The document defines basic concepts about sets including:
- A set is a collection of distinct objects called elements. Sets can be represented using curly brackets or the set builder method.
- Common set symbols are defined such as belongs to (∈), is a subset of (⊆), and is not a subset of (⊄).
- Types of sets like empty sets, singleton sets, finite sets, and infinite sets are described.
- Operations between sets such as union, intersection, difference, and complement are explained using Venn diagrams.
- Laws for sets like commutative, associative, distributive, double complement, and De Morgan's laws are listed.
- An example problem calculates
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
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
Know the basics on sets such as the methods of writing sets, the cardinality of a set, null and universal sets, equal and equivalents sets, and many more.
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
Introduction to Sets and Set Operations. The presentation include contents of a KWLH Chart, Essential Questions, and Self-Assessment Questions. With exploration and formative assessments.
Know the basics on sets such as the methods of writing sets, the cardinality of a set, null and universal sets, equal and equivalents sets, and many more.
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
Introduction to Sets and Set Operations. The presentation include contents of a KWLH Chart, Essential Questions, and Self-Assessment Questions. With exploration and formative assessments.
The presentation mainly focus on effective Math
Study skills, beginning with general study skills, and even when and how to begin your studies. It also tells ways to memorize formulae using mnemonics.
Exploring the world of mathematics Dr. Farhana ShaheenFarhana Shaheen
The presentation is about some information of Math softwares, and Apps that are helpful for teachers and students. It focuses on teaching Maths in different ways to enhance Math learning skills.
Sets & Set Operation
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 11, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
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
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
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Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
1. A = {1, 3, 2, 5}
n(A) = 4
Sets use “curly” brackets
The number of elements
in Set A is 4
Sets are denoted by
Capital letters
A3∈
A7∉
3 is an element of A 7 is not an element of A
SETS
A set is a collection of well defined
distinct objects.
The objects of the set are called elements.
2. {1, 3, 2, 3, 5, 2} We never repeat elements in a set.{1, 2, 3, 5}
Sets can be represented by two
methods:
Rooster or Tabular form:
In this the elements are separated by commas.
e.g. set A of all odd natural numbers less than 10.
A = {1,3,5,7,9}
Set builder method:
In this the common property of the elements is
specified.
e.g. set A of all odd natural numbers less than 10.
A = {x : x is odd natural number less than 10}
3. Symbols Meaning
{ } enclose elements in set
∈ belongs to
⊆ is a subset of (includes equal
sets)
⊂ is a proper subset of
⊄ is not a subset of
⊃ is a superset of
4. Empty set: If a set doesn't contain any elements it is called
the empty set or the null set. It is denoted by ∅ or { }.
Singleton set: It is a set which contains only one element.
e.g. A = {0}
Finite set: It is a set which contains finite number of
different elements.
e.g. A = {a,e,i,o,u}
Infinite set: It is a set which contains infinite number of
different elements.
e.g. A = {x : x ∈ set of natural numbers}
Equal sets: If two or more sets contain the same elements,
they are called equal sets irrespective of the order.
e.g. If A = {1,2,3} and B = {2,3,1}
Then A = B
5. Number of
Elements in Set
Possible Subsets Total Number of
Possible Subsets
{a} {a} ; ∅2
{a , b} {a , b} ; {a} , {b} , ∅4
{a , b , c} {a , b , c} , {a , b} , {a , c} ,
{b , c} , {a} , {b} , {c} , ∅
8
{a , b , c, d}
{a , b , c , d} , {a , b, c} , {a , b ,
d} , {a , c , d} , {b , c , d} , {a , b}
, {a , c} , {a , d} , {a , b} ……
{D} , ∅
The number of possible subsets of a set of size n is 2n
16
6. The Power Set (P)
The power set is the set of all subsets that can be
created from a given set.
Example:
A = {a, b, c}
P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, φ}
Cardinal Number of A Set
It is the number of elements in a set.
Example:
A = {a, b, c, d}
n(A) = 4
7. A B
VENN DIAGRAM
Representation of sets by means of diagrams known as:
Venn Diagrams are named after the English logician, John Venn.
These diagrams consist of rectangles and closed curves usually
circles. The universal set is represented usually by a rectangle
and its subsets by circles.
8. A B
A ∩ B
A ∪ B
A B
This is the union symbol. It means
the set that consists of all
elements of set A and all elements
of set B.
This is the intersect symbol. It
means the set containing all
elements that are in both A and B
9. A B
A - B A B
A B
B- A
A B
Difference Of Sets
The difference of two sets A and B Is the
set of elements which belongs to A but
which do not belong to B. It is denoted
by A – B.
A – B = {x:x A and x B}∈ ∉
B – A = {x:x B and x A}∈ ∉
10. A BDisjoint Sets
ACompliment of a set
The two sets which do not
have any elements in common
are called disjoint sets.
A ∪ B = ∅
U
If U is the universal set, A is a
subset, then compliment of A is
A = {x:x U, x A}∈ ∉
or U - A
11. U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
A = {3,4,5,7,12,15}
B = {2,3,4,6,7,10,14}
A ∩ B = {3,7,4}
A ∪ B = {2,3,4,5,6,7,10,12,14,15}
A - B = {5,12,15}
B – A = {2,6,10,14}
2
3
4
7
5
6
8
9
10
11
12
13
1415
A B
1
VENN DIAGRAM
U
12. A B
C
A ∩ B ∩ C
Only A Only B
Only C
Only A ∩ B not C
Only B ∩ C not AOnly A ∩ C not
13. Commutative Laws:
A ∩ B = A ∩ B and A ∪ B = B ∪ A
Associative Laws:
(A ∩ B) ∩ C = A ∩ (B ∩ C) and (A ∪ B) ∪ C = A ∪ (B ∪ C)
Distributive Laws:
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩C)
Double Complement Law:
(Ac
)c
= A
De Morgan’s Laws:
(A ∩ B)c
= Ac
∪ Bc
and(A ∪ B)c
= Ac
∩ Bc
14. 100 people were surveyed. 52 people in a survey owned a cat.
36 people owned a dog. 24 did not own a dog or cat.
universal set is 100 people surveyed
C D
Set C is the cat owners and Set D is the dog
owners. The sets are NOT disjoint. Some
people could own both a dog and a cat.
24
Since 24
did not own
a dog or
cat, there
must be 76
that do own
a cat or a
dog.
52 + 36 = 88 so
there must be
88 - 76 = 12
people that own
both a dog and
a cat.
1240 24
100
Example: