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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
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
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
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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
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Chi-squared Goodness of Fit Test Project Overview and.docxbissacr
Chi-squared Goodness of Fit Test Project
Overview and Rationale
This assignment is designed to provide you with hands-on experience in generating
random values and performing statistical analysis on those values.
Course Outcomes
This assignment is directly linked to the following key learning outcomes from the course
syllabus:
• Use descriptive, Heuristic and prescriptive analysis to drive business strategies and
actions
Assignment Summary
Follow the instructions in this project document to generate a number of different random
values using random number generation algorithm in Excel, the Inverse Transform. Then
apply the Chi-squared Goodness of Fit test to verify whether their generated values belong
to a particular probability distribution. Finally, complete a report summarizing the results
in your Excel workbook. Submit both the report and the Excel workbook.
The Excel workbook contains all statistical work. The report should explain the
experiments and their respective conclusions, and additional information as indicated in
each problem. Be sure to include all your findings along with important statistical issues.
Format & Guidelines
The report should follow the following format:
(i) Introduction
(ii) Analysis
(iii) Conclusion
And be 1000 - 1200 words in length and presented in the APA format
Project Instructions:
The project consists of 4 problems and a summary set of questions. For each problem, tom
hints and theoretical background is provided.
Complete each section in a separate worksheet of the same workbook (Excel file). Name
your Excel workbook as follows:
ALY6050-Module 1 Project – Your Last Name – First Initial.xlsx
In the following set of problems, r is the standard uniform random value (a continuous
random value between 0 and 1).
Problem 1
Generate 1000 random values r. For each r generated, calculate the random value 𝑿𝑿 by:
𝒙𝒙 = −𝑳𝑳𝑳𝑳(𝒓𝒓),
where “Ln“ is the natural logarithm function.
Investigate the probability distribution of X by doing the following:
1. Create a relative frequency histogram of X.
2. Select a probability distribution that, in your judgement, is the best fit for X.
3. Support your assertion above by creating a probability plot for X.
4. Support your assertion above by performing a Chi-squared test of best fit with a 0.05
level of significance.
5. In the word document, describe your methodologies and conclusions.
6. In the word document, explain what you have learned from this experiment.
Hints and Theoretical Background
A popular method for generating random values according to a certain probability
distribution is to use the inverse transform method. In this method, the cumulative
function of the distribution (F(x)) is used for such a random number generation. More
specifically, a standard uniform random value r is generated first. Most software
environments are capable of generating such a value. In E
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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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.
2. Circle the name of your Tutorial Instructor:
Ashish Carole Christos Eric
George Jack Nick Tina
• This quiz is closed book. There is an Appendix giving the definitions of standard
properties of relations.
• There are four (4) problems totaling 100 points. Problems are labeled with their point
values.
• Put your name on the top of every page – these pages may be separated for grading.
• Write your solutions in the space provided. If you need more space, write on the back
of the sheet containing the problem. Please keep your entire answer to a problem on
that problem’s page.
• You may assume any of the results presented in class or in the lecture notes.
• Be neat and write legibly. You will be graded not only on the correctness of your
answer, but also on the clarity with which you express it.
Problems
statisticshomeworkhelper.com
3. Problem 1 Consider the following system specifications1.
1. The system is in multiuser state if it is operating normally.
2. If the system is operating normally, then the kernel is functioning.
3. The kernel is not functioning or the system is in interrupt mode.
4. If the system is not in multiuser state, then it is in interrupt mode.
5. The system is not in interrupt mode.
(a) To make sense of these confusing conditions, let’s introduce four Boolean variables.
statisticshomeworkhelper.com
4. M ::= in Multiuser state (1)
N ::= operating Normally (2)
K ::= Kernel is functioning (3)
I ::= in Interrupt mode (4)
Translate the five statements in the specification into propositional
logic notation: ∧,∨,¬, −→
statisticshomeworkhelper.com
5. (b) Are these system specifications consistent? . Prove it!
Problem 2
For each of the following logical formulas with domain of discourse the natural numbers,
N, indicate whether it is a possible formulation of I:
the Induction Axiom,
S: the Strong Induction Axiom,
L: the Least Number Principle (also known as Wellordering),
or N: None of these.
For example, the ordinary Induction Axiom could be expressed by the following formula, so
it gets labelled “I”.
statisticshomeworkhelper.com
6. This is a multiple choice problem: do not explain your answer.
(a) (P(b) ∧ [∀k ≥ b P(k) −→ P(k + 1)]) −→ ∀k ≥ b P(k)
(b) (P(b) ∧ [∀k ≤ b P(k) −→ P(k + 1)]) −→ ∀k ≤ b P(k)
(c) [∀b (∀k < b P(k)) −→ P(b)] −→ ∀k P(k) (
(d) (∃n P(n)) −→ ∃n ∀k < n P(k)
(e) ∀n [P(n) −→ (∃n P(n) ∧ ∀k < n P(k))]
Problem 3 Classify each of the following binary relations as
E: An equivalence relation.
T: A Total order,
P: A Partial order that is not total.
S: A Symmetric relation that is not transitive.
N: None of the above..
statisticshomeworkhelper.com
7. This is a multiple choice problem: do not explain your answer.
(a) The relation xRy between times of day such that x and y are at most twenty minutes
apart.
(b) The relation xRy between times of day such that x is more than twenty minutes later
than y.
(c) The relation xRy over all words in this sentence such that x does not appear after y.
(Consider “x”, “y”, and “xRy” to be words.)
(d) The relation xRy over all words in this sentence such that word x does not appear
after word y.
(e) The relation xRy over all words in this sentence such that the final appearance of y
occurs after x.
Problem 4 To encourage collaborative study, the 6.042 staff is considering assigning
each student to a study group with two or three other students. Prove that as long as the
enrollment is large enough, the class can always be divided into such study groups.
statisticshomeworkhelper.com
8. Appendix
A binary relation, R, on a set A is
• reflexive iff xRx,
• symmetric iff xRy −→ yRx,
• antisymmetric iff xRy ∧ yRx −→ x = y,
• transitive iff xRy ∧ yRz −→ xRz,
for all x, y, z ∈ A.
• R is an equivalence relation iff it is reflexive, symmetric and transitive.
• R is a partial order iff it is transitive and antisymmetric.
• R is a total order iff it is a partial order and for all x = y ∈ A either xRy or yRx.
statisticshomeworkhelper.com
9. Problem 1 Consider the following system specifications1.
1. The system is in multiuser state iff it is operating normally.
2. If the system is operating normally, then the kernel is functioning.
3. The kernel is not functioning or the system is in interrupt mode.
4. If the system is not in multiuser state, then it is in interrupt mode.
5. The system is not in interrupt mode.
(a) To make sense of these confusing conditions, let’s introduce four Boolean
variables.
M ::= in Multiuser state (1)
N ::= operating Normally (2)
K ::= Kernel is functioning (3)
I ::= in Interrupt mode (4)
statisticshomeworkhelper.com
10. Translate the five statements in the specification into propositional logic notation: ∧,∨,¬,
−→ ,←→
Solution.
M N ←→ (5)
N −→ K (6)
¬K ∨ I (7)
¬M −→ I (8)
¬I (9)
(b) (0 points) Are these system specifications consistent? . Prove it!
Solution. There are several ways to approach this problem. One is to construct a truth
table with sixteen lines—one for each way of assigning truth values to the four variables
M, N, K, and I.
We can avoid the cumbersome truthtable if we reason by cases. Case 1: I is true. Then
the last formula (9) is false, and the whole specification is false. Case 2: I is false. Now
formula (8) can be true only if ¬M is false, that is, only if M is true. Likewise, formula (7)
can be true only if ¬K is true, that is, K is false.
Since K is false, formula (6) can be true only if N is false. Thus, we have deduced that in
order to be consistent in this case, we must have
statisticshomeworkhelper.com
11. I = false
M = true
K = false
N = false.
But now formula (5) is false, so it is impossible for all the formulas to be true: the system is
inconsistent.
Problem 2 For each of the following logical formulas with domain of discourse the
natural numbers, N, indicate whether it is a possible formulation of
I: the Induction Axiom,
S: the Strong Induction Axiom,
L: the Least Number Principle (also known as Wellordering), or
N: None of these.
For example, the ordinary Induction Axiom could be expressed by the following formula, so
it gets labelled “I”.
(P(0) ∧ [∀k P(k) −→ P(k + 1)]) −→ ∀k P(k) I
statisticshomeworkhelper.com
12. This is a multiple choice problem: do not explain your answer.
(a) (P(b) ∧ [∀k ≥ b P(k) −→ P(k + 1)]) −→ ∀k ≥ b P(k)
Solution. I. This is a perfect formulation of the Induction Axiom. b is used for the base
case; P(k) −→ P(k + 1) is the inductive case.
(b) (P(b) ∧ [∀k ≤ b P(k) −→ P(k + 1)]) −→ ∀k ≤ b P(k) Solution. N. The two occurences
of k <= b should have been k >= b
(c) [∀b (∀k < b P(k)) −→ P(b)] −→ ∀k P(k) Solution. S. Since you are assuming P(k)
for all k < b, this is strong induction.
(d) (∃n P(n)) −→ ∃n ∀k < n P(k) Solution. N. This statement is in fact always true;
when n = 0, ∀k < n P(k). It should say P(n) ∧ ∀k < n; P(k)
(e) ∀n [P(n) −→ (∃n P(n) ∧ ∀k < n P(k))] Solution. L. This is a valid formulation; it does
not have that problem in (2d).
Problem 3 . Classify each of the following binary relations as
statisticshomeworkhelper.com
13. E: An equivalence relation.
T: A Total order,
P: A Partial order that is not total
S: A Symmetric relation that is not transitive.
N: None of the above.
This is a multiple choice problem: do not explain your answer.
(a) The relation xRy between times of day such that x and y are at most twenty
minutes apart.
Solution. S: This relation is reflexive and symmetric. It is not transitive; Consider the
counterexample: 1:00R 1:15, 1:15R 1:22 but ¬1:00R 1:22.
(b) The relation xRy between times of day such that x is more than twenty minutes later
than y.
Solution. P: This relation is antisymmetric and transitive but not reflexive (since a time
isn’t 20 minutes after itself). This is not a total ordering because some times are
incomparable to each other. For example, 1:15 is incomparable to 1:22
statisticshomeworkhelper.com
14. Note: This answer assumes that the question was referring to the moments in time in
a single day. Otherwise, one could argue that 1:00 precedes 1:22 on tuesday, but
1:22 on tuesday precedes 1:00 on wednesday, and then the relation would not be
antisymmetric.
(c) The relation xRy over all words in this sentence such that x does not appear after y.
(Consider “x”, “y”, and “xRy” to be words.)
Solution. T: Because all the words in the sentence are unique, this relation is transitive,
antisymmetric, and reflexive. This makes the relation a partial order. However, the
relation is also a total order, because any two elements are comparable.
(d) The relation xRy over all words in this sentence such that word x does not
appear after word y.
Solution. P: This is the same as saying that all appearances of a word x occur
before all appearances of a word y. While apparently very similar to part c, this
relation is not reflexive because the word word appears twice in the sentence. It is
transitive and antisymmetric, but not a total order because all the words between
the two occurrences of word are incomparable to word.
statisticshomeworkhelper.com
15. (e) The relation xRy over all words in this sentence such that the final appearance of y
occurs after x.
Solution. N: This relation is not reflexive and transitive. It is neither symmetric nor
antisymmetric; the word the occurs twice, all the others once. All words w, between the
two occurrences of the satisfy wR the and theRw. So R is not antisymmetric. But “x” is
maximal, so it’s not symmetric either
Problem 4 (35 points). To encourage collaborative study, the 6.042 staff is
considering assigning each student to a study group with two or three other
students. Prove that as long as the enrollment is large enough, the class can
always be divided into such study groups.
Solution. Proof. The proof is by strong induction. The induction hypothesis is
that a class with n ≥ 6 students can be divided into teams of 3 or 4. More
precisely
P(n) ::= n ≥ 6 −→ ∃x, y ∈ N 3x + 4y = n.
For any n ≥ 0, we may assume P(6), . . . , P(n − 1) to prove P(n).
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16. Case 1: (n < 6). P(n) holds because the hypothesis n ≥ 6 is false.
Case 2: (n = 6, 7, or 8) P(6) is true because there could be two teams of 3, P(7) is
true because there could be a team of 3 and a team of 4, and P(8) is true because
there could be two teams of 4.
Case 3: (n ≥ 9). Of course n > n−3 so P(n−3) holds by the strong induction
hypothesis. But n − 3 ≥ 6, so P(n − 3) implies 3x + 4y= n − 3 for some x, y ∈ N,
and therefore 3x + 4y = n where x ::= x + 1 and y ::= y . So P(n) holds, as required.
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