MATH3066: Semester 1, 2022

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Units MATH3066 Semester 1 2022 [Normal day]

Unit of study_

MATH3066: Algebra and Logic

Semester 1, 2022 [Normal day] - Camperdown/Darlington, Sydney

Overview

This unit of study unifies and extends mathematical ideas and techniques that most participants will have met in their first and second years, and will be of general interest to all students of pure and applied mathematics. It combines algebra and logic to present and answer a number of related questions of fundamental importance in the development of mathematics, from ancient to modern times. The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem the undecidability of First Order Logic, and a discussion of Godel's Incompleteness Theorem. Classical and novel arithmetics are introduced, unified and described abstractly using field and ring axioms and the language of field extensions. Quotient rings are introduced, which are used to construct different finite and infinite fields. A construction of the real numbers, by factoring out rings of Cauchy sequences of rationals by the ideal of sequences, is presented. Axiomatics are placed in the context of reasoning within first order logic and set theory.

Unit details and rules

Unit code	MATH3066
Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prohibitions ?	MATH3062 or MATH3065
Prerequisites ?	6 credit points of MATH2XXX
Corequisites ?	None
Assumed knowledge ?	Introductory knowledge of group theory. For example as in MATH2X22
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	David Easdown, david.easdown@sydney.edu.au

Type	Description	Weight	Due	Length
Final exam (Record+)	Final Exam Final Exam	60%	Formal exam period	2 hours
Outcomes assessed:
Assignment	First assignment Written assignment	20%	Week 06	2 weeks
Outcomes assessed:
Assignment	Second assignment Written assignment	20%	Week 12	2 weeks
Outcomes assessed:
= Type B final exam ?

Assessment summary

Detailed information for each assessment can be found on Canvas.

Final exam: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator.

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name	Mark range	Description
High distinction	85 - 100	Representing complete or close to complete mastery of the material.
Distinction	75 - 84	Representing excellence, but substantially less than complete mastery.
Credit	65 - 74	Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.
Pass	50 - 64	Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.
Fail	0 - 49	When you don’t meet the learning outcomes of the unit to a satisfactory standard.

Final exam: The exam will cover all material in the unit from both lectures and tutorials. The exam will have a mixture of multiple choice questions and short answer questions.
Assignment: This assignment will require you to integrate information from lectures and tutorials to create coherent logical arguments, and find innovative solutions to problems.

For more information see sydney.edu.au/students/guide-to-grades.

For more information see guide to grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

Deduction of 5% of the maximum mark for each calendar day after the due date.
After ten calendar days late, a mark of zero will be awarded.

Academic integrity

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Weekly schedule

WK	Topic	Learning activity
Week 01	Introduction to logic and impossibility. Mathematical Implication. Truth tables	Lecture and tutorial (4 hr)
Week 02	Russell’s Paradox. Cardinality. Uncountability. Integers and modular arithmetic	Lecture and tutorial (4 hr)
Week 03	Polynomials and factorisation. Introduction to groups, rings and fields	Lecture and tutorial (4 hr)
Week 04	Deduction and proof in the propositional calculus	Lecture and tutorial (4 hr)
Week 05	Metaproofs. Soundness in the propositional calculus	Lecture and tutorial (4 hr)
Week 06	Syntax and semantics. Completeness of the propositional calculus	Lecture and tutorial (4 hr)
Week 07	Quantifiers. Predicate calculus	Lecture and tutorial (4 hr)
Week 08	Deduction and proof in the Predicate calculus	Lecture and tutorial (4 hr)
Week 09	Soundness and completeness of the predicate calculus	Lecture and tutorial (4 hr)
Week 10	Quotient constructions. Field extensions. Irreducible polynomials and finite fields	Lecture and tutorial (4 hr)
Week 11	Construction of the real numbers, the completeness axiom	Lecture and tutorial (4 hr)
Week 12	Unsolvability of ancient problems of the Greeks	Lecture and tutorial (4 hr)
Week 13	The Halting Problem and undecidability of First Order Logic	Lecture and tutorial (4 hr)

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

Outcomes
Graduate qualities

At the completion of this unit, you should be able to:

LO1. be fluent in analysing and constructing logical arguments
LO2. be conversant with the Propositional and Predicate Calculi, and related notions of syntax (deduction) and semantics (completeness)
LO3. be informed about the historical underpinnings of abstract algebra that lead to axiomatic theories of groups, rings, integral domains and fields, and their use in exploring questions of decidability or impossibility
LO4. be fluent with a range of standard and exotic arithmetics and ring constructions, and be able to prove elementary propositions and theorems about them
LO5. understand the Halting Problem and Turing’s use of it to prove the undecidability of first order logic.

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1	Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.
GQ2	Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.
GQ3	Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.
GQ4	Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.
GQ5	Inventiveness Generating novel ideas and solutions.
GQ6	Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.
GQ7	Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.
GQ8	Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.
GQ9	Influence Engaging others in a process, idea or vision.

Outcome map

Learning outcomes	Graduate qualities
Learning outcomes

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered

Additional information

Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General Laboratory Safety Rules

No eating or drinking is allowed in any laboratory under any circumstances
A laboratory coat and closed-toe shoes are mandatory
Follow safety instructions in your manual and posted in laboratories
In case of fire, follow instructions posted outside the laboratory door
First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory
As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

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