SESSION-1 PPT (1).pptx on discrete structures

CREATED BY K. VICTOR BABU
DISCRETE STRUCTURES
23MT1001
Topic:
Sets , Subsets,Power Set, Venn Diagram Set Operations, and
Cartesian Product
Department of Mathematics
Session - 1

AIM OF THE SESSION
To familiarize students with the concepts Set Theory and its Operations
INSTRUCTIONAL OBJECTIVES
This Session is designed to Set Theory , operation of set Theory and Cartesian Product
.
LEARNING OUTCOMES
At the end of this sessions, concept of Set Theory and operation of set Theory and also , Cartesian
Product and Venn Diagram

SESSION INTRODUCTION
01/31/2025 Koneru Lakshmaiah Education Foundation 3
Introduction:
Sets: The concept of a set emerged in mathematics at the end of the 19th century.[7]
The
German word for set, Menge, was coined by Bernard Bolzano in his work
Paradoxes of the Infinite
Georg Cantor, one of the founders of set theory, gave the following definition at
the beginning of his BeiträgezurBegründung der transfinitenMengenlehre
Set Theory Origin
Georg Cantor (1845-1918), a German mathematician, initiated the concept
‘Theory of sets’ or ‘Set Theory’. While working on “Problems on Trigonometric
Series”, he encountered sets, that have become one of the most fundamental
concepts in mathematics

Definition of Sets:
Sets are represented as a collection of well-defined objects or elements and it does not change
from person to person. A set is represented by a capital letter. The number of elements in the
finite set is known as the cardinal number of a set..
Representation of Sets
Sets can be represented in two ways:
1.Roster Form or Tabular form
2.Set Builder Form
Roster Form
In roster form, all the elements of the set are listed, separated by commas and enclosed between curly braces { }.
Example:1 If set represents all the leap years between the year 1995 and 2015, then it would be described using
Roster form as:
A ={1996,2000,2004,2008,2012}

Set Builder Form
In set builder form, all the elements have a common property. This property is not applicable to the objects that do not
belong to the set.
Example 1:
Express the given set in set-builder form: A = {2, 4, 6, 8, 10, 12, 14}
Solution: Given: A = {2, 4, 6, 8, 10, 12, 14}
Using sets notations, we can represent the given set A in set-builder form as,
A = {x | x is an even natural number less than 15}
Example:2
F = {p: p is a set of two-digit perfect square numbers}
F = {16, 25, 36, 49, 64, 81}
We can see, in the above example, 16 is a square of 4, 25 is square of 5, 36 is square of 6, 49 is square of 7, 64 is square of 8
and 81 is a square of 9}.
Even though, 4, 9, 121, etc., are also perfect squares, but they are not elements of the set F, because the it is limited to only
two-digit perfect square.

Types of Sets
The sets are further categorised into different types, based on elements or types of elements. These different types of sets
in basic set theory are:
· Finite set: The number of elements is finite
· Infinite set: The number of elements are infinite
· Empty set: It has no elements
· Singleton set: It has one only element
· Equal set: Two sets are equal if they have same elements
· Equivalent set: Two sets are equivalent if they have same number of elements
· Power set: A set of every possible subset.
· Universal set: Any set that contains all the sets under consideration.
· Subset: When all the elements of set A belong to set B, then A is subset of B
Number sets
• ℕ = “natural numbers”; i.e., {0, 1, 2, 3, ...}, e.g., 86 ∈ ℕ
• ℤ = set of all integers, i.e. {..., -3, -2, -1, 0, 1, 2, 3, ...}, e.g., +86 ∈ ℤ
• ℚ = set of all rational numbers, e.g., 86/1 ∈ ℚ
• ℝ = set of all real numbers, e.g., 86π ∈ ℝ
• ℂ = set of all complex numbers, e.g., 86 + 12i C
∈

What is a Venn diagram?
A Venn diagram uses overlapping circles or other shapes to illustrate the logical relationships
between two or more sets of items. Often, they serve to graphically organize things, highlighting
how the items are similar and different.
Venn diagrams, also called Set diagrams or Logic diagrams, are widely used in mathematics,
statistics, logic, teaching, linguistics, computer science and business. Many people first encounter
them in school as they study math or logic, since Venn diagrams became part of “new math”
curricula in the 1960s.
Venn diagrams are named after British logician John Venn. He wrote about them in an 1880
paper entitled “On the Diagrammatic and Mechanical Representation of Propositions and
Reasoning’s” in the Philosophical Magazine and Journal of Science.

Example Venn diagram
• Say our universe is pets, and we want to compare which type of pet our family might agree on.
• Set A contains my preferences: dog, bird, hamster.
• Set B contains Family Member B’s preferences: dog, cat, fish.
• Set C contains Family Member C’s preferences: dog, cat, turtle, snake.
• The overlap, or intersection, of the three sets contains only dog. Looks like we’re getting a dog.
• Of course, Venn diagrams can get a lot more involved than that, as they are used extensively in various
fields.

Venn diagram purpose and benefits
To visually organize information to see the relationship between sets of items, such as commonalities
and differences. Students and professionals can use them to think through the logic behind a concept
and to depict the relationships for visual communication. This purpose can range from elementary to
highly advanced.
To compare two or more choices and clearly see what they have in common versus what might
distinguish them. This might be done for selecting an important product or service to buy.
To solve complex mathematical problems. Assuming you’re a mathematician, of course.
To compare data sets, find correlations and predict probabilities of certain occurrences.
To reason through the logic behind statements or equations, such as the Boolean logic behind a word
search involving “or” and “and” statements and how they’re grouped.

SESSION DESCRIPTION (Cont..)
Set Operations
Union of two sets: The union of two sets A and B is the set whose elements are all of the elements in
A or in B or in both. The union of sets A and B denoted by A È B is read as ‘’A union B”
Intersection of two sets: The intersection of two sets A and B is the set whose elements are all of the
elements common to both A and B. The intersection of the sets of’ A ’and ‘B’ is denoted by A I B and is
read as “A intersection B”.

Difference of sets: If A and B are subsets of the universal set U, then the relative complement of B in
A is the set of all elements in A which are not in B. It is denoted by A – B thus: A – B = {x | x ∈ A and x
Ï B}
A-B B-A
Complement of a set: If U is a universal set containing the set A, then U – A is called the
complement of A. It is denoted by A1
. Thus A 1
= {x: x Ï A}
Complement of A

Symmetric difference of two sets: If A and B are subsets of the universal set U, then the
symmetric difference of A and B, also known as the disjunctive union, is the set of elements
which are in either of the sets, but not in their intersection and denoted by AB
AB=(A-B)U(B-A)
Operations on sets
• Operations
▪ Binary: union, intersection, difference
▪ Unary: complement
▪ Other: cartesian product
• Context for operations
▪ Operations are defined within set S, aka universal set
▪ Operands and results are elements of (S), i.e., subsets of S
℘
▪ Binary operations act on any two subsets of S to produce a subset of S
▪ Unary operations act on any subset of S to produce a subset of S

Algebraic Properties of Set Operations
Property 1. Commutative property
Intersection and union of sets satisfy the commutative property.
A B = B A
⋂ ⋂
A B = B A
⋃ ⋃
Property 2. Associative property
Intersection and union of sets satisfy the associative property.
(A B) C = A (B C)
⋂ ⋂ ⋂ ⋂
(A B) C = A (B C)
⋃ ⋃ ⋃ ⋃
Property 3. Distributive property
Intersection and union of sets satisfy the distributive property.
A (B C) = (A B) (A C)
⋃ ⋂ ⋃ ⋂ ⋃
A (B C) = (A B) (A C)
⋂ ⋃ ⋂ ⋃ ⋂
Property 4. Identity
A = A
⋃∅
A U = A
⋂
Property 5. Complement
A A
⋃ C
= U
A A
⋂ C
= ∅

Property 6. Idempotent
A A = A
⋂
A A = A
⋃
Property 7. De Morgan’s Laws
For any two finite sets A and B;
(i) A – (B ∩ C) = (A – B) U (A – C)
(ii)A – (B U C) = (A – B) ∩ (A – C)
De Morgan’s Laws can also be written as:
(i)(A ∩ B)’ = A’ U B’
(ii) (A U B)’ = A’ ∩ B’
Cartesian Product of sets
If set A and set B are two sets then the cartesian product of set A and set B is a set of all ordered pairs (a,b),
such that a is an element of A and b is an element of B. It is denoted by A × B.
We can represent it in set-builder form, such as:
A × B = {(a, b) : a A and b B}
∈ ∈
Example: set A = {1,2,3} and set B = {Bat, Ball}, then;
A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}

Problems and Solutions
Q.1: If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find (A ∩ B) ∪ (A ∩ C).
Solution: A ∩ B = {a, b, c} ∩ {c, d, e, f}
A ∩ B = { c }
A ∩ C = { a, b, c } ∩ { c, d, e }
A ∩ C = { c }
∴ (A ∩ B) ∪ (A ∩ C) = { c }
Q.2: Give examples of finite sets.
Solution: The examples of finite sets are:
Set of months in a year
Set of days in a week
Set of natural numbers less than 20
Set of integers greater than -2 and less than 3
Q.3: If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {3, 5, 7, 9, 11} and B = {7, 8, 9, 10, 11}, Then find (A – B) .
′
Solution: A – B is a set of member which belong to A but do not belong to B
∴ A – B = {3, 5, 7, 9, 11} – {7, 8, 9, 10, 11}
A – B = {3, 5}
According to formula,
(A − B) = U – (A – B)
′
∴ (A − B) = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} – {3, 5}
′
(A − B) = {2, 4, 6, 7, 8, 9, 10, 11}.
′

Self-assessment questions:
1.Question: The set of intelligent students in a class is
(a) A null set
(b) A singleton set
(c) A finite set
(d) Not a well defined collection
2. QuestionWhich of the following is the empty set?
(a) {x : x is a real number and x2
– 1 = 0} (b) {x : x is a real number and x2
+ 1 = 0}
(c) {x : x is a real number and x2
– 9 = 0 (d) {x : x is a real number and x2
= x + 2
3 Question: . Verify using the Venn diagram:
(i) A – B = A ∩ BC
(ii) (A ∩ B)C
= AC
B
∪ c
4.Question:
.If A = {3, 7}, find (a)P( A) (b) What is | A|? (c) What is | P( A)|?

CLASSROOM DELIVARY PROBLEMS
1.Find the power set of the Set {a, b, c ,d}
2.If U = {1,2,3,4,5,6,7,8}, A = {1,2,4}, B = {3,4,5,6}, C = {3,4,7}, find
a) (A ∩ B) (A ∩ C).
∪
b) (A – B)1
c) A-(B U C)
d) (A U B)1
e)(A-B) X (A-C)
3.Draw the Venn diagrams for following
a) (A-B) U (A-C)
b) A-(B U C)
C) (AU B)1
4. What will be the Cardinality of the Power Set of {a,b,c,d,e}

Tutorial Problems
1. Find the power set of the Set {a, b, c }
2.If U = {1,2,3,4,5,6,7,8}, A = {1,2,3}, B = {3,4,5,6}, C =
{3,4,8}, find
a) (A ∩ B) (A ∩ C).
∪
b) (A – B)1
c) A-(B U C)
d) (A U B)1
3.Draw the Venn diagrams for following
a) A-B
b) A-(B U C)
c)AI
U BI
4. If A = {1,2,3,4}; B = {2,3,4,5} and c={4,5,6,7}then find
a) A × (B∩C) = (A×B) ∩ (A×C),
b)A × (B C) = (A×B) (A×C), and
∪ ∪
c)A × (BC) = (A×B) (A×C)

HOME ASSIGNMENT PROBLEMS
1.Find the power set of the Set {a, b, c ,d}
2. Verify using the Venn diagram:
(i) A – B = A ∩ BC
(ii) (A ∩ B)C = AC BC
∪
3. For given two sets P and Q, n(P – Q) = 24, n(Q – P) = 19 and n(P ∩ Q) = 11, find:
(i) n(P)
(ii) n(Q)
(iii) n (P Q)
∪
4..If A = {1,2,4}, B = {3,4,5,}, C = {3,4,7}, find
a) (A ∩ B) (A ∩ C).
∪
b) (A X B)
c) A X (B U C)
d) (A U B) X (A U C)

REFERENCES FOR FURTHER LEARNING OF THE SESSION
Reference Books:
Text Books:1.Kenneth H. Rosen, Discrete mathematics and its applications, McGraw Hill
Publication, 2022.
2.Bernard Kolman, Robert Busby, Sharon C. Ross, Discrete Mathematical Structures, Sixth Edition
Pearson Publications, 2015
Reference Books:
1.Joe L Mott, Abraham Kandel, Theodore P Baker, Discrete Mathematics for Computer Scientists and
Mathematicians, Printice Hall of India, Second Edition, 2008.
2. Tremblay J P and Manohar R, Discrete Mathematical Structures with Applications to Computer
Science, Tata McGraw Hill publishers, 1st edition, 2001,India.
Sites and Web links:
https://www.youtube.com/watch?v=tyDKR4FG3Yw
https://www.youtube.com/watch?v=3SjAnxpVAPg

THANK YOU
Team – Discrete Structures

SESSION-1 PPT (1).pptx on discrete structures

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