example: The power set P (S) of the set S under the operation of intersection and union is a distributive
function. Since, a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∪ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) for any sets a, b and c of P(S).
• The lattice shown in fig II is a distributive. Since, it satisfies the distributive properties for all ordered triples
which are taken from 1, 2, 3, and 4.

11/11/2022
3
41
Types of Lattice
Complements and complemented lattices:
Let L be a bounded lattice with lower bound o and upper bound I. Let a be an element if L. An
element x in L is called a complement of a if a ∨ x = I and a ∧ x = 0
A lattice L is said to be complemented if L is bounded and every element in L has a complement.
Example: Determine the complement of a and c in fig:
Solution: The complement of a is d. Since, a ∨ d = 1 and a ∧ d = 0
Question : Find out complemented and Distributive Lattice from the Figure given below.

11/11/2022
3
42
Types of Lattice

After learning Boolean algebra students will be able to
• Perform the three basic logic operations.
• Describe the operation of and construct the truth tables for the AND, NAND, OR,
and NOR gates, and the NOT (INVERTER) circuit.
• Implement logic circuits using basic AND, OR, expression for the logic gates and
combinations of logic gates. and NOT gates.
• Appreciate the potential of Boolean algebra to simplify complex logic expressions.
• Use De-Morgan’s theorems to simplify logic expressions.
11/11/2022
Topic Objective: Boolean Algebra (CO3)
43

• An algebra which is use for simplifying the presentation & manipulation of
propositional logic.
• It is also known as Switching Algebra.
• Invented by George Boole in 1854.
• Boolean algebra is mainly use for simplify and manipulate electronic logic
circuits in computers.
11/11/2022
Boolean Algebra(CO3)
44

• Boolean Algebra:
• A complemented distributive lattice is known as a Boolean Algebra. It is denoted by (B, ∧,∨,',0,1), where B is a
set on which two binary operations ∧ (*) and ∨(+) and a unary operation (complement) are defined. Here 0 and
1 are two distinct elements of B.
• Since (B,∧,∨) is a complemented distributive lattice, therefore each element of B has a unique complement.
• Sub-Algebra:
• Consider a Boolean-Algebra (B, *, +,', 0,1) and let A ⊆ B. Then (A,*, +,', 0,1) is called a sub-algebra or Sub-
Boolean Algebra of B if A itself is a Boolean Algebra i.e., A contains the elements 0 and 1 and is closed under
the operations *, + and '.
• Example: Consider the Boolean algebra D70 whose Hasse diagram is shown in fig
• A= {1, 7, 10, 70} and B = {1, 2, 35, 70} is a sub-algebra of D70. Since both A and B are closed under operation
∧,∨and '.
11/11/2022
Boolean Algebra(CO3)
45

• It is denoted by (B, +, ., ‘, 0, 1), where
 B = non-empty set
 + is the binary operation denoting logical OR
 . is the binary operation denoting logical AND
 ‘ is the unary operation denoting logical NOT
 0 and 1 are distinct elements known as identity elements of + and . operation
• It deals with the binary number system {0,1}.
• Truth Table -It is a tabular representation of all the combination of values for input and
their corresponding outputs.
11/11/2022
Basic concepts of Boolean algebra (CO1)
46

• AND operation is used for logical multiplication.
• The dot symbol (“.”)used for representing AND operation.
• AND operator has an output 1 if all the inputs are 1 otherwise the output is 0.
11/11/2022
AND Operation(CO1)
47
=A.B

• OR operation is use for logical addition.
• The symbol (“+”) used for represent OR operation.
• OR operator has an output 0 if all of the inputs are 0 otherwise the output is 1.
11/11/2022
OR Operation(CO1)
48
=A+B

 NOT gate performs the logical complementation operation.
 It is an electronic circuit that generates the reverse of the input signal as output
signal.
11/11/2022
NOT operation (CO1)
49
=A’

If a, b, c ∈ B, then
1. Closure Law
i. a + b ∈ B ii. a.b ∈ B
2. Commutative Law
i. a + b = b + a ii. a.b = b.a
3. Distributive Law
i. a.(b + c) = a.b + a.c ii. a + (b.c) = (a + b).(a + c)
4. Identity Law
i. a + 0 = a ii. a.1 = a
5. Complement Law
i. a + a’ = 1 ii. a.a’ = 0
11/11/2022
Postulates/Axioms of Boolean Algebra (CO1)
50

6. Idempotent Law
i. a + a = a ii. a.a = a
7. Boundness Law
i. a + 1 = 1 ii. a.0 = 0
8. Absorption Law
i. a + a.b = a ii. a.(a + b) = a
9. Associative Law
i. a + (b + c) = (a + b) + c ii. a.(b.c) = (a.b).c
11/11/2022
Postulates/Axioms of Boolean Algebra (CO1)
51

Boolean function
• A Boolean function is a special kind of mathematical function f:Xn→X
f:Xn→X of degree n, where X={0,1}X={0,1} is a Boolean domain and n is a
non-negative integer. It describes the way how to derive Boolean output
from Boolean inputs.
• Example − Let, F(A,B)=A′B′F(A,B)=A′B′. This is a function of degree 2 from
the set of ordered pairs of Boolean variables to the
set {0,1}{0,1} where F(0,0)=1,F(0,1)=0,F(1,0)=0F(0,0)=1,F(0,1)=0,F(1,0)=0 an
d F(1,1)=0
•
11/11/2022
3
52

11/11/2022
Minimization of Boolean Algebra using laws (CO1)
53
1.

11/11/2022
54
2.

11/11/2022
55
(Complement law)
(Distributive law)
(Complement law)
(Complement law)
(Identity Law)
(Identity law)
(Complement law)

If a, b, c ∈ B, then
1. Uniqueness of complement
If a + x = 1 and a.x = 0, then x = a’
2. Involution Law
(a’)’ = a
3. De-Morgan’s Law
i. (a + b)’ = a’.b’ ii.(a.b) = a’ + b’
11/11/2022
Other laws of Boolean Algebra (CO1)
56

11/11/2022
Example of De-Morgan Law(CO1)
57

• Boolean function is an expression formed with binary variable.
• It is the combination of Boolean operators such as AND, OR & NOT operators.
X.( X’+Y)=X.Y
11/11/2022
Boolean Function(CO1)
58
Truth table for Boolean function.

• Computer carries out all operations by the combination of signals that pass
through standard blocks of built-in circuits.
• Logic gate is an elementary building block of digital electronic circuits that
operates one or more input signals to produce standard output.
• The common use of logic gate elements is to act as switch.
• In computer logic gates are use to implement Boolean functions.
11/11/2022
Logic Gate (CO1)
59

 AND Gate
 OR Gate
 NOT Gate
11/11/2022
Logic Gates Symbol (CO1)
60

 AND gate is use for logical multiplication input signals.
 If the input values for an AND gate are 1 then the output is 1
otherwise the output is 0.
11/11/2022
AND Gate (CO1)
61

 OR gate performs the logical addition operation.
 It is an electronic circuit that generates an output signal 1 if any of the input
signal is 1.
11/11/2022
OR Gate (CO1)
62

 NOT gate performs the logical complementation operation.
 It is an electronic circuit that generates the reverse of the input signal as output
signal.
11/11/2022
NOT Gate (CO1)
63

11/11/2022
Exclusive OR gate(CO1)
64

11/11/2022
Exclusive NOR gate(CO1)
65

11/11/2022
Universal Gates(CO1)
66
NAND & NOR gates

11/11/2022
Summary of Logic Gates(CO1)
67
Input Output
A A’ or Ā
0 1
1 0

This principle states that any algebraic equality derived from these axioms
will still be valid whenever the OR and AND operators, and identity
elements 0 and 1, have been interchanged. i.e. changing every OR into
AND and vice versa, and every 0 into 1 and vice versa.
Example:
1. X + 0 = X 2. X • 1 = X (dual of 1)
3. X + 1 = 1 4. X • 0 = 0 (dual of 3)
5. X + X = X 6. X • X = X (dual of 5)
7. X + X’ = 1 8. X • X’ = 0 (dual of 8)
11/11/2022
Principle of Duality(CO1)
68

We need to consider formal techniques for the simplification of Boolean
functions.
– Identical functions will have exactly the same canonical form.
– Minterms and Maxterms.
– Sum-of-Minterms and Product-of- Maxterms.
– Product and Sum terms.
– Sum-of-Products (SOP) and Product-of-Sums (POS)
11/11/2022
Canonical/Standard forms(CO1)
69

• Literal: A variable or its complement.
• Product term: literals connected by •
• Sum term: literals connected by +
• Minterm: a product term in which all the variables appear exactly once,
either complemented or un-complemented.
• Maxterm: a sum term in which all the variables appear exactly once,
either complemented or un-complemented.
11/11/2022
Terms of Canonical/standard forms(CO1)
70

• Represents exactly one combination in the truth table.
• Denoted by mj, where j is the decimal equivalent of the minterm ’s
corresponding binary combination (bj).
• A variable in mj is complemented if its value in bj is 0, otherwise is u-
ncomplemented.
• Example: Assume 3 variables (A,B,C), and j=3. Then, bj = 011 and its
corresponding minterm is denoted by mj = A’BC
11/11/2022
Minterms(CO1)
71

• Represents exactly one combination in the truth table.
• Denoted by Mj, where j is the decimal equivalent of the maxterm’s
corresponding binary combination (bj).
• A variable in Mj is complemented if its value in bj is 1, otherwise is
uncomplemented.
• Example: Assume 3 variables (A,B,C), and j=3. Then, bj = 011 and its
corresponding maxterm is denoted by Mj = A+B’+C’
11/11/2022
Maxterms(CO1)
72

• An implementation of a Boolean Function requires the use of logic
gates.
• A smaller number of gates, with each gate (other then Inverter) having
less number of inputs, may reduce the cost of the implementation.
• There are 2 methods for simplification of Boolean functions: algebraic
method and graphical method (K-maps).
11/11/2022
Simplification of Boolean Functions (CO1)
73

• Simplification of Boolean functions and expressions using K-map.
• K-map implementation for circuit designing.
11/11/2022
Topic objective: K-maps(CO1)
74

Prerequisite
• Boolean functions.
• Binary logic.
• SOP form
• POS form
Recap
• K-map is a graphical representation to solve the Boolean functions to
reduce redundancy. It is easy to find the minimized expression of lengthy
Boolean functions using K-maps.
11/11/2022
Prerequisite and recap of Topic: K-maps(CO1)
75

• The Karnaugh map (KM or K-map) is a method of simplifying Boolean
algebra expressions. Maurice Karnaugh introduced it in 1953.
• The Karnaugh map reduces the need for extensive calculations by taking
advantage of humans' pattern-recognition capability. It also permits the rapid
identification and elimination of potential race conditions.
• The required Boolean results are transferred from a truth table onto a two-
dimensional grid where, in Karnaugh maps, the cells are ordered in Gray
code, and each cell position represents one combination of input conditions.
• Cells are also known as minterms, while each cell value represents the
corresponding output value of the boolean function.
• Optimal groups of 1s or 0s are identified, which represent the terms of
a canonical form of the logic in the original truth table. These terms can be
used to write a minimal Boolean expression representing the required logic.
11/11/2022
K-Map (CO1)
76

• Groups may not contain zero.
• We can group 1,2,4,8 ,…cells.
• Each group should be a large as possible.
• Groups may overlap.
• Opposite grouping and corner grouping are allowed.
• Grouping can not possible in diagonally manner.
• Cells contains 1 must be grouped.
11/11/2022
Rules for K-Map Simplification(CO1)
77

11/11/2022
Grouping :(CO1)
78
0 1 3 2
4 5 7 6
00 01 11 10
0
1
BC
A

 Cell numbers are written in the cells.
 2-variable K-map
11/11/2022
K-Maps Numbering (CO1)
79
0 1
2 3
B
A
0 1
0
1

11/11/2022
Examples of 2-variable K-Maps (CO1)
80
B
A
0 1
0
1

11/11/2022
2-Variable K-Map SOP & POS:(CO1)
81
0 1 3 2
4 5 7 6
00 01 11 10
0
1
BC
A

11/11/2022
3-Variable K-Map Numbering:(CO1)
82
0 1 3 2
4 5 7 6
00 01 11 10
0
1
BC
A

11/11/2022
3-Variable K-Map Examples:(CO1)
83
0 1 3 2
4 5 7 6
00 01 11 10
0
1
BC
A

11/11/2022
4-variable K-map Numbering (CO1)
84
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
00 01 11 10
00
01
11
10
CD
AB

11/11/2022
4-variable K-map Examples (CO1)
85
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
00 01 11 10
00
01
11
10
AB

11/11/2022
4-variable K-map simplification(CO1)
86
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
00 01 11 10
00
01
11
10
CD
AB

11/11/2022
Don’t cares in K map(CO1)
87
A don't-care term (abbreviated DC, historically also known as
redundancies, irrelevancies, optional entries, invalid combinations,
vacuous combinations, forbidden combinations, or unused states) for a
function is an input-sequence (a series of bits) for which the function
output does not matter.

11/11/2022
Don’t cares in K map(CO1)
88

11/11/2022
Practice Problems on K-map (CO1)
89
Simplify using SOP
1) Z= (A,B,C)=∑(1,3,6,7) Ans: (A’C+AB)
2) F(P,Q,R,S)=∑(0,2,5,7,8,10,13,15) Ans: (QS+Q’S’)
Simplify using POS
1) F(A,B,C)=π(0,3,6,7)
Ans: (A’ + B’ + C) (B’ + C’) (A + B + C)
2) F(A,B,C,D)=π(3,5,7,8,10,11,12,13)
Ans: (C+D’+B’).(C’+D’+A).(A’+C+D).(A’+B+C’)

1. Algebra of logic is termed as ______________
a) Numerical logic
b) Boolean algebra
c) Arithmetic logic
d) Boolean number
2. Boolean algebra can be used ____________
a) For designing of the digital computers
b) In building logic symbols
c) Circuit theory
d) Building algebraic functions
3. What is the definition of Boolean functions?
a) An arithmetic function with k degrees such that f:Y–>Yk
b) A special mathematical function with n degrees such that f:Yn–>Y
c) An algebraic function with n degrees such that f:Xn–>X
d) A polynomial function with k degrees such that f:X2–>Xn
11/11/2022 90
Daily Quiz (CO1)

4. F(X,Y,Z,M) = X`Y`Z`M`. The degree of the function is ________
a) 2
b) 5
c) 4
d) 1
5. K-map is used for _______
a) logic minimization
b) expression maximization
c) summing of parity bits
d) logic gate creation
6. A Poset in which every pair of elements has both a least upper bound and a greatest lower bound is
termed as _______
a) sublattice
b) lattice
c) trail
d) walk
11/11/2022 91
Daily Quiz (CO1)

7. Which of the following is a Simplification law?
a) M.(~M+N) = M.N
b) M+(N.O) = (M+N)(M+O)
c) ~(M+N) = ~M.~N
d) M.(N.O) = (M.N).O
8. What are the canonical forms of Boolean Expressions?
a) OR and XOR
b) NOR and XNOR
c) MAX and MIN
d) SOM and POM
9. Which of the following is/are the universal logic gates?
a) OR and NOR
b) AND
c) NAND and NOR
d) NOT
11/11/2022 92
Daily Quiz (CO3)

11. The ___________ of all the variables in direct or complemented from is a maxterm.
a) addition
b) product
c) moduler
d) subtraction
12. _________ is used to implement the Boolean functions.
a) Logical notations
b) Arithmetic logics
c) Logic gates
d) Expressions
13. Inversion of single bit input to a single bit output using _________
a) NOT gate
b) NOR gate
c) AND gate
d) NAND gate
11/11/2022 93
Daily Quiz (CO1)

14. There are _________ numbers of Boolean functions of degree n.
a) n
b) 2(2*n)
c) n3
d) n(n*2)
15. Minimization of function F(A,B,C) = A*B*(B+C) is _________
a) AC
b) B+C
c) B`
d) AB
16. The set for which the Boolean function is functionally complete is __________
a) {*, %, /}
b) {., +, -}
c) {^, +, -}
d) {%, +, *}
11/11/2022 94
Daily Quiz (CO1)

Q1. Define Poset.
Q2. Explain Join and Meet.
Q3. Draw Karnaugh map and simplify the Boolean expression
A’B’C’D’+ A’B C’ D + A’ B’ C D + A’B’C D’ + A’B C D
11/11/2022
3
95
Weekly Assignment
(CO3)

17. ______ and _______ are the two binary operations defined for lattices.
a) Join, meet
b) Addition, subtraction
c) Union, intersection
d) Multiplication, modulo division
18. The graph given below is an example of
a) non-lattice poset
b) semilattice
c) partial lattice
d) bounded lattice
19. In the poset (Z+, |) (where Z+ is the set of all positive integers and | is the divides relation) are the
integers 9 and 351 comparable?
a) comparable
b) not comparable
c) comparable but not determined
d) determined but not comparable
11/11/2022 96
Daily Quiz (CO1)

17. ______ and _______ are the two binary operations defined for lattices.
a) Join, meet
b) Addition, subtraction
c) Union, intersection
d) Multiplication, modulo division
18. The graph given below is an example of
a) non-lattice poset
b) semilattice
c) partial lattice
d) bounded lattice
19. In the poset (Z+, |) (where Z+ is the set of all positive integers and | is the divides relation) are the
integers 9 and 351 comparable?
a) comparable
b) not comparable
c) comparable but not determined
d) determined but not comparable
11/11/2022 97
Daily Quiz (CO1)

1. Algebra of logic is termed as ______________
a) Numerical logic b) Boolean algebra
c) Arithmetic logic d) Boolean number
2. Boolean algebra can be used ____________
a) For designing of the digital computers
b) In building logic symbols
c) Circuit theory
d) Building algebraic functions
a) An arithmetic function with k degrees such that f:Y–>Yk
b) A special mathematical function with n degrees such that f:Yn–>Y
c) An algebraic function with n degrees such that f:Xn–>X
d) A polynomial function with k degrees such that f:X2–>Xn
11/11/2022 98
MCQs (CO1)

4. F(X,Y,Z,M) = X`Y`Z`M`. The degree of the function is ________
a) 2 b) 5
c) 4 d) 1
5. K-map is used for _______
a) logic minimization b) expression maximization
c) summing of parity bits d) logic gate creation
6. Simplify the expression: XY’ + X’ + Y’X’.
a) X’ + Y
b) XY’
c) (XY)’
d) Y’ + X
7. Find the simplified term Y’ (X’ + Y’) (X + X’Y)?
a) XY’
b) X’Y
c) X + Y
d) X’Y’
11/11/2022 99
MCQs (CO1)

8. To display time in railway stations which digital circuit is used?
a) seven segment decoder
b) eight segment encoder
c) 8:3 multiplexer
d) 9 bit segment driver
9. When designing a circuit to emulate a truth table, both Product-of-Sums (POS) expressions and
Sum-of-Products (SOP) expressions can be derived from?
a) k-map
b) NAND gate
c) NOR gate
d) X-NOR gate
10. Addition of two or more bits produces how many bits to construct a logic gate?
a) 108
b) 2
c) 32
d) 64
11/11/2022 100
MCQs (CO1)

11. Who has invented K-map?
a) Maurice Karnaugh
b) Edward Veitch
c) George Boole
d) Adam Smith
12. In Gray coding, the adjacent code values differ by _______
a) single bit
b) 3 bits
c) 10 bits
d) 0 bit
13. Simplify the expression using K-maps: F(A,B,C) = π(0,2,4,5,7).
a) (x+y)(y+z)(x+z)(x’+z’)
b) (x+z’)(y+z)(x+y)
c) (x+y’+z)(x+z’)
d) (y’+z’)(x’+y)(z+y’)
11/11/2022 101
MCQs (CO1)

1. Draw Karnaugh map and simplify the Boolean expression
A’B’C’D’+ A’B C’ D + A’ B’ C D + A’B’C D’ + A’B C D
2. Simplify the following Boolean function using K-map:
F(x,y,z)=Σ(0,2,3,7)
3. Find the values of the Boolean function represented by F (x, y, z) = xy + z’.
4. Simplify the given 2-variable Boolean equation by using K-map F = X Y’ + X’Y + X’Y’.
5. Simplify the given 4-variable Boolean equation by using k-map. F (W, X, Y, Z) = (1, 5, 12, 13).
6. What is De-Morgan’s Law. Explain with example.
7. What are Boolean axioms?
8. Minimize the Boolean function without using K map: A+AB.
9. How many NAND gates are required for XOR gate?
10. Describe Boolean duality principle.
11/11/2022 102
Old Question Papers (CO1)

11/11/2022 103
Old Question Papers (CO1)
11. Design logic circuit using AND, OR, Not gate to solve following problem:
(i) input two bits x,y and output two bits representing x-y(1-1=00,1-0=01,0-0=00,0-1=11)
(ii) input three bits x,y,z and output one bit which is majority of three input bits.
For more Previous year Question papers:
https://drive.google.com/drive/folders/1xmt08wjuxu71WAmO9Gxj2iDQ0lQf-so1

1. Draw Karnaugh map and simplify the Boolean expression: - A’B’C’D’+ A’B C’ D + A’ B’ C D +
A’B’C D’ + A’B C D
2. Simplify the following Boolean function using K-map:
F(x,y,z)=Σ(0,2,3,7)
3. Simplify the given 4-variable Boolean equation by using k-map. F (W, X, Y, Z) = (1, 5, 12, 13).
5. Describe Boolean duality principle.
6. Define SOP and POS.
7. Design logic circuit using AND, OR, Not gate to solve following problem:
(i) input two bits x,y and output two bits representing x-y(1-1=00,1-0=01,0-0=00,0-1=11)
(ii) input three bits x,y,z and output one bit which is majority of three input bits.
10. Simplify the expression: A’(A + BC) + (AC + B’C).
11/11/2022 104
Expected Questions for University Exam (CO1)

11/11/2022 105
UNIT Recap (CO3)
• Boolean algebra simplifies logic circuits to increase work efficiency of
digital device.
• Logic circuits can be built for any binary electric or electronic devices
including switches, relays, electron tubes and transistors.
• The subject enhances one’s ability to develop logical thinking and ability
to problem solving.

11/11/2022 106

Unit3_DIS.ppt

More Related Content

Similar to Unit3_DIS.ppt (20)

Recently uploaded (20)

Unit3_DIS.ppt

Editor's Notes