Ідентифікація багатофакторних залежностей на основі нечіткої бази знань з рі...Роман Тилець
Розглядається ідентифікація багатофакторних залежностей за допомогою гібридної нечіткої бази знань, в яку входять правила різних форматів – Мамдані, Сугено, Ларсена тощо. Такий формат дозволяє описати досліджувану залежність в різних зонах факторного простору за допомогою нечітких правил найбільш релевантного формату.
Ідентифікація багатофакторних залежностей на основі нечіткої бази знань з рі...Роман Тилець
Розглядається ідентифікація багатофакторних залежностей за допомогою гібридної нечіткої бази знань, в яку входять правила різних форматів – Мамдані, Сугено, Ларсена тощо. Такий формат дозволяє описати досліджувану залежність в різних зонах факторного простору за допомогою нечітких правил найбільш релевантного формату.
Logical Abduction and an Application on Business Rules ManagementTobias Trapp
A tour d'horizon about last developments in Artifical Intelligence, my personal opinion about the future role of AI in ERP systems and a small application of established theories in Business Rules Management.
Logical Abduction and an Application on Business Rules ManagementTobias Trapp
A tour d'horizon about last developments in Artifical Intelligence, my personal opinion about the future role of AI in ERP systems and a small application of established theories in Business Rules Management.
Now we have learnt the basics in logic.
We are going to apply the logical rules in proving mathematical theorems.
1-Direct proof
2-Contrapositive
3-Proof by contradiction
4-Proof by cases
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
Analysis of algorithms is the determination of the amount of time and space resources required to execute it. Usually, the efficiency or running time of an algorithm is stated as a function relating the input length to the number of steps, known as time complexity, or volume of memory, known as space complexity.
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than propositional logic.
The topics on first order logic are:
1-Quantifiers
2-Negation
3-Multiple quantifiers
4-Arguments of quantified statements
Proof Techniques
There are some of the most common proof techniques.
1. Direct Proof
2. Proof by Contradiction
3. Proof by Contapositive
4. Proof by Cases
A complete and enhanced presentation on mathematical induction and divisibility rules with out any calculation.
Here are some defined formulas and techniques to find the divisibility of numbers.
Internet of Things (IoT) two-factor authentication using blockchainDavid Wood
Presented at the Ethereum Engineering Group Meetup in Brisbane, Australia, on 13 Nov 2019. We report on research to use an Ethereum blockchain as an MFA and/or MPA device to secure command channels on IoT networks, even when the underlying network may be compromised.
Methods for Securing Spacecraft Tasking and Control via an Enterprise Ethereu...David Wood
Presentation at ICSSC 2019 (see http://www.kaconf.org) associated with the following academic paper:
David Hyland-Wood, Peter Robinson, Roberto Saltini, Sandra Johnson, Christopher Hare. Method for Securing Spacecraft Tasking and Control via an Enterprise Ethereum Blockchain. Proc. 37th International Communications Satellite Systems Conference (ICSSC), 29 October - 1 November 2019.
Implementing the Verifiable Claims data modelDavid Wood
The W3C Verifiable Claims data model arguably requires a decentralised, distributed database controllable by three types of parties; issuers, inspectors, and holders. This presentation explores the benefits of implementing the Verifiable Claims data model using the RDF and Linked Data technology stack.
Metaphors define civilized life. They are all around us in the stories that we teach our children and tell each other to justify our actions. But social metaphors have a dark side. They can cause entire civilizations to self destruct. Metaphors can kill. This presentation explores the power, and danger, of metaphors as social memes.
These slides are from a talk given to the Fredericksburg Secular Humanists (FSH) in Fredericksburg, Virginia, on 8 November 2015. FSH is sub-chapter of the United Coalition of Reason (unitedcor.org). The talk compared the secular societies of the United States and Australia.
Building a writer's platform with social mediaDavid Wood
This presentation reports on my progress in trying to build my writer's platform using social media. It focuses on Twitter, but the advice is generally applicable. Kudos to my mentors @DanCitizen and @RayneHall.
A summary of the Hero's Journey, Joseph Campbell's formulation of the "monomyth" in mythology and literature. Originally presented to the Fredericksburg Writing as a Business Meetup, 24 January 2015.
Open Data is the idea that "certain data should be freely available to everyone to use and republish as they wish, without restrictions from copyright, patents or other mechanisms of control”. Open Data follows similar “open” concepts that have proven to be valuable in the information economy such as Open Standards, Open Source Software, Open Content and has been followed more recently by variations on the theme such as Open Science and Open Government.
Open Data allows information of common value to be reused without needing to be recreated. The economic benefits of Open Data include cost reduction, organizational efficiencies and the facilitation of commonly held understanding. The costs of implementing Open Data deployment strategies tend to be iterative on top of existing information infrastructure.
This presentation will describe Open Data and its place in the ecosystem of economic and governmental discourse.
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
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
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.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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.
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.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
2. Proof Techniques
● Informal proof methods :
■ Inductive reasoning
■ Deductive reasoning
■ Proof by exhaustion
■ Direct proof
■ Proof by contraposition
■ Serendipity
● A few terms to remember:
■ Axioms: Statements that are assumed true.
• Example: Given two distinct points, there is exactly one line that
contains them.
■ Definitions: Used to create new concepts in terms of existing ones.
■ Theorem: A proposition that has been proved to be true.
• Two special kinds of theorems: Lemma and Corollary.
• Lemma: A theorem that is usually not too interesting in its own right
but is useful in proving another theorem.
• Corollary: A theorem that follows quickly from another theorem.
Section 2.1 Proof Techniques 1
3. Deductive Reasoning: Counter Example
● Inductive Reasoning: Drawing a conclusion from a hypothesis
based on experience. Hence the more cases you find where P
follows from Q, the more confident you are about the conjecture
P → Q.
● Usually, deductive reasoning is also applied to the same
conjecture to ensure that it is indeed valid.
● Deductive reasoning either looks for a counterexample that
disproves the conjecture (i.e. a case when P is true but Q is
false) or a to construct a proof (theorem).
● Example: Prove that “For every positive integer n, n! ≤ n2.”
■ Start testing some cases say, n = 1, 2, 3 etc.
■ It might seem like it is true for some cases but how far do you test,
say n = 4.
■ We get n! = 24 and n2 = 16 which is a counter example for this
theorem. Hence,even finding a single case that doesn’t satisfy the
condition is enough to disprove the theorem.
Section 2.1 Proof Techniques 2
4. Counterexample
● More examples of counterexample:
■ All animals living in the ocean are fish.
■ Every integer less than 10 is bigger than 5.
● Counter example is not trivial for all cases, so we have
to use other proof methods.
Section 2.1 Proof Techniques 3
5. Exhaustive Proof
● If dealing with a finite domain in which the proof is to be shown to be
valid, then using the exhaustive proof technique, one can go over all
the possible cases for each member of the finite domain.
● Final result of this exercise: you prove or disprove the theorem but you
could be definitely exhausted.
● Example: For any positive integer less than or equal to 5, the square of
the integer is less than or equal to the sum of 10 plus 5 times the
integer.
n
n2
10+5n
n2 < 10+5n
0
0
10
yes
1
1
15
yes
2
4
20
yes
3
9
25
yes
4
16
30
yes
5
25
35
yes
Section 2.1 Proof Techniques 4
6. Example: Exhaustive Proof
● Example 1: If an integer between 1 and 20 is divisible by 6, then
it is also divisible by 3.
Section 2.1 Proof Techniques 5
7. Direct Proof
● Direct Proof:
● Used when exhaustive proof doesn’t work. Using the rules of
propositional and predicate logic, prove P → Q.
● Hence, assume the hypothesis P and prove Q. Hence, a formal proof
would consist of a proof sequence to go from P to Q.
● Consider the conjecture
x is an even integer Λ y is an even integer → the product xy is an even integer.
● A complete formal proof sequence might look like the following:
1. x is an even integer Λ y is an even integer
hyp
2. (∀x)[x is even integer →(∃k)(k is an integer Λ x = 2k)]
number fact
(definition of even integer)
3. x is even integer →(∃k)(k is an integer Λ x = 2k)
2,ui
4. y is even integer →(∃k)(k is an integer Λ y = 2k)
2,ui
5. x is an even integer
1,sim
6. (∃k)(k is an integer Λ x = 2k)
3,5,mp
7. m is an integer Λ x = 2m
6, ei
8. y is an even integer
1,sim
Section 2.1 Proof Techniques 6
8. Direct Proof Example (contd.)
9. (∃k)(k is an integer Λ y = 2k)
4,8,mp
10. n is an integer and y = 2n
9, ei
11. x = 2m
7,sim
12. y = 2n
10, sim
13. xy = (2m)(2n)
11, 12, substitution of equals
14. xy = 2(2mn)
13, multiplication fact
15. m is an integer
7,sim "
16. n is an integer
10, sim
17. 2mn is an integer
15, 16, number fact
18. xy = 2(2mn) Λ 2mn is an integer
14, 17, con
19. (∃k)(k is an integer Λ xy = 2k)
18,eg
20. (∀x)((∃k)(k is an integer Λ x = 2k) → x is even integer)
number fact
(definition of even integer)
21. (∃k)(k an integer Λ xy = 2k) → xy is even integer
20, ui
22. xy is an even integer
19, 21, mp
Section 2.1 Proof Techniques 7
9. Indirect Proof: Proof by Contradiction
● Derived from the definition of contradiction which says
P Λ Qʹ′ → 0
● Hence, (P Λ Qʹ′ → 0) → (P → Q) is a tautology.
● In a proof of contradiction, one assumes that the hypothesis and
the negation of the conclusion are true and then to deduce some
contradiction from these assumptions.
● Example 1: Prove that “If a number added to itself gives itself,
then the number is 0.”
■ The hypothesis (P) is x + x = x and the conclusion (Q) is x = 0.
Hence, the hypotheses for the proof by contradiction are:
■ x + x = x and x ≠0 (the negation of the conclusion)
■ Then 2x = x and x ≠0, hence dividing both sides by x, the result is 2
= 1, which is a contradiction. Hence, (x + x = x) → (x = 0).
Section 2.1 Proof Techniques 8
10. Proof by Contradiction
● Example 2: Prove “For all real numbers x and y, if x + y ≥ 2,
then either x ≥1 or y ≥1.”
■ Proof: Say the conclusion is false, i.e. x < 1 and y < 1. (Note: or
becomes and in negation.)
■ Adding the two conditions, the result is x + y < 2.
■ At this point, this condition is Pʹ′ if P = x + y ≥ 2, hence, P Λ Pʹ′
which is a contradiction. Hence, the statement above is true.
● Example 3: The sum of even integers is even.
■ Proof: Let x = 2m, y = 2n for integers m and n and assume that
x + y is odd.
■ Then x + y = 2m + 2n = 2k + 1 for some integer k.
■ Hence, 2*(m + k - n) = 1, where m + n - k is some integer.
■ This is a contradiction since 1 is not even.
Section 2.1 Proof Techniques 9
11. Proof by Contraposition
● Use the variants of P → Q to prove the conjecture. From the
inference rules of propositional logic, we know a rule called
contraposition (termed “cont” in short).
● It states that (Qʹ′ → Pʹ′) → (P → Q) (a Tautology).
● Qʹ′ → Pʹ′ is the contrapositive of P → Q.
● Hence, proving the contrapositive implies proving the
conjecture.
● Example: Prove that if the square of an integer is odd, then the
integer must be odd
■ Hence, prove n2 odd → n odd
■ To do a proof by contraposition prove n even → n2 even
■ If n is even, then it can be written as 2m where m is an integer, i.e.
even/odd.
■ Then, n2 = 4m2 which is always even no matter what m2 is because
of the factor 4.
Section 2.1 Proof Techniques 10
12. Serendipity
● Serendipity: Fortuitous happening or something by chance or
good luck.
● Not a formal proof technique.
● Interesting proofs provided by this method although other
methods can be used as well.
● Example: 342 players in a tennis tournament. One winner in the
end. Each match is between two players with exactly one winner
and the loser gets eliminated. Prove the total number of matches
played in the tournament are 341.
■ Solution: Only one champion at the end of the tournament, hence
341 losers at the end, hence 341 matches should have been played
to have 341 losers.
Section 2.1 Proof Techniques 11
13. Summarizing Proof techniques
Proof Technique
Approach to prove P → Q
Remarks
Exhaustive Proof
Demonstrate P → Q
May only be used for
for all cases
finite number of cases
Direct Proof
Assume P, deduce Q
The standard approach-
usually the thing to try
Proof by Assume Qʹ′, derive Pʹ′
Use this Qʹ′ if as a
Contraposition
hypothesis seems to
give more ammunition
then P would
Proof by Assume P Λ Qʹ′ , deduce a Use this when Q says
Contradiction
contradiction
something is not true
Serendipity
Not really a proof Fun to know
technique
Section 2.1 Proof Techniques 12
14. Class Exercise
1. Product of any 2 consecutive integers is even.
2. The sum of 3 consecutive integers is even.
3. Product of 3 consecutive integers is even.
4. The square of an odd integer equals 8k+1 for some
integer k.
5. The sum of two rational numbers is rational.
6. For some positive integer x, x + 1/x ≥ 2.
Section 2.1 Proof Techniques 13