Unit of study_

# MATH3066: Algebra and Logic

## 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 Mathematics and Statistics Academic Operations 6 MATH3062 or MATH3065 6 credit points of MATH2XXX None Introductory knowledge of group theory. For example as in MATH2X22 Yes

### Teaching staff

Coordinator Oded Yacobi, oded.yacobi@sydney.edu.au

## Assessment

Type Description Weight Due Length
Final exam (Take-home short release) Final Exam
Final Exam
60% Formal exam period 3 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 D final exam

### Assessment summary

Detailed information for each assessment can be found on Canvas.

### 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.

### 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.

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 Learning outcomes
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.

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.

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

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## 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

### 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