This document provides an overview of key concepts in set theory including:
- The definition of a set as an unordered collection of distinct elements
- Common ways to describe and represent sets such as listing elements, set-builder notation, and Venn diagrams
- Important set terminology including subset, proper subset, set equality, cardinality (size of a set), finite vs infinite sets, power set, and Cartesian product
The document uses examples and explanations to illustrate each concept over 34 pages. It appears to be lecture material introducing students to the basic foundations of set theory.
The document summarizes key concepts in discrete mathematics including sets, operations on sets, functions, sequences, and counting techniques. It defines what a set is, ways to describe sets, and set operations like unions and intersections. Examples are given of common sets like integers, rational numbers, and real numbers. Subsets, the empty set, cardinality (size) of sets, and Venn diagrams are also explained.
Sections Included:
1. Collection
2. Types of Collection
3. Sets
4. Commonly used Sets in Maths
5. Notation
6. Different Types of Sets
7. Venn Diagram
8. Operation on sets
9. Properties of Union of Sets
10. Properties of Intersection of Sets
11. Difference in Sets
12. Complement of Sets
13. Properties of Complement Sets
14. De Morgan’s Law
15. Inclusion Exclusion Principle
This document introduces some basic concepts of set theory, including:
1) Defining sets by listing elements or describing properties. Common sets include real numbers, integers, etc.
2) Basic set operations like union, intersection, difference, and complement.
3) Relationships between sets like subset, proper subset, and equality.
4) Other concepts like partitions, power sets, and Cartesian products involving ordered pairs from multiple sets.
The document introduces basic concepts of set theory, including:
- A set is a collection of distinct objects called elements or members.
- Special sets include the natural numbers, integers, rational numbers, and real numbers.
- Types of sets include subsets, equal sets, empty sets, singleton sets, finite sets, infinite sets, disjoint sets, power sets, and universal sets.
- Cardinal numbers represent the number of elements in a set.
1. Set theory deals with operations between, relations among, and statements about sets.
2. A set is an unordered collection of distinct objects that can be defined by listing its elements or using set-builder notation.
3. Basic set operations include union, intersection, difference, and complement. The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements that are in both A and B.
1. Set theory deals with operations between, relations among, and statements about sets.
2. A set is an unordered collection of distinct objects that can be defined by listing its elements or using set-builder notation.
3. Basic set operations include union, intersection, difference, and complement. The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements that are in both A and B.
A set is a structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.
Set theory deals with operations between, relations among, and statements about sets
1. Set theory deals with operations between, relations among, and statements about sets.
2. A set is an unordered collection of distinct objects that can be defined by listing its elements or using set-builder notation.
3. Basic set operations include union, intersection, difference, and complement. The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements that are in both A and B.
1) The document summarizes key concepts from a discrete mathematics course including sets, operations on sets, functions, sequences, and sums.
2) It covers topics like basic set theory, operations on sets like union and intersection, subsets, power sets, Cartesian products, and cardinality.
3) Examples are provided to illustrate concepts like Venn diagrams, disjoint sets, complements, and set differences.
A set is an unordered collection of objects called elements. A set contains its elements. Sets can be finite or infinite. Common ways to describe a set include listing elements between curly braces or using a set builder notation. The cardinality of a set refers to the number of elements it contains. Power sets contain all possible subsets of a given set. Cartesian products combine elements of sets into ordered pairs. Common set operations include union, intersection, difference, and testing if sets are disjoint.
The document provides notes on set theory concepts including:
1. Sets are well-defined collections of objects called elements or members. Examples of sets include collections of boys in a class or numbers within a specified range.
2. For a collection to be considered a set, the objects must be well-defined so it is clear which objects are members.
3. There are different ways to represent sets including roster form listing elements, statement form describing characteristics of elements, and rule form using logical statements.
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.
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.
The document defines basic set theory concepts including:
- A set is a collection of distinct objects called elements.
- Sets can be represented using curly brackets or set builder notation.
- The empty set, subsets, proper subsets, unions, intersections and complements are defined.
- Cardinality refers to the number of elements in a finite set.
- Power sets are the set of all subsets of a given set.
A study on drug utilization evaluation of bronchodilators using DDD method
The abstract was published as a conference proceeding in a Newsletter after being presented as an e-posture and secured 2nd prize during the scientific proceedings of "National Conference on Health Economics and Outcomes Research (HEOR) to Enhance Decision Making for Global Health" held at Raghavendra Institute of Pharmaceutical Education and Research (RIPER)- Autonomous in association with the International Society for Pharmacoeconomics and Outcomes Research (ISPOR)-India Andhra Pradesh Regional Chapter during 4th& 5th August 2023.
Nasir A. A study on drug utilization evaluation of bronchodilators using the DDD method. RIPER - PDIC Bulletin ISPOR India Andhra Pradesh Regional Chapter Newsletter [Internet]. 2023 Sep;11(51):14. Available from: www.riper.ac.in
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
This document provides an overview of key concepts in set theory, including:
1) Sets can be represented in roster or set-builder form. Common sets used in mathematics include the natural numbers, integers, rational numbers, and real numbers.
2) The empty set contains no elements. Finite sets have a definite number of elements, while infinite sets have an unlimited number of elements.
3) Two sets are equal if they contain exactly the same elements. A set is a subset of another set if all its elements are also elements of the other set.
4) The power set of a set contains all possible subsets of that set.
Set and Set operations, UITM KPPIM DUNGUNbaberexha
This document defines sets and common set operations such as union, intersection, difference, complement, Cartesian product, and cardinality. It begins by defining a set as a collection of distinct objects and provides examples of sets. It then discusses ways to represent and visualize sets using listings, set-builder notation, Venn diagrams, and properties of subsets, supersets, equal sets, disjoint sets, and infinite sets. The document concludes by defining common set operations and identities using membership tables and examples.
The document summarizes key concepts in discrete mathematics including sets, operations on sets, functions, sequences, and counting techniques. It defines what a set is, ways to describe sets, and set operations like unions and intersections. Examples are given of common sets like integers, rational numbers, and real numbers. Subsets, the empty set, cardinality (size) of sets, and Venn diagrams are also explained.
Sections Included:
1. Collection
2. Types of Collection
3. Sets
4. Commonly used Sets in Maths
5. Notation
6. Different Types of Sets
7. Venn Diagram
8. Operation on sets
9. Properties of Union of Sets
10. Properties of Intersection of Sets
11. Difference in Sets
12. Complement of Sets
13. Properties of Complement Sets
14. De Morgan’s Law
15. Inclusion Exclusion Principle
This document introduces some basic concepts of set theory, including:
1) Defining sets by listing elements or describing properties. Common sets include real numbers, integers, etc.
2) Basic set operations like union, intersection, difference, and complement.
3) Relationships between sets like subset, proper subset, and equality.
4) Other concepts like partitions, power sets, and Cartesian products involving ordered pairs from multiple sets.
The document introduces basic concepts of set theory, including:
- A set is a collection of distinct objects called elements or members.
- Special sets include the natural numbers, integers, rational numbers, and real numbers.
- Types of sets include subsets, equal sets, empty sets, singleton sets, finite sets, infinite sets, disjoint sets, power sets, and universal sets.
- Cardinal numbers represent the number of elements in a set.
1. Set theory deals with operations between, relations among, and statements about sets.
2. A set is an unordered collection of distinct objects that can be defined by listing its elements or using set-builder notation.
3. Basic set operations include union, intersection, difference, and complement. The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements that are in both A and B.
1. Set theory deals with operations between, relations among, and statements about sets.
2. A set is an unordered collection of distinct objects that can be defined by listing its elements or using set-builder notation.
3. Basic set operations include union, intersection, difference, and complement. The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements that are in both A and B.
A set is a structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.
Set theory deals with operations between, relations among, and statements about sets
1. Set theory deals with operations between, relations among, and statements about sets.
2. A set is an unordered collection of distinct objects that can be defined by listing its elements or using set-builder notation.
3. Basic set operations include union, intersection, difference, and complement. The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements that are in both A and B.
1) The document summarizes key concepts from a discrete mathematics course including sets, operations on sets, functions, sequences, and sums.
2) It covers topics like basic set theory, operations on sets like union and intersection, subsets, power sets, Cartesian products, and cardinality.
3) Examples are provided to illustrate concepts like Venn diagrams, disjoint sets, complements, and set differences.
A set is an unordered collection of objects called elements. A set contains its elements. Sets can be finite or infinite. Common ways to describe a set include listing elements between curly braces or using a set builder notation. The cardinality of a set refers to the number of elements it contains. Power sets contain all possible subsets of a given set. Cartesian products combine elements of sets into ordered pairs. Common set operations include union, intersection, difference, and testing if sets are disjoint.
The document provides notes on set theory concepts including:
1. Sets are well-defined collections of objects called elements or members. Examples of sets include collections of boys in a class or numbers within a specified range.
2. For a collection to be considered a set, the objects must be well-defined so it is clear which objects are members.
3. There are different ways to represent sets including roster form listing elements, statement form describing characteristics of elements, and rule form using logical statements.
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.
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.
The document defines basic set theory concepts including:
- A set is a collection of distinct objects called elements.
- Sets can be represented using curly brackets or set builder notation.
- The empty set, subsets, proper subsets, unions, intersections and complements are defined.
- Cardinality refers to the number of elements in a finite set.
- Power sets are the set of all subsets of a given set.
Similar to Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx (20)
A study on drug utilization evaluation of bronchodilators using DDD methodDr. Afreen Nasir
The abstract was published as a conference proceeding in a Newsletter after being presented as an e-posture and secured 2nd prize during the scientific proceedings of "National Conference on Health Economics and Outcomes Research (HEOR) to Enhance Decision Making for Global Health" held at Raghavendra Institute of Pharmaceutical Education and Research (RIPER)- Autonomous in association with the International Society for Pharmacoeconomics and Outcomes Research (ISPOR)-India Andhra Pradesh Regional Chapter during 4th& 5th August 2023.
Nasir A. A study on drug utilization evaluation of bronchodilators using the DDD method. RIPER - PDIC Bulletin ISPOR India Andhra Pradesh Regional Chapter Newsletter [Internet]. 2023 Sep;11(51):14. Available from: www.riper.ac.in
Developing Strategies for Adoption of Sustainable Development Goals in Insti...Amgad Morgan
PHD research document summary file about "Developing Strategies for Adoption of Sustainable Development Goals in Institutions", study consists of developing strategies for different organizations and measure the implementation process.
Air New Zealand OSL Terminal (1). PDF...argen tina
Greetings from Oslo Airport to Air New Zealand OSL Terminal! Our terminal is easily accessible at Oslo Airport (OSL) and provides easy boarding and check-in for your travels. Savor welcomes Kiwi hospitality, well-run lounges, and prompt service. For a seamless and enjoyable journey, rely on Air New Zealand whether you're visiting New Zealand or somewhere else. Good luck on your journey!
stackconf 2024 | On-Prem is the new Black by AJ JesterNETWAYS
In a world where Cloud gives us the ease and flexibility to deploy and scale your apps we often overlook security and control. The fact that resources in the cloud are still shared, the hardware is shared, the network is shared, there is not much insight into the infrastructure unless the logs are exposed by the cloud provider. Even an air gap environment in the cloud is truly not air gapped, it’s a pseudo-private network. Moreover, the general trend in the industry is shifting towards cloud repatriation, it’s a fancy term for bringing your apps and services from cloud back to on-prem, like old school how things were run before the cloud was even a thing. This shift has caused what I call a knowledge gap where engineers are only familiar with interacting with infrastructure via APIs but not the hardware or networks their application runs on. In this talk I aim to demystify on-prem environments and more importantly show engineers how easy and smooth it is to repatriate data from cloud to an on-prem air gap environment.
4. Page 4
Set Basics
Examination [5]
1. What is Set?
2. State whether the sets in each pair are equal or not.
a) {a, b, c, d} and {a, c, d, b}
b) {2, 4, 6} and {x | x is an even number, 0<x<8}
5. Page 5
Set Basics
Definition
A set is an unordered collection of objects, called elements or members
of the set. A set is said to contain its elements.
Example
People in a class: {Jui, Sujit, Salman, Koni}
Districts in the BD : {Rajshahi, Dhaka, Nator, … }
Sets can contain non-related elements: {3, a, Potato}
All positive numbers less than or equal to 5: {1, 2, 3, 4, 5}
6. Page 6
Set Basics
Definition
A set is an unordered collection of objects, called elements or members
of the set. A set is said to contain its elements.
Example
People in a class: {Soumita, Moumita, Taohid, Shahriar….}
Districts in the BD : {Rajshahi, Dhaka, Nator, … }
Sets can contain non-related elements: {3, a, Potato}
All positive numbers less than or equal to 5: {1, 2, 3, 4, 5}
7. Page 7
Set Basics
Definition
A set is an unordered collection of objects, called elements or members
of the set. A set is said to contain its elements.
• We write a ∈ A to denote that a is an element of the set A. (∈ = belongs to)
• The notation a ∈ A denotes that a is not an element of the set A. ( ∉ =
not belongs to)
8. Page 8
Set Basics
Definition
A set is an unordered collection of objects, called elements or members
of the set. A set is said to contain its elements.
• We write a ∈ Ato denote that a is an element of the set A. (∈ = belongs to)
• The notation a ∈ A denotes that a is not an element of the set A. ( ∉ =
not belongs to)
• It is common for SETS to be denoted using uppercase letters.
• Lowercase letters are usually used to denote elements of sets.
9. Page 9
Set and Elements
9
Let, A = { 1, a, e, u, i, o, 2, 3}
• Name of the Set?
• 1 ∉ 𝐴 (true or false)
• a ∈ A (true or false)
10. Page 10
How to describe a Set?
10
Three popular methods
1. Word description
Set of even counting numbers less than 10
2. The listing method / Roster method
{2, 4, 6, 8}
3. Set-builder notation
{x | x is an even counting number less than 10}
11. Page 11
How to describe a Set?
11
1. Word description
• Make a word description of the set.
1. Multiples of ten between ten and hundred inclusively
={10, 20,30,40,50,60,70,80,90,100}
2. The counting number multiples of 5 that are less than 35
={5,10,15,20,25,30}
12. Page 12
How to describe a Set?
12
2. The Listing/Roster Method
• Represented by listing its elements between braces {}
• Example : 𝐴 = { 1, 2, 3, 4}
• Sometime use ellipses (...) rather than listing all elements.
• The set of positive integers less than 100 can be denoted by
{1,2,3,...,99}.
13. Page 13
How to describe a Set?
13
3. Set-builder notation
• characterize all elements in the set by stating the property or properties they must have
to be members.
• the set O of all odd positive integers less than 10 can be written as
O = { x | x is an odd positive integer less than 10 }
O = { x ∈ Z+ | x is odd and x < 10 }
Example: B = {x | x is an even integer, x > 0}
• Read as- “B is the set of x such that x is an even integer and x is greater than 0”
• | is read as “such that” and comma as “and”.
14. Page 14
How to describe a Set?
14
3. Set-builder notation with interval
• the notation for intervals of real numbers. When a and b are real
numbers with a < b, we write
• [a, b] = {x | a ≤ x ≤ b}
• [a, b) = {x | a ≤ x < b}
• (a, b] = {x | a < x ≤ b}
• (a, b) = {x | a < x < b}
• Note that [a, b] is called the closed interval from a to b and (a, b) is
called the open interval from a to b.
15. Page 15
• N = {0, 1, 2, 3, …} is the set of natural numbers
• Z = {…, -2, -1, 0, 1, 2, …} is the set of integers
• Z+ = {1, 2, 3, …} is the set of positive integers (a.k.a whole numbers)
– Note that people disagree on the exact definitions of whole numbers and natural numbers
• Q = {p/q | p Z, q Z, q ≠ 0} is the set of rational numbers
– Any number that can be expressed as a fraction of two integers (where the bottom one is not zero)
• R is the set of real numbers
• R+ the set of positive real numbers
• C the set of complex numbers.
15
Often used sets
17. Page 17
Specifying Sets (cont.)
• A = {a, e, i, o, u}
• B = {x | x is an even integer, x > 0}
• E = {x | 𝑥2
− 3𝑥 + 2 = 0}
17
A = {x | x is a letter in English, x is a vowel}
B = {2, 4, 6, …….}
E = {1, 2}
Specifying Set
18. Page 18
Specifying Sets (cont.)
• A = {x: x Z, x is even, x <15 }
• B = {x: x Z, x + 4 = 3 }
• C = {x: x Z, x2 + 2 = 6 }
18
A = {… -8, -6, -4, -2, 0, 2, 4, …., 14}
B = {-1}
E = {-2, +2}
Specifying Set
19. Page 19
Order does not matter-
{1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1}
19
Set - properties
Frequency does not matter
- Consider the list of students in this class
- It does not make sense to list somebody twice
{1,2,2,2,3,3,4,4,4,4,5} is equivalent to {1,2,3,4,5}
20. Page 20
Set Terminology : The universal set
20
Definition
U is the universal set – the set of all of elements (or the “universe”)
from which given any set is drawn.
• For the set {-2, 0.4, 2}, U would be the real numbers
• For the set {0, 1, 2}, U could be the N, Z, Q, R depending on the context
• For the set of the vowels of the alphabet, U would be all the letters of the
alphabet
21. Page 21
Set Terminology : The Empty Set
21
Definition
If a set has zero elements, it is called the empty (or null) set
• Written using the symbol
• Thus, = { } VERY IMPORTANT
• It can be a element of other sets
{ , 1, 2, 3, x } is a valid set
• ≠ { }
The first is a set of zero elements
The second is a set of 1 element [A set with one element is called a singleton set]
22. Page 22
• Represents sets graphically
– The box represents the universal set
– Circles represent the set(s)
• Consider set S, which is the set of all
vowels in the alphabet
• The individual elements are usually not
written in a Venn diagram
22
a e i
o u
b c d f
g h j
k l m
n p q
r s t
v w x
y z
U
S
Venn diagrams
23. Page 23
Set Terminology : Subset
23
Definition
The set A is a sub set of B if and only if every element of A is also an
element of B.
• We use the notation A ⊆ B to indicate that A is a subset of the set B.
We see that A ⊆ B if and only if the quantification ∀x (x∈ A → x ∈ B) is true
24. Page 24
Set Terminology : Subset
24
Example
• If A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6, 7}; A is a subset of B
• If A = {1, 2, 3, 4} and B = {1, 2, 3, 4}; A is a subset of B
• Every nonempty set S has at least two subset
For any set S, S S (S S S)
For any set S, S (S S)
25. Page 25
Set Terminology : Proper Subset
25
Definition
When a set A is a subset of a set B but that A ≠ B, we write A ⊂ B and
say that A is a proper subset of B.
• For A ⊂ B to be true, it must be the case that A ⊆ B and there must exist an
element y of B that is not an element of A.
That is, A is a proper subset of B if and only if
∀x (x ∈ A → x ∈ B) ∧ ∃y (y ∈ B ∧ y ∉A) is true
26. Page 26
Set Terminology : Proper Subset
26
Example
• If A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6, 7}; A is a subset of B and also proper subset
A ⊂ B and A ⊆ B both are true.
• If A = {1, 2, 3, 4} and B = {1, 2, 3, 4}; A is not a proper subset of B but subset.
A ⊆ B but A ⊄ B.
27. Page 27
Set Terminology : Set Equality
27
Definition
Two sets are equal if and only if they have the same elements. We write
A = B if A and B are equal sets.
• Therefore, if A and B are sets, then A and B are equal if and only if
∀x (x ∈ A ↔ x ∈ B)
28. Page 28
Set Terminology : Set Equality
28
Example
• Let two sets A = {1, 2, 3} and B = {3, 2, 1}
then A = B (true or false?)
• Let two sets A = {1, 2, 3} and B = {3, 3, 2, 1, 2, 1}
then A = B (true or false?)
A = {x: x is an odd positive integer less than 10}
B = {1, 3, 5, 7, 9}
A = B ?
29. Page 29
Set Terminology : Set Cardinality
29
Definition
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|.
The term cardinality comes from the common usage of the term cardinal number as
the size of a finite set.
30. Page 30
Set Terminology : Set Cardinality
30
Example
• Let A be the set of odd positive integers less than 10. Then |A| =
• Let S be the set of letters in the English alphabet. Then |S| =
• Let R = {1, 2, 3, 4, 5}. Then |R| =
• || =
• | 𝜙 | =
5
0
5
26
1
31. Page 31
Set Terminology : Finite Set and Infinite Set
31
Definition : Finite 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
• R = {1, 2, 3, 4, 5} finite set
Definition : Infinite Set
A set is said to be infinite if it is not finite.
• The set of positive integers is infinite.
32. Page 32
Set Terminology : Power Set
32
Definition
Given a set S, the power set of S is the set of all subsets of the set S. The
power set of S is denoted by P(S).
• What is the power set of the set {0,1,2}?
• What is the power set of the empty set?
• What is the power set of the set{∅}?
P({})={{}}
P({∅})={∅,{∅}}
33. Page 33
Set Terminology : Cartesian Product
33
Definition
Let A and B be sets. The Cartesian product of A and B, denoted by A x B,
is the set of all ordered pairs (a, b) where a A and b B.
Hence A×B = {(a, b) | a ∈ A ∧ b ∈ B}.
Let, A = {1, 2} and b = {a, b, c}
A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
B x A = ?