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.
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.
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
In detail and In very simple method That can any one understand.
If you read this all you doubts about function will be clear.
because i have used very simple example and simple English words that you can pick quickly concept about functions.
#inshallah.
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.
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.
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
In detail and In very simple method That can any one understand.
If you read this all you doubts about function will be clear.
because i have used very simple example and simple English words that you can pick quickly concept about functions.
#inshallah.
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
This presentation includes detailed information on Hypothesis testing for large and small samples, for two sample means. Briefed computational procedure with various case studies.
This presentation includes topics related to sampling and its distributions, estimates related to large samples and small samples using Z test and T test respectively. Also when to use Finite Population Multiplier is explained in detail.
It includes introduction to quantitative techniques; Meaning, Importance applications and Limitations of statistics. Primary vs Secondary Data and their collection methods, Different graphs and their examples. Classification of data, types of data/series etc.
Brief description of the concepts related to correlation analysis. Problem Sums related to Karl Pearson's Correlation, Spearman's Rank Correlation, Coefficient of Concurrent Deviation, Correlation of a grouped data.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
It includes concepts of logarithms including properties and log tables. The methodology to find out values using log tables and anti log tables is also mentioned in a detailed manner. Moreover, questions related to logarithms are mentioned for practice.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
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”.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
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.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
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.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2. SET THEORY
Set: A collection of well defined objects.
Sets are usually denoted by capital letters A, B, C
etc. and their elements are denoted by a, b, c etc.
Few examples:
The collection of vowels in English alphabets. This
set contains five elements i.e. a,e,i,o,u.
The set of 3 cycle companies of India. This set
contains 3 elements i.e. Hero, Avon, Suncross.
The set of 4 rivers in India. This set contains 4
elements i.e. Ganga, Yamuna, Beas, Narmada,
Kaveri.
The collection of good cricket players of India is
not a set
BirinderSingh,AssistantProfessor,PCTE
3. REPRESENTATION OF A SET
Roster
Form Set Buider
Form
BirinderSingh,AssistantProfessor,PCTE
4. ROSTER FORM
This method is also called Tabular Method.
In this, a set is described by listing elements,
separated by commas, within braces { }
The collection of vowels in English alphabets. This
set contains five elements i.e. {a,e,i,o,u }
If A is the set of even natural numbers, then
A = {2, 4, 6, …….. }
If A is the set of all prime numbers less than 11, then
A = {2, 3, 5, 7}
Note:
The order of writing the elements of a set is
immaterial.
An element of a set is not written more than once.
BirinderSingh,AssistantProfessor,PCTE
5. SET BUILDER FORM
This form is also called Property Form.
In this, a set is represented by stating all the
properties P(x) which are satisfied by the
elements x of the set and not by other element
outside the set.
If A is the set of even natural numbers, then
A = {x: x ϵ N, x = 2n, n ϵ N}
A = {x: x is a natural number and x = 2n for n ϵ N}
If A = {0, 1, 4, 9, ……}
A = {x2 : x ϵ N}
If B = The set of all real numbers greater than -3 and
less than 3
B = {-3<x<3 : x ϵ R}
BirinderSingh,AssistantProfessor,PCTE
6. TYPES OF SETS
Empty Set: A set is said to be empty or null or void
set if it has no element and it is denoted by φ.
In Roster Method, it is denoted by { }
Examples:
A = {x: x ϵ N, 7 < x < 8} = φ
B = {x ϵ R : x2 = -2} = φ
C = Any Indian company which is into Automobiles,
Clothing, Plastics, Paper, Processed Food
Note:
{φ} is not a null set, since it contains φ as an element.
{0} is not a null set, since it contains 0 as an element.
BirinderSingh,AssistantProfessor,PCTE
7. TYPES OF SETS
Singleton Set: A set is said to be a singleton set
as it contains only one element.
Examples: {5}, {0}, {-15}, {Mukesh Ambani}
Finite Set: A set whose elements can be listed
or counted.
Examples: {1,2,3}, {5, 10, 15, 20}, {a, e, i, o, u}
Infinite Set: A set whose elements can’t be
listed or counted.
Examples: {1, 2, 3, ………}, All real numbers
BirinderSingh,AssistantProfessor,PCTE
8. TYPES OF SETS
Equivalent Sets: Two finite sets A and B are
equivalent if their cardinal numbers are same.
i.e. n(A) = n(B).
Example: {5, 10, 15, 20, 25}, & {a, e, i, o, u} are
equivalent.
Equal Sets: Two sets A and B are said to be
equal if every element of A is a member of B and
every element of B is a member of A.
Example: If A is the set of even natural numbers
B = {2, 4, 6, …….. }
BirinderSingh,AssistantProfessor,PCTE
9. SUBSETS
In two sets A & B, if every element of A is an
element of B, then A is called subset of B.
If A is a subset of B, we write A ⊆ B.
Thus, A ⊆ B if a 𝜖 A implies a 𝜖 B.
If A is a subset of B, then B is called Super Set of
A.
BirinderSingh,AssistantProfessor,PCTE
10. PROPERTIES OF SUBSETS
The null set is subset of every set i.e. φ ⊆ A
Every set is subset of every set i.e. A ⊆ A
If A ⊆ B and B ⊆ C, then A ⊆ C.
The total number of subsets of a finite set
containing n elements is 2n.
BirinderSingh,AssistantProfessor,PCTE
11. SUBSETS - TYPES
Proper Subset: A set is said to be proper subset
of B, if A is a subset of B, but A is not equal to B.
It is denoted by A ⊂ B. Ex: N ⊂ W ⊂ Z ⊂ Q ⊂ R
Universal Set: It is the set which contains all
the sets under consideration i.e. it is a super set
of each of the given sets. It is denoted by U. Ex: R
Power Set: The collection of all the subsets of a
given set is called the power set. It is denoted by
P(A).
Q1: Find the power set of A = 𝑎, 𝑏, 𝑐
Q2: If A = 1, 2 , find P(A)
BirinderSingh,AssistantProfessor,PCTE
12. PRACTICE PROBLEMS
Q3: Which of the following are set:
i. The collection of all the prime numbers between 23 and
37.
ii. Collection of all factors of 64 which are greater than 8.
iii. The collection of rich persons in India.
Q4: Describe the following sets in Roster Form:
i. A = 𝑥: 𝑥 𝑖𝑠 𝑎 𝑙𝑒𝑡𝑡𝑒𝑟 𝑏𝑒𝑓𝑜𝑟𝑒 𝑓 𝑖𝑛 𝑡𝑒 𝐸𝑛𝑔𝑙𝑖𝑠 𝑎𝑙𝑝𝑎𝑏𝑒𝑡
ii. B = 𝑥 ∈ 𝑁: 𝑥 = 3𝑛, 𝑛 ∈ 𝑁
iii. C = 𝑥 ∈ 𝑍: 𝑥2 + 5𝑥 + 6 = 0
iv. D = 𝑥 ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒, 𝑥 < 30
v. E = 𝑥 ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑑𝑖𝑣𝑖𝑠𝑜𝑟 𝑜𝑓 60
BirinderSingh,AssistantProfessor,PCTE
13. PRACTICE PROBLEMS
Q5: Describe the following sets in Set Builder Form:
i. A = 1 ,
1
2
,
1
3
,
1
4
,
1
5
, … …
ii. B = 0, 3, 8, 15, 24, 35
iii.C = 0
iv. D = 3, 5, 7, 9, … … . . , 47, 49
v. E = 0, 2, 6, 12, 20, 30
Q6: Which of the following are null sets:
i. A = 𝑥: 𝑥 𝑖𝑠 𝑎 𝑙𝑒𝑡𝑡𝑒𝑟 𝑏𝑒𝑓𝑜𝑟𝑒 𝑓 𝑖𝑛 𝑡𝑒 𝐸𝑛𝑔𝑙𝑖𝑠 𝑎𝑙𝑝𝑎𝑏𝑒𝑡
ii. B = 𝑥 ∈ 𝑁: 𝑥 = 3𝑛, 𝑛 ∈ 𝑁
iii.C = 𝑥 ∈ 𝑍: 𝑥2 + 5𝑥 + 6 = 0
iv. D = 𝑥 ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒, 𝑥 < 30
v. E = 𝑥 ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑑𝑖𝑣𝑖𝑠𝑜𝑟 𝑜𝑓 60
BirinderSingh,AssistantProfessor,PCTE
14. OPERATION ON SETS – SET UNION
𝐴 ∪ 𝐵
“A union B” is the set of all elements that are in
A, or B, or both.
This is similar to the logical “or” operator.
15. COMBINING SETS – SET INTERSECTION
𝐴 ∩ 𝐵
“A intersect B” is the set of all elements that are
in both A and B.
This is similar to the logical “and”
16. SET COMPLEMENT
“A complement,” or “not A” is the set of all
elements not in A.
The complement operator is similar to the logical
not, and is reflexive, that is,
A
A A
17. SET DIFFERENCE
The set difference “A minus B” is the set of
elements that are in A, with those that are in B
subtracted out. Another way of putting it is, it is
the set of elements that are in A, and not in B, so
A B
A B A B
22. MUTUALLY EXCLUSIVE AND EXHAUSTIVE
SETS
Definition. We say that a group of sets is
exhaustive of another set if their union is equal to
that set. For example, if
we say that A and B are exhaustive with respect
to C.
Definition. We say that two sets A and B are
mutually exclusive if , that is, the sets
have no elements in common.
A B C
A B
23. VENN DIAGRAM
A – B B – A
BirinderSingh,AssistantProfessor,PCTE
A∩B
A∪B
24. SYMMETRIC DIFFERENCE OF TWO SETS
It is defined as the union of sets A – B and B – A.
It is denoted by AΔB.
A Δ B = (A – B) U (B – A)
BirinderSingh,AssistantProfessor,PCTE
25. PRACTICE PROBLEMS
Q: Find 𝐴 ∪ 𝐵, 𝐴 ∩ 𝐵, 𝐴 − 𝐵, 𝐵 − 𝐴, 𝐴 ∆ 𝐵
i. A = ∅ , B = 1, 2, 3, 4
ii. A = 𝑥: 𝑥 = 3𝑛 + 1, 𝑛 ≤ 5, 𝑛 ∈ 𝑁 ,
B = 𝑥: 𝑥 = 4𝑛 − 5, 𝑛 ≤ 5, 𝑛 ∈ 𝑁
iii. A = 501, 502, 503 , B = 502, 504, 506
iv. A = 𝑥 ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒, 𝑥 < 30, 𝑥 ∈ 𝑁 ,
B = 𝑥 ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑑𝑖𝑣𝑖𝑠𝑜𝑟 𝑜𝑓 60
v. A = 𝑥: 𝑥 = 3𝑛, 𝑛 ≤ 3, 𝑛 ∈ 𝑁 , B = 3, 6, 9, 12, 15
vi. A = 13, 14, 15, 16 , B =
vii.A = 𝑎, 𝑐, 𝑒, 𝑔 , B = 𝑏, 𝑐, 𝑑
BirinderSingh,AssistantProfessor,PCTE
26. PROPERTIES OF A COMPLEMENT
U′ = ∅ & ∅′ = 𝑈
(A′)′ = A
A U A′ = U
A ∩ A′ = ∅
If 𝐴 ⊆ B, then B′ ⊆ A′
Q: A = 1, 3, 5, 8, 9 , B = 5, 10, 11, 12 , U = 1, 2 , … , 12
Verify that A – B = A∩B′ = B′ – A′
BirinderSingh,AssistantProfessor,PCTE
28. APPLICATION OF SETS
𝑛(𝐴 ∪ 𝐵) = n(A) + n(B) if 𝐴 ∩ 𝐵 = 𝜙
𝑛(𝐴 ∪ 𝐵) = n(A) + n(B) – n 𝐴 ∩ 𝐵 if 𝐴 ∩ 𝐵 ≠ 𝜙
𝑛(𝐴 ∪ 𝐵) = n(A – B) + n(B – A) + n 𝐴 ∩ 𝐵
𝑛(𝐴 ∪ 𝐵 ∪ 𝐶) = n(A) + n(B) + n(C) – n 𝐵 ∩ 𝐶 –
n 𝐶 ∩ 𝐴 + n 𝐴 ∩ 𝐵 ∩ 𝐶
𝑛(𝐴) = n(A – B) + n 𝐴 ∩ 𝐵
𝑛(𝐵) = (B – A) + n 𝐴 ∩ 𝐵
BirinderSingh,AssistantProfessor,PCTE
29. PRACTICE PROBLEMS – APPLICATIONS OF
SETS
Q1: In a class of 25 students, 12 have taken
economics, 8 have taken economics but not politics.
Find the number of students who have taken:
i. Politics
ii. Politics but not economics
iii. Both politics & economics
Q2: In a class of 60 boys, there are 45 boys who play
cards & 30 boys who play carom. How many play:
i. Both the games
ii. Cards only
iii. Carom only
BirinderSingh,AssistantProfessor,PCTE
30. PRACTICE PROBLEMS – APPLICATIONS OF
SETS
Q3: In a group of 52 persons, 16 take tea but not coffee &
33 drink tea. Find the number of persons who take:
i. Both tea & coffee
ii. Take coffee but not tea
Q4: For a certain test, a candidate could offer English or
Hindi or both. Total no. of students were 500, from whom
350 appeared in English & 90 appeared in both the
subjects. Find how many:
i. Appeared in English only
ii. Appeared in Hindi
iii. Appeared in Hindi only
BirinderSingh,AssistantProfessor,PCTE
31. PRACTICE PROBLEMS – APPLICATIONS OF
SETS
Q5: A town has a total population of 60000. Out of it,
32000 read HT, 35000 read TOI & 7500 read both
newspapers. Find how many read neither HT not
TOI.
Q6: In a joint family of 12 persons, 7 take tea, 6 take
milk and two take neither. How many members take
both tea & milk?
Q7: In a survey of 60 people, it was found that 25
read Magazine A, 26 read B & 26 read C. 9 read both
A & C, 11 read both A & B, 8 read both B & C and 8
read no magazine at all. Find the number of people:
i. Who read all three magazines
ii. Who read exactly one magazine
BirinderSingh,AssistantProfessor,PCTE