CHAPTER 2: SET THEORY

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CHAPTER 2 SET THEORY

• Fundamental Discrete Structure
• BCT 1073

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CONTENT

• 2.1 Set Terminologies and Concepts
• 2.2 Operation on Sets
• 2.3 Cartesian Products
• 2.4 Power Sets
• 2.5 Applications of Set Theory

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OBJECTIVES

• At the end of this chapter you should be able to
• Write and define a set in different notation
• Identify the element of a set, empty set, set
  equality, subset, and cardinality of a set.
• Use set operation and identities to solve problem
  in set theory
• Identify the Cartesian product of two or more
  sets.
• Identify the power set of a given set and its
  number of elements.
• Apply the knowledge of set theory into real world
  problem

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2.1 SET TERMINOLOGIES AND CONCEPTS
Lesson Learn 1. Write and define a set in
different notation 2. Identify the element of a
set, empty set, set equality, subset, and
cardinality of a set.
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Introduction

• One of the most basic human impulses is to sort
  and classify things
• Example
• consider yourself how many different categories
  are you a member of
• Every categories have different element
  (information) which describe the characteristics
  of each categories.
• In Mathematics, all these categories are called
  SETS
• We encounter sets in many different ways every
  day of our lives

A SET is an unordered collection of well defined
objects, which called elements or members of the
set. (1 character 1 category)
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Set Notation

• Set generally named with capital letters
• 3 ways to indicate a set description, roster
  set-builder

Example
Set notation
Definition
The set of days in a week containing the elements
Monday, Tuesday, Wednesday, Thursday, Friday,
Saturday and Sunday
Describes the set and it elements in a word
description
Listing the elements of a set inside a pair of
braces
Set A is the set of the days in a week, A
Monday, Tuesday, Wednesday, Thursday, Friday,
Saturday, Sunday
roster
A rule is given that describe the definite
properties an object x must satisfy to qualify
for membership in the set
A xx is the day in a week Read as A is a
set of all element in x such that x is the day in
a week
Set-builder
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Elements of sets

• The objects in a set called the elements, or
  members, of the set.
• Example
• The set of all vowels in the English alphabet
  a,e,i,o,u can be written as
• The set of odd positive integers less than 10 can
  be expressed by
• The set of positive integers less than 100 can be
  denoted by

a is an element of a set A read a belongs to A
or a is an element of A
description notation
Roster notation
Set builder notation
elements
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Elements of sets

• Although sets are usually used to group together
  elements with common properties, there is nothing
  that prevents a set from having seemingly
  unrelated elements.
• For instance, a, 2, Amir, Kuala Lumpur
• is the set containing the four elements
• ? a, 2, Amir and Kuala Lumpur.

a is not an element of a set A read a does not
belongs to A or a is not element of A
Example Given set A 1,2,3, Thus,
Empty set / null set The set that contains no
elements
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Sets of Real Numbers

• These sets play an important role in discrete
  mathematics.

Set of Real Numbers
Natural Numbers / Counting Numbers
Whole Numbers
Integers
Rational Numbers
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The Set of Real Numbers R
Rational Numbers
Irrational Numbers
Nonterminating nonrepeating decimal number
Integers
Fractions
Terminating or repeating decimal numbers
Proper, Improper, Mixed Number
Whole Numbers
Negative Integers
Prime number
Zero
Natural/Counting Numbers/ positive integers
Composite number
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Venn Diagram
U
A
- A useful technique for picturing set
relationship - The Universal Set U is
represented by a rectangle, and subsets of U
are represented by regions lying inside the
rectangle
Universal set The set that contains all the
elements for any specific discussion
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Set Equality

• Two sets are equal if and only if they have the
  same elements.
• The order in which the elements of a set are
  listed does not matter.
• 1,3,5 3,5,1
• If an element of a set is listed more than once,
  it does not matter.
• 1,3,3,3,5,5,5,5 1,3,5

U
A,B
Set equality . Read set A equal set B The two
sets A and B are equal if and only if they have
exactly the same elements. The order in which the
elements are displayed is immaterial.
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Subset
Read A subset B. Set A is a subset of set B if
element of a set A is also an element of a set B
For any set S, (i) (ii)
Empty set is subset of All set
Read A not subset B. Set A is a not a subset of
set B if element of a set A is not an element of
a set B
Example Given set A 2,3, B
1,2,3,4,5 and C 2,3,5 Thus,
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Relationship among various sets of number
Real Numbers R
Rational Numbers Q
Irrational numbers
Irrational Numbers H
Integers Z
Whole numbers W
Natural Numbers N
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Proper Subset
Read A is proper subset of B. Set A is a
proper subset of set B if 1. A subset B
2. there exists at least one element in set B
that is not in set A Set A is
properly smaller than set B
U
A
B
U
U
B
A
B
Read A is not a proper subset of B.
A
Example Given set A 2,3, B 1,2,3,4,5
and C 1,2,3,4,5 Thus,
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Distinct Subset, Finite Infinite Set
Example
Finite set The set that contains finite number
of elements
Example
Infinite set The set that contains infinite
number of elements
The number of distinct subset of a finite set A
where n is the number of elements in set A
Example Given set A 4,d, thus the number of
distinct subset are Which given by , 4, 4,
and 4,d The proper subsets are , 4, and4
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Cardinal Number Equivalent Set

• Let S be a set. If there are exactly n distinct
  elements in S where n is a nonnegative integer,
  we say that S is a finite set and that n is the
  cardinality of S. The cardinality of S is denoted
  by S .
• Example
• Let A be the set of odd positive integers less
  than 10. Then, A 5
• Let S be the set of letters in the English
  alphabet. Then S 26
• Since the null set has no elements, it follows
  that

The Cardinal number of set S. The number of
elements in set S.
Equivalent set. Set A is equivalent to set B if
and only if
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Exercise 2.1

• Write set B 1, 2, 3, 4, 5 in set builder
  notation if N 1, 2, 3, is a
  set of natural number
• Write
  in roster form
• List all subsets of the set A a, b, c
• Given C S, L, A, B
• Determine the number of distinct subsets for the
  set C
• List all the distinct subsets for the set C
• How many of the distinct subset are proper
  subset?

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Exercise 2.1

• 5. Determine whether the following are true or
  false.

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Exercise 2.1

• Which of these statements are TRUE? Why?
• Sets 1,2,3,4, 2,1,4,3 and 3,2,1,4,2,1,1,3,4,2
  ,2 are same
• 1,2,3,4,5 2,5
• a, b, c a, b, c
• 0 0
• What is the cardinality of each of the following
  sets?
• a
• a
• a, a
• a, a, a, a

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2.2 OPERATION ON SET
Lesson Learn Use set operation and identities to
solve problem in set theory
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Union and Intersection
Set Intersection The intersection of sets A and B
is the set containing elements that common to
both set A and set B
Set Union The union of sets A and B is the set
containing all elements that are belong either
set A or set B or both
or
and
U
B
A
A
Examples
Examples
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Exercise 2.2

• Given that U 1,2,3,4,5,6,7
• A 1,2,3, B
  3,4,5,6, C 2,3,4
• List the elements of

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Compliment Disjoint
Compliment Set , A The set of all elements in
the Universal Set that are not in set A
Disjoint Set The set A and B are disjoint if they
have no elements in common
U
U
A
B
A
A
Examples
Examples
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Exercise 2.2

• Given that U is the universal set and
• Shade the sets of
• in separate Venn diagrams

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Exercise 2.2

• Given that U 2,3,4,5,6,7,8,9,10
• A x x is even number
• B x 7 lt 3x lt 25
• C x x is multiple of
  3
• a. By drawing a Venn diagram list the
  elements of the sets
• b. Find an element x such that

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Set Identities
Let U be universal set. If A, B and C are
arbitrary subsets of U, then
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Set Complementation De Morgans Law
Set Complementation If U is a Universal set and
A is a subset of U, then
De Morgans Law Let A and B be the set, then
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Addition Principle
For any finite sets A, B and C

• Example
• 1. Given A and B are disjoint sets with A 25
  and B 33, so
• 2. Given A and B are not disjoint sets with A
  25 and B 33, and A n B 17, so

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Exercise 2.2

• Let A and B be subsets of a universal set U and
  suppose that
• Compute

• Let
• Show that

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Difference Symmetric Different
Difference Set , A - B The set containing of all
elements in A but not in B
Symmetric Difference of A and B The set
containing those elements in either A or B but
not in both A and B
U
U
A
B
A
B
A
Examples
Examples
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Exercise 2.2

• Given that U 2,3,4,5,6,7,8,9,10
• A x x is even number
• B x 7 lt 3x lt 25
• C x x is multiple of
  3
• Find

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Generalized Union Intersection
The Union of a collection of sets The set that
contains those elements that are members of at
least one set in the collection
The Intersection of a collection of sets The set
that contains those elements that are members of
all the sets in the collection
Examples
Answer
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Exercise 2.2
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2.3 CARTESIAN PRODUCTS
Lesson Learn Identify the Cartesian product of
two or more sets.
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Ordered n -tuples

• What is order of element in a collection (set)?
• Since sets are unordered, a different structure
  is needed to represent ordered collections
• This provide by n-tuples.
• The ordered n-tuples is the ordered collection
  of

First element
nth element
Second element
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Ordered Pairs

• Two ordered n-tuples are equal if and only if
  each corresponding pair of their elements is
  equal.
• 2-tuples are called ordered pairs
• The ordered pairs (a, b) and (c, d) are equal if
  and only if a c and b d.
• (a, b) and (b, a) are equal if a b.

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Cartesian Products of 2 Sets

• Let A and B be sets. The Cartesian product of two
  sets A and B, denoted by A B is the set of all
  ordered pairs (a, b) where a is element of A and
  b is element of B.
• It means the set of all ordered pairs with the
  first element of each pair selected from A and
  the second element selected from B.

Cartesian product of A and B
Ordered pairs
and
1, 2 3, 4 (1, 3), (1, 4), (2, 3), (2, 4)
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Number of elements in Cartesian Products

• For all finite sets A and B,

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Cartesian Products of more than 2 Sets

• The Cartesian product of more than two sets is
  defined by

• For all finite sets A and B,

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Exercise 2.3
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2.4 POWER SETS
Lesson Learn Identify the power set of a given
set and its number of elements.
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Power Sets

• Power sets - the set of all subsets.
• The power set of S is denoted by P(S).
• Use in many problems involve testing all
  combinations of elements of a set to see if they
  satisfy some property.
• The empty set and the set itself are members of
  this set of subsets, i.e.
• If a set has n elements, then its power set has
  elements

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Example of Power Set
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Exercise 2.4
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2.5 APPLICATIONS OF SET THEORY
Lesson Learn Apply the knowledge of set theory
into real world problem
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Example

• In a survey of 100 coffee drinkers, it was found
  that 70 take sugar, 60 take cream, and 50 both
  takes sugar and cream with their coffee. How many
  coffee drinkers takes sugar or cream with their
  coffee?

Solution
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Exercise 2.5

• Each of 35 girls in a class takes part in at
  least one of the following activities. Jogging,
  Swimming and Dancing.
• Of the 15 girls who chose jogging,
• 4 also choose swimming and dancing
• 2 choose Jogging only
• 7 choose Swimming but not dancing
• Of the 20 girls who do not choose Jogging,
• x choose both swimming and
  dancing
• 2x choose only dancing
• 2 choose only swimming
• Draw a Venn Diagram to illustrate this
  information
• Find the value of x
• How many girls choose jogging and dancing but not
  swimming?

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Exercise 2.5

• Of 24 students in a class, 18 like to play
  basketball and 12 like to play volleyball. It is
  given that
• U Students in the class
• B Students who like to play Basketball
• V Students who like to play Volleyball
• Let ,
• Draw a Venn Diagram to illustrate this
  information.
• Express in term of x
• Find the smallest possible value of x
• Find the largest possible value of x

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Exercise 2.5

• A leading cosmetic manufacturer advertises its
  products in 3 magazines Cosmo, McCalls and
  Ladies. A survey of 500 customers by the
  manufacturer reveals the following information.
• 180 learned of its products from Cosmo
• 200 learned of its products from McCalls
• 192 learned of its products from Ladies
• 84 learned of its products from Cosmo and
  McCalls
• 52 learned of its products from Cosmo and
  Ladies
• 64 learned of its products from McCalls and
  Ladies
• 38 learned of its products from all three
  magazines
• How many of the customers saw the manufacturers
  advertisement in
• At least one magazines
• Exactly one magazines

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Thank You