# CHAPTER1THEORY SET

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### DCT1043 Chapter 1 :

1 DCT1043 Chapter 1 Theory Set

### Content :

2 Content 1.1 Introduction 1.2 Set terminologies and Concepts 1.3 Venn Diagram 1.4 Operation on Set 1.5 Applications of Set Theory

### Objectives :

3 Objectives Write and define a set in different notation Identify the element of a set, set equality, subset and empty set Use set complementation, set operation, and De Morgan’s Law to solve problem in set theory Demonstrate and use the Venn diagram to solve problem in set theory Apply the knowledge of set theory into real world problem At the end of this chapter, you should be able to;

### 1.1 Introduction :

4 1.1 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 a collection of well defined objects, which called elements or members of the set.

### Set Notation :

5 Set Notation Set generally named with capital letters 3 ways to indicate a set – description, roster & set-builder description roster Set-builder The set of days in a week containing the elements Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday Example Describes the set and it elements in a word A rule is given that describe the definite properties an object x must satisfy to qualify for membership in the set 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} A = {x|x 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 notation Definition

### 1.2 Venn Diagram :

6 1.2 Venn Diagram - 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 U A Universal set The set that contains all the elements for any specific discussion

### 1.3 Set Terminologies & Concepts :

7 1.3 Set Terminologies & Concepts a is an element of a set A read “a belongs to A” or “a is an element of A” a is not an element of a set A read “a does not belongs to A” or “a is not element of A” 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. U A,B

### 1.3 Set Terminologies & Concepts :

8 1.3 Set Terminologies & Concepts 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 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 U A

### 1.3 Set Terminologies & Concepts :

9 1.3 Set Terminologies & Concepts Empty set / null set The set that contains no elements Is a subset of every set The Cardinal number of set A. The number of elements in set A. Equivalent set. Set A is equivalent to set B if and only if The number of distinct subset of a finite set A where n is the number of elements in set A

### 1.3 Set Terminologies & Concepts :

10 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 1.3 Set Terminologies & Concepts U B A U A U A B B

### Examples :

11 Examples List all subsets of the set A = {a, b, c} 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 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?

### Examples :

12 Examples Determine whether the following are true or false.

### 1.3 Set Terminologies & Concepts :

13 Finite set The set that contains finite number of elements Natural Numbers / Counting Numbers Infinite set The set that contains infinite number of elements Set of Real Numbers (will be discussed more in Chapter 2) Integers Whole Numbers Rational Numbers 1.3 Set Terminologies & Concepts

### 1.4 Operation on Sets :

14 1.4 Operation on Sets 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 Set Intersection The intersection of sets A and B is the set containing elements that common to both set A and set B U A B

### Example :

15 Example 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

### 1.4 Operation on Sets :

16 Compliment Set , A’ The set of all elements in the Universal Set that are not in set A U A A’ Disjoint Set The set A and B are disjoint if they have no elements in common U A B 1.4 Operation on Sets

### Example :

17 Example Given that U = {x: x is an integer, 0 < x < 11} A = {x: 2x >7} B = {x: 3x < 20} List the elements of The sets A and The sets B and State whether each of the following true or false

### Example :

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

### Example :

19 Example Given that U = {2,3,4,5,6,7,8,9,10} A = {x: x is even number} B = {x: 7 < 3x < 25} C = {x: x is multiple of 3} By drawing a Venn diagram list the elements of the sets

### Slide 20:

20 Set Complementation If U is a Universal set and A is a subset of U, then De Morgan’s Law Let A and B be the set, then 1.4 Operation on Sets

### Slide 21:

21 Set Operations Let U be universal set. If A, B and C are arbitrary subsets of U, then Counting the elements in a set For any finite sets A and B 1.4 Operation on Sets

### Example :

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

### 1.5 Applications of Set Theory :

23 1.5 Applications of Set Theory

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