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
Relations and their Properties
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 9, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
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.
Relations and their Properties
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 9, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
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.
A quotient construction defines an abstract type from a concrete type, using an equivalence relation to identify elements of the concrete type that are to be regarded as indistinguishable. The elements of a quotient type are equivalence classes: sets of equivalent concrete values. Simple techniques are presented for defining and reasoning about quotient constructions, based on a general lemma library concerning functions that operate on equivalence classes. The techniques are applied to a definition of the integers from the natural numbers, and then to the definition of a recursive datatype satisfying equational constraints.
Published in ACM Trans. on Computational Logic 7 4 (2006), 658–675.
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
A quotient construction defines an abstract type from a concrete type, using an equivalence relation to identify elements of the concrete type that are to be regarded as indistinguishable. The elements of a quotient type are equivalence classes: sets of equivalent concrete values. Simple techniques are presented for defining and reasoning about quotient constructions, based on a general lemma library concerning functions that operate on equivalence classes. The techniques are applied to a definition of the integers from the natural numbers, and then to the definition of a recursive datatype satisfying equational constraints.
Published in ACM Trans. on Computational Logic 7 4 (2006), 658–675.
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
the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown
Matrices
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 30, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Closures of Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 20, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Representing Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 14, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben
Recursive Definition & Algorithms, and Program Correctness
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 23, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Mathematical Induction
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 18, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Integer Representations & Algorithms
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Growth of Functions
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 6, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Predicates & Quantifiers
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 5, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Proofs Methods and Strategy
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 10, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Predicates & Quantifiers
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 1, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Propositional Equivalences
CMSC 56 | Discrete Mathematical Structure for Computer Science
August 23, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Propositional Logic
CMSC 56 | Discrete Mathematical Structure for Computer Science
August 17, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
La Solidaridad and the Propaganda Movement
PI100 Life & Works of Rizal
March 2018
by: Allyn Joy Calcaben, & Jemwel Autor
University of the Philippines Visayas
Computer Vision: Feature matching with RANSAC Algorithmallyn joy calcaben
Computer Vision: Feature matching with RANdom SAmple Consensus Algorithm
CMSC197.1 Introduction to Computer Vision
April 2018
by: Allyn Joy Calcaben, Jemwel Autor, & Jefferson Butch Obero
University of the Philippines Visayas
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.
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.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
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.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
5. Equivalence
Definition 2
Two elements a and b that are related by an equivalence relation
are called equivalent.
The notation a ~ b is often used to denote that a and b are
equivalent elements with respect to a particular equivalence
relation.
6. Example
Let R be the relation on the set of integers such that aRb if and
only if a = b or a = -b. Is R an equivalence relation?
7. Solution
Let R be the relation on the set of integers such that aRb if and
only if a = b or a = -b. Is R an equivalence relation?
YES
8. Example
Let R be the relation on the set of real numbers such that aRb if
and only if a − b is an integer. Is R an equivalence relation?
9. Solution
Let R be the relation on the set of real numbers such that aRb if
and only if a − b is an integer. Is R an equivalence relation?
YES
10. Example
Let R be the relation on the set of real numbers such that aRb if
and only if a − b is a positive integer. Is R an equivalence relation?
11. Solution
Let R be the relation on the set of real numbers such that aRb if
and only if a − b is a positive integer. Is R an equivalence relation?
NO
14. Equivalence Class
Definition 3
Let R be an equivalence relation on a set A. The set of all elements
that are related to an element a of A is called the equivalence
class of a.
15. Equivalence Class
Definition 3
Let R be an equivalence relation on a set A. The set of all elements
that are related to an element a of A is called the equivalence
class of a.
The equivalence class of a with respect to R is denoted by [a]R.
When only one relation is under consideration, we can delete the
subscript R and write [a] for this equivalence class.
16. Equivalence Class
In other words, if R is an equivalence relation on a set A, the
equivalence class of the element a is [a]R = { b | (a, b) ∈ R }.
If b ∈ [a]R, then b is called a representative of this equivalence
class
17. Example
Given the equivalence relation R = {(1,1), (1,2), (1,3), (2,1), (2,2),
(2,3), (3,1), (3,2), (3,3), (4,4), (5,5)} on set {1, 2, 3, 4, 5}.
Find [1], [2], [3], [4], and [5].
18. Solution
Given the equivalence relation R = {(1,1), (1,2), (1,3), (2,1), (2,2),
(2,3), (3,1), (3,2), (3,3), (4,4), (5,5)} on set {1, 2, 3, 4, 5}.
Find [1], [2], [3], [4], and [5].
[1] = {1, 2, 3} [4] = {4}
[2] = {1, 2, 3} [5] = {5}
[3] = {1, 2, 3}
19. Example
What is the equivalence class of an integer for the equivalence
relation such that aRb if and only if a = b or a = −b?
20. Solution
What is the equivalence class of an integer for the equivalence
relation such that aRb if and only if a = b or a = −b?
[a] = { –a, a }.
This set contains two distinct integers unless a = 0.
21. Solution
What is the equivalence class of an integer for the equivalence
relation such that aRb if and only if a = b or a = −b?
[a] = { –a, a }.
This set contains two distinct integers unless a = 0.
For instance, [ 7 ] = { −7, 7 },
[−5] = {−5, 5 }, and
[0] = {0}
23. Partial Orderings
Definition 1
A relation R on a set S is called a partial ordering or partial order if
it is reflexive, antisymmetric, and transitive.
24. Partial Orderings
Definition 1
A relation R on a set S is called a partial ordering or partial order if
it is reflexive, antisymmetric, and transitive.
A set S together with a partial ordering R is called a partially
ordered set, or poset, and is denoted by (S,R). Members of S are
called elements of the poset.
26. Solution
Is the “divides” relation a partial ordering on the set of positive
integers?
(Z+, |) is a poset
27. Partial Ordering
Definition 2
The elements a and b of a poset (S, ≼ ) are called comparable if
either a ≼ b or b ≼ a.
When a and b are elements of S such that neither a ≼ b nor b ≼ a,
a and b are called incomparable.
29. Solution
Are all the elements in the poset (Z+, |) comparable?
NO
For instance: 3 | 7 and 7 | 3
30. Partial Ordering
The adjective “partial” is used to describe partial orderings
because pairs of elements may be incomparable.
When every two elements in the set are comparable, the relation
is called a total ordering.
31. Partial Ordering
Definition 3
If (S, ≼ ) is a poset and every two elements of S are comparable, S
is called a totally ordered or linearly ordered set, and ≼ is called a
total order or a linear order. A totally ordered set is also called a
chain.
33. Solution
Are all the elements in the poset (Z, ≤) comparable?
Yes
The poset (Z, ≤) is totally ordered.
34. Hasse Diagrams
Consider the directed graph for the partial ordering { (a, b) | a ≤ b } on the
set {1, 2, 3, 4}
35. Hasse Diagrams
Consider the directed graph for the partial ordering { (a, b) | a ≤ b } on the
set {1, 2, 3, 4}
1. Start with the directed graph for this relation
36. Hasse Diagrams
Consider the directed graph for the partial ordering { (a, b) | a ≤ b } on the
set {1, 2, 3, 4}
1. Start with the directed graph for this relation
2. Because a partial ordering is reflexive, a loop (a, a) is present at every
vertex a. Remove these loops.
37. Hasse Diagrams
Consider the directed graph for the partial ordering { (a, b) | a ≤ b } on the
set {1, 2, 3, 4}
1. Start with the directed graph for this relation
2. Because a partial ordering is reflexive, a loop (a, a) is present at every
vertex a. Remove these loops.
3. Next, remove all edges that must be in the partial ordering because of
the presence of other edges and transitivity.
38. Hasse Diagrams
Consider the directed graph for the partial ordering { (a, b) | a ≤ b } on the
set {1, 2, 3, 4}
1. Start with the directed graph for this relation
2. Because a partial ordering is reflexive, a loop (a, a) is present at every
vertex a. Remove these loops.
3. Next, remove all edges that must be in the partial ordering because of
the presence of other edges and transitivity.
4. Finally, arrange each edge so that its initial vertex is below its terminal
vertex. Remove all the arrows on the directed edges, because all edges
point “upward” toward their terminal vertex.
39. Example
Draw the Hasse diagram representing the partial ordering { (a, b) |a divides
b } on {1, 2, 3, 4, 6, 8, 12}.