This document provides an overview of basic set concepts and notations. It begins by explaining that a set is a collection of distinct objects, which can be anything. Sets are represented using curly brackets and elements are separated by commas. A set can be finite or infinite depending on the number of elements. There are various relationships between sets such as subsets, supersets, disjoint sets, equivalent sets and more. Set operations like union and intersection are demonstrated using Venn diagrams. The document concludes by providing examples and exercises to solidify understanding of fundamental set concepts.
After going through this module, you are expected to:
• define well-defined sets and other terms associated to sets
• write a set in two different forms;
• determine the cardinality of a set;
• enumerate the different subsets of a set;
• distinguish finite from infinite sets; equal sets from equivalent sets
• determine the union, intersection of sets and the difference of two sets
After going through this module, you are expected to:
• define well-defined sets and other terms associated to sets
• write a set in two different forms;
• determine the cardinality of a set;
• enumerate the different subsets of a set;
• distinguish finite from infinite sets; equal sets from equivalent sets
• determine the union, intersection of sets and the difference of two sets
A power point presentation on the topic SETS of class XI Mathematics. it includes all the brief knowledge on sets like their intoduction, defination, types of sets with very intersting graphics n presentation.
Discrete Mathematics - Sets. ... He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.
This slide help in the study of those students who are enrolled in BSCS BSC computer MSCS. In this slide introduction about discrete structure are explained. As soon as I upload my next lecture on proposition logic.
About sets , definition example, and some types of set. Explained the some operation of set like union of set and intersection of set with usual number example
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
A power point presentation on the topic SETS of class XI Mathematics. it includes all the brief knowledge on sets like their intoduction, defination, types of sets with very intersting graphics n presentation.
Discrete Mathematics - Sets. ... He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.
This slide help in the study of those students who are enrolled in BSCS BSC computer MSCS. In this slide introduction about discrete structure are explained. As soon as I upload my next lecture on proposition logic.
About sets , definition example, and some types of set. Explained the some operation of set like union of set and intersection of set with usual number example
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
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Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
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Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
1. Chapter 1
Basic Concepts of Sets
Objectives:
At the end of this module, the student will be able to:
1. understand the concept of sets and its notations;
2. distinguish the different set representations schemes;
3. identify subsets and equality of sets; and
4. draw the Venn diagrams.
1.1 Introduction
George Cantor, in the early 1870’s founded the study of sets and its application.
Newmann, Frege and other mathematicians further developed it.
The theory of sets is one of the most important tool and foundation for higher
mathematics. We study algebra, geometry and almost every other area of
contemporary mathematics, which involved sets.
1.2 Sets
A fundamental concept in all branches of mathematics is that of a set. Basically,
a set is a well-defined list, collection, or class of objects. By the term "well-defined",
we mean that there is some criterion that enables us to say that an object belongs to
the given set or otherwise. The objects in a set can be anything such as, people, letters
in the English alphabet, list of cities in the world, etc. These objects are called the
elements of the set. The following examples illustrate the above definition.
Example 1.1 The numbers 1,2,3,4, and 5.
Example 1.2 The distinct letters in the word " mathematics ".
Example 1.3 The students enrolled in CSC 141-ESET class.
Example 1.4 The vowels in the English alphabet.
Example 1.5 The integers between 9 and 21.
Lomesindo T. Caparida, PhD C1 - Basic Concept of Sets
2. 1.3 Set Notations and its Representation
Sets will be denoted by capital letters such as A, B, C, X, Y, etc. The members in
our set are called elements. These elements can be represented by lower case letters
such as a, b, c, …. The elements of the set are separated by a comma and enclosed by
a pair of curly braces { }.
Example 1.6 As in Example 1.1, we can write this as A={1, 2, 3, 4, 5}.
Example 1.7 As in Example 1.4, we can also write this as B={a, e, i, o, u}.
Example 1.8 As in Example 1.5, we can represent this as C={x|9<x<21, x is an integer}.
The elements of a set can either be enumerated or be described in accordance
with their behavior or in terms of their characteristics. Listing explicitly the elements in
the set is called enumeration or tabular method. If the elements in the set are
described or grouped in accordance with their properties or behavior then the set
representation is called set-builder form or rule method.
A shown in Examples 1.6 and 1.7, the sets A and B are represented in tabular
form while in Example 1.8, set C is represented in set-builder form. It is common in
mathematics that the vertical line "|" is read as "such that", as show in Example 1.8.
If an object x is a member of a set A, i.e., A contains x as one of its elements,
then we can write .
A
x∈ However, if x does not belongs to set A, then we write .
A
x∉
Consider set A={1, 2, 3, 4, 5}. Then we can write A
∈
1 , A
∈
2 , A
∈
3 , A
∈
4 ,
and A
∈
5 . Clearly, if "10 does not belongs to set A", then we can write it as " A
∉
10 ".
1.4 Universal Set and Null Set
All sets under investigation will be likely be subsets of a fixed set called the
universal set. We call a universal set as universe of discourse, denoted by U. Also, a
null set (or empty set) is a set that contains no element. This set is denoted by the
symbol ∅ or { }.
Example 1.9 In plane geometry, the Universal set consists of all points in the plane.
Example 1.10 In the English alphabet, the Universal set consists of all letters in the
English alphabet.
Example 1.11 Let H be a set of people who are brainless. Then H is an empty set since
no person is brainless.
Example 1.12 Let D={x| x =13 and x is even}. Then D is ∅ because this set does not
exist.
Lomesindo T. Caparida, PhD C1 - Basic Concept of Sets
2
3. 1.5 Subset and Proper Subset
Let A and B be two non-empty sets. Set A is a subset of set B, denoted by
B
A ⊆ , if every element of A also belongs to set B. That is, A is a subset of B or B
A =
Mathematically,
B
x
A
x
B
A ∈
→
∈
∀
→
⊆ .
We can also say that A is a proper subset of B, denoted by B
A ⊂ , if every
element of A is found in B and there exists at least an element in B which is not found
in A. Mathematically, A
x
B
x
and
B
x
A
x
B
A ∉
∋
∈
∃
∈
→
∈
∀
→
⊂ .Here, B is a
superset that contains A, i.e. A
B ⊃ .
However, if there is an element of A that is not found in B then A is not a subset
of B and it is denoted by B
A ⊄ .
Remarks:
1. B
x
A
x ∈
→
∈
∀ reads as " for every element x in set A implies x belongs to set B.
2. A
x
B
x ∉
∋
∈
∃ reads as " there exists at least an element x in B such that x is not
found in A.
3. An empty set ∅ is always a subset to any set.
Example 1.13 Let E={1, 3, 5} and F={5, 3, 1}. Then E is a subset of F denoted by
F
E ⊆ .
Example 1.14 Let A={3, 5} and B={1, 2, 3, 4, 5}. Then A is a proper subset of B, i.e.,
B
A ⊂ or A
B ⊃ .
Example 1.15 Let D={a, b, c, d} and G={a, c, d}. Then D is not a subset of G, i.e.,
G
D ⊄ since G
b∉ .
1.6 Cardinality of a Set and Power Set
Let A be a non-empty set. The cardinality of A, denoted by n(A) or |A|, gives
the number of elements that belongs to A. Then the power set of A, denoted by )
(A
P
, is the set of all possible subsets of A and its cardinality is given by
)
(
2
))
(
( A
n
A
P
n = .
Example 1.16 Let A={a, b}. Then n(A)=2. Thus, set A contains two elements. The
cardinality of the power set of A is 4. Thus, P(A)={∅, {a}, {b}, {a, b}, A}.
Lomesindo T. Caparida, PhD C1 - Basic Concept of Sets
3
4. 1.7 One-To-One Correspondence
Let two sets A and B be defined as A={a1,a2,…,an} and B={b1,b2,…, bn}. Then A
and B are said to be in one-to-one correspondence when there exists a pairing of
the a's and the b's such that each a corresponds to one and only one b and each b
corresponds to one and only one a.
Example 1.18 Let A={1, 2, 3, 4} and set B={a, b, c, d }. Then there is a one-to-one
correspondence with element of A and B as shown below.
1.8 Finite and Infinite Sets
Sets can be finite or infinite. A set is finite if it consists of a specific number of
different elements, that is, if in counting the different members of the set then the
counting process can come to and end. Otherwise, a set is infinite. The following
examples below illustrate the idea about finite and infinite sets.
Example 1.18 Let O={x| 5< x<12, x is an odd integer}. Then O is finite.
Example 1.19 Let M be the set of months of the year. Then M is finite.
Example 1.20 Let E={ 2, 4, 6, …}. Then E is infinite.
Example 1.21 Let N be the set of positive integers. Then N is infinite.
1.9 Equality of Sets
Let A and B be two non-empty sets. Sets A and B are said to be equal if an only
if both sets have the same elements, that is, A is a subset of B and B is a subset of A.
Mathematically, A
B
and
B
A
B
A ⊆
⊆
↔
= .
Example 1.23 Sets E and F in Example 1.13 are equal sets because both sets E and F
have exactly the same elements.
Example 1.24 Let A={x|2<x<10, x is an odd integer} and B={3, 5, 7}. Then B
A ≠ .
Lomesindo T. Caparida, PhD C1 - Basic Concept of Sets
4
A B
1 a
2 b
3 c
4 d
5. 1.10 Equivalent Sets
Let A and B be two non-empty sets. Then A and B are said to be equivalent,
denoted by A∼B, if there exists a one-to-one correspondence between the elements of
the two sets.
Example 1.25 Let A={a, b, c} and B={apple, ball, cart}. Sets A and B are equivalent
sets because n(A)=n(B)=3. Hence, we can have a one-to-one
correspondence between sets A and B.
1.11 Disjoint Sets
Let A and B be two non-empty sets. Then A and B are disjoint sets if and only
if they have no elements in common. This concept is illustrated in Figure 1 below.
Figure 1. A and B are disjoint sets.
Example 1.26 Let A={0, 1} and B={1, 2}. Then A and B are not disjoint because there
is an element, namely 1, is found in both sets.
Example 1.27 Let C={a, b, c, d, e} and D={1, 2, 3, 4}. Sets C and D are disjoint
because there are no elements in common.
1.12 Venn-Euler Diagrams
A graphical representation of the relationship of sets is the so-called Venn-
Euler diagrams or simply, Venn diagrams. In this diagram, sets are represented in
circles. Area within each circle represents elements in that particular set.
Figure 1.2 Venn Diagram of B
A ∪ .
Lomesindo T. Caparida, PhD C1 - Basic Concept of Sets
5
U
A
U
A B
6. Chapter 1: Sets and Subsets
Exercises 1
Name: ___________________ Date:________
Year/Section:______________ ID#:________
Answer the following:
1. Translate the following statements using set notation. Write the answers on the
space provided.
a. x belongs to A __________
b. y does not belong to U __________
c. A is a proper subset of B __________
d. S is not a subset of T __________
e. B is a superset of A __________
2. Let S = {1, 2, 3, 4, 5} and T = {1, 3, 5}. Write T when the statement is true and
F when the statement is false on the space provided.
a. S
∈
3 __________
b. T
⊂
5 __________
c. S
⊆
}
4
,
1
{ __________
d. S
T ⊂ __________
e. T
∈
φ __________
f. S
∈
}
2
{ __________
g. T
S = __________
h. T
S ⊆ __________
i. T
∈
2
,
1 __________
j. S
∉
}
{φ __________
3. Given A={a, b, c}. Find the power set of A.
4. Give an example of two equivalent sets.
5. Given two sets A and B. Use Venn diagram to show that
B
A ∩ B
A ⊂
Lomesindo T. Caparida, PhD C1 - Basic Concept of Sets
6