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.
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.
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.
It is related to Analysis and Design Of Algorithms Subject.Basically it describe basic of topological sorting, it's algorithm and step by step process to solve the example of topological sort.
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
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It is related to Analysis and Design Of Algorithms Subject.Basically it describe basic of topological sorting, it's algorithm and step by step process to solve the example of topological sort.
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
Propositional calculus (also called propositional logic, sentential calculus, sentential logic, or sometimes zeroth-order logic) is the branch of logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components. Logical connectives are found in natural languages.
Combinatorics - Possible Solutions for given variables2IIM
To find the number of pairs of positive integer values of two variables, a and b such that one is greater than the other, under the given constraints/equation.
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
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
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
MLDM provides an original scientific position in Europe on problems related to pattern recognition, machine learning, classification, modelling, knowledge extraction and data mining. These issues have a strong employability potential for students trained in the field of modelling, prediction or decision support, as well as in the area of the Web, image and video processing, health informatics, etc.
For graphs of mathematical functions, see Graph of a function. For other uses, see Graph (disambiguation). A drawing of a graph. In mathematics graph theory is the study of graphs, which are mathematical structures used.In mathematics, and more specifically in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any acyclic connected graph is a tree. A forest is a disjoint union of trees.
Synchronous Optical Networking (SONET) and Synchronous Digital Hierarchy (SDH) are standardized multiplexing protocols that transfer multiple digital bit streams over optical fiber using lasers or light-emitting diodes (LEDs). Lower data rates can also be transferred via an electrical interface.
The Parallel RLC Circuit is the exact opposite to the series circuit we looked at in the previous tutorial although some of the previous concepts and equations still apply.
The Parallel RLC Circuit is the exact opposite to the series circuit we looked at in the previous tutorial although some of the previous concepts and equations still apply.
Discrete Mathematics - Relations. ... Relations may exist between objects of the same set or between objects of two or more sets. Definition and Properties. A binary relation R from set x to y (written as x R y o r R ( x , y ) ) is a subset of the Cartesian product x × y .
Propositional calculus (also called propositional logic, sentential calculus, sentential logic, or sometimes zeroth-order logic) is the branch of logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components. Logical connectives are found in natural languages.
In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized.
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
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.
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.
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.
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.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
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
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?
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 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.
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2. A set is said to contain its elements.
A set is an unordered collection of objects, called elements or
members of the set.
{1, 2, 3} is the set containing “1” and “2” and “3.” list the members
between braces.
{1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant.
{1, 2, 3} = {3, 2, 1} since sets are unordered.
{1,2,3, …, 99} is the set of positive integers less than 100
{1, 2, 3, …} is a way we denote an infinite set (in this case, the
natural numbers).
∅ = {} is the empty set, or the set containing no elements.
Note: ∅ ≠ {∅}
3. Some examples
• The set V of all vowels in the English alphabet V = {a,
e, i, o, u}.
• The set of positive integers less than 100 can be
denoted by {1, 2, 3, . . . , 99}. ellipses (. . .) are used
when the general pattern of the elements is obvious.
• {a, 2, Fred, New Jersey} is the set containing the four
elements a, 2, Fred, and New Jersey.
4. Element of Set
• A set is an unordered collection of objects
referred to as elements.
• A set is said to contain its elements. We write
a A to denote that a is an element of the set∈
A.
• The notation a A denotes that a is not an∈
element of the set A.
5. Try Yourself
• Let A = {1, 3, { { 1,2}, ø},{ø } }. State whether
the following statements are true or not. Give
reason.
• {{1, 2},{ø } } Aϵ
• {1, 4,{ø } } Aϵ
• ø Aϵ
6. Some Sets
N = {0,1,2,3,…}, the set of natural numbers, non negative integers, (occasionally IN)
Z = { …, -2, -1, 0, 1, 2,3, …), the set of integers
Z+
= {1,2,3,…} set of positive integers
Q = {p/q | p ∈ Z, q ∈Z, and q≠0}, set of rational numbers
R, the set of real numbers
R+, the set of positive real numbers
C, the set of complex numbers.
7. Set builder notation
• Another way to describe a set is to use set
builder notation.
• O = {x | x is an odd positive integer less than
10}
• or, specifying the universe as the set of
positive integers, as
• O = {x Z+ | x is odd and x < 10}.∈
8. Empty Set
• There is a special set that has no elements. This set is called the empty
set,or null set, and is denoted by . The empty set can also be denoted by∅
{ }
Common error is to confuse the empty set with the set { }∅ ∅
• The empty set can be thought of as an empty folder and the set consisting
of just the empty set can be thought of as a folder with exactly one folder
inside, namely, the empty folder.
• Determine whether these statements are true or false.
• a) { } b) {∅ ∈ ∅ ∅ ∈ ∅, { }}∅
• c) { } { } d) { } {{ }}∅ ∈ ∅ ∅ ∈ ∅
9. Subset
• The set A is a subset 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.⊆
11. Try Yourself
• Let A = {1, 5, { { 1,2}, ø},{ø } }. State whether
the following statements are true or not. Give
reason.
• {1, 3, ø} A⊆
• {1, 5, ø} A⊆
• { } A⊆
12. Proper subset
• When we wish to emphasize that 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.
∀x(x A → x B) x(x B x A)∈ ∈ ∧ ∃ ∈ ∧ ∈
13. Try Yourself
A is the set of prime numbers less than 10 , B is the set of odd
numbers less than 10, C is the set of even numbers less than
10.
How many of the following statements are true? Explain
• i. A B⊂ Is ∀x(x A → x B) x(x B x A) true?∈ ∈ ∧ ∃ ∈ ∧ ∈
• ii. B A⊂
• iii. A C⊂
• iv. C A⊂
• v. B C⊂
• vi. C A⊂
• Prime numbers less than 10 are: 2,3,5,7
14. Set Theory - Definitions and notation
A few more:
Is {a} ⊆ {a}?
Is {a} ∈ {a,{a}}?
Is {a} ⊆ {a,{a}}?
Is {a} ∈ {a}?
Yes
Yes
Yes
No
15. Power set
• The power set of S is denoted by P(S).
• The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence,
• P({0, 1, 2}) = { , {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}.∅
• Note that the empty set and the set itself are members of this set of
subsets.
16. Examples
If S is a set, then the power set of S is 2S
= { x : x ⊆ S }.
If S = {a},
If S = {a,b},
If S = ∅,
If S = {∅,{∅}},
We say, “P(S) is
the set of all
subsets of S.”
2S
= {∅, {a}}.
2S
= {∅, {a}, {b}, {a,b}}.
2S
= {∅}.
2S
= {∅, {∅}, {{∅}}, {∅,{∅}}}.
Fact: if S is finite, |2S
| = 2|S|
. (if |S| = n, |2S
| = 2n
) Why?
17. Set Theory - Definitions and notation
Quick examples:
{1,2,3} ⊆ {1,2,3,4,5}
{1,2,3} ⊂ {1,2,3,4,5}
Is ∅ ⊆ {1,2,3}?
Yes! ∀x (x ∈ ∅) → (x ∈ {1,2,3}) holds (for all
over empty domain)
Is ∅ ∈ {1,2,3}? No!
Is ∅ ⊆ {∅,1,2,3}? Yes!
Is ∅ ∈ {∅,1,2,3}? Yes!
18. Set operators
The union of two sets A and B is:
A ∪ B = { x : x ∈ A v x ∈ B}
If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi},
then
A ∪ B = {Charlie, Lucy, Linus, Desi}
A
B
19. Set Theory -Operators
The intersection of two sets A and B is:
A ∩ B = { x : x ∈ A ∧ x ∈ B}
If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi},
then
A ∩ B = {Lucy}
A
B
20. Set Theory -Operators
The intersection of two sets A and B is:
A ∩ B = { x : x ∈ A ∧ x ∈ B}
20
If A = {x : x is a US president}, and B = {x : x is in this room}, then
A ∩ B = {x : x is a US president in this room} = ∅
A
B
Sets whose
intersection is
empty are called
disjoint sets
21. Set Theory -Operators
The complement of a set A is:
A = { x : x ∉ A}
If A = {x : x is bored}, then
A = {x : x is not bored}
A ∅= U
and
U = ∅
U
I.e., A = U – A, where U is the universal set.
“A set fixed within the framework of a theory and consisting of all objects
considered in the theory. “
22. Try yourself
• (i) Identify the area ¬ R(x) I(x) ¬ M(x)˄ ˄
• Determine True or False: ∀x R(x) M(x) I(x) → D(x)˄ ˄
23. Rules of Inference
• We can always use a truth table to show that
an argument form is valid. We do this by
showing that whenever the premises are true,
the conclusion must also be true.
24. Example
• “If you have a current password, then you can log onto the network.”
“You have a current password.”
• Therefore, “You can log onto the network.”
• We would like to determine whether this is a valid argument.
• That is, we would like to determine whether the conclusion “You can log onto the
network” must be true when the premises “If you have a current password, then
you can log onto the network” and “You have a current password” are both true.
• Use p to represent “You have a current password” and q to represent “You can log
onto the network.”
25. • Then, the argument has the form
• p → q
• p
• ∴ q
• where is the symbol that denotes “therefore.” the statement∴ ((p → q) ∧
p) → q is a tautology
• This argument is valid because when (premises) p → q and p are both
true, then the (conclusion) q is true.
p q p q
T T T
T F F
F T T
F F T
26. Try Yourself
• . Verify whether the following argument is valid or not.
a) ¬ q
p → q
∴ ¬ p
b) p → q
q → p
∴ p q˅