Introduction to discrete mathematics and its use in mathematical modelling. Emphasis will be placed on developing facility, technique and use in applications. Modelling of processes and phenomena which occur in the physical, environmental and life sciences, especially computer science, will be used as a vehicle throughout. Topics to be covered include: logic and set theory, combinatorics and counting, probability, induction and recurrence relations, graph theory and networks, matrix arithmetic and Markov chains.

Note: This course is co-taught with undergraduate students but assessed separately.

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

1. Recall, invent or interpret examples of motivation for mathematical constructs used in discrete mathematics as models of processes in the world.
2. Recognise, define, explain and use terminology and notation from discrete mathematics.
3. Identify the logical structure of a statement, and then identify the logical structure of an argument that may be used to prove or disprove the statement.
4. Competently perform mathematical calculations in discrete mathematics using methods presented in the course.
5. Write simple proofs/construct explicit counterexamples for statements relating to discrete mathematics topics covered in the course.
6. Use their deep knowledge and understanding of the material presented in the course to formulate responses to complex concrete and abstract problems.
7. Communicate their understanding and skills in discrete mathematics with colleagues and non-experts and apply their knowledge in an occupational situation.

Research-Led Teaching

Where appropriate, indication will be given of current research areas related to topics in the course.

Examination Material or equipment

In the final exam the permitted materials will include:

• A non-annotated translation dictionary (for ESL students).

Required Resources

Course Notes: PDF copies of all lecture slides will be made available on the course website, in addition to the ECHO lecture recordings.

Recommended Resources

Optional Text: Susanna Epp: Discrete Mathematics with Applications; 5th ed. Metric Edition, Cengage.

An e-book of this text is available to all students for free through the ANU Library https://anu.primo.exlibrisgroup.com/permalink/61ANU_INST/1csbe8o/cdi_askewsholts_vlebooks_9780357121467

References to this text will be provided for all course topics except the last (Random Walks).

The text does not cover every single subtopic in the course, and does cover some subtopics not required for the course, but is nonetheless a good match to the course in both level and content.

There are a variety of online platforms you will use to participate in your study program at ANU, across all of your courses. These could include videos for lectures and other instruction, two-way video conferencing for interactive learning, email and other messaging tools for communication, interactive web apps for formative and collaborative activities, print and/or photo/scan for handwritten work and drawings, and home-based assessment.

ANU outlines recommended student system requirements to ensure you are able to participate fully in your learning. Other information is also available about the various Learning Platforms you may use.

Staff Feedback

Students will be given feedback in the following forms in this course:

• Workshops: Demonstrators will give individual guidance and correction to student work on, and presentation of, worksheet problems.
• Midsemester and final exam: Students will have an opportunity to view their exam scripts, view official solutions and rubrics, and ask the convener about any grading issues using a grade appeal process.

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). Feedback can also be provided to Course Conveners and teachers via the Student Experience of Learning & Teaching (SELT) feedback program. SELT surveys are confidential and also provide the Colleges and ANU Executive with opportunities to recognise excellent teaching, and opportunities for improvement.

Other Information

Use of Artificial Intelligence (AI): unless explicitly allowed in the assessment instructions, assessment tasks for this course should be completed without the assistance of generative AI (for example, without using ChatGPT).

Class Schedule

Week/Session	Summary of Activities	Assessment
1	A1Logic: Statements and Predicates. Valid Arguments.	No workshops in Week 1.
2	A2Sets: Set Operations and identities. Russell's Paradox.	Workshop with participation assessed.
3	A3Relations and Functions: Definition & Properties of Relations and Functions.	Workshop with participation assessed.
4	B1Numbers: N,Z,Q,R. Base n. Computer & Modular Arithmetic.	Workshop with participation assessed.
5	B2Sequences and Induction: Implicit to Explicit Sequence Definition by Induction. Sorting.B3Matrices: Matrix & Vector Operations. Linear Functions.	Workshop with participation assessed.
6	C1Counting: Cardinality. Permutations & Combinations. Stars & Bars. Pigeonhole Principle.	Workshop with participation assessed.
7	C2Probability: Probability Properties. Distributions. Random Variables.	Mid-semester exam held in-class during Wednesday lecture. No workshops. Video recording to replace Tuesday lecture.
8	C3Markov Processes: Markov States &Transition Matrices. Steady State.	Workshop with participation assessed.
9	D1Graph Theory: Graphs & Digraphs. Degree. Euler & Hamilton Graphs. Trees.	Workshop with participation assessed.
10	D2Weighted Graphs: Minimum Span. Travelling Sales Person Problem. Shortest Path. Max Flow. Matching.	Workshop with participation assessed.
11	D3Random Walks: Graph 'Walking'. Webgraphs & PageRank Algorithm.	Workshop with participation assessed.
12	Revision	Workshop with participation assessed.

Tutorial Registration

Students are required to enrol in one of the available weekly workshop groups using the ANU MyTimetable system. Each workshop is 90 minutes long; the 30 minutes following the end of the scheduled workshop time is an optional consultation session for students to ask questions after their workshop has ended. Please refer to the course Wattle site for more information. ANU utilises MyTimetable to enable students to view the timetable for their enrolled courses, browse, then self-allocate to small teaching activities / tutorials so they can better plan their time. Find out more on the Timetable webpage.

Assessment Summary

Assessment task	Value	Due Date	Learning Outcomes
MATH6005 Workshop Participation (best 8 of 10)	20 %	*	1,2,3,4,5,6,7
MATH6005 Mid-semester examination	20 %	16/04/2025	1,2,3,4,5,6,7
MATH6005 Final Examination	60 %	*	1,2,3,4,5,6,7

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines , which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Integrity Rule before the commencement of their course. Other key policies and guidelines include:

• Academic Integrity Policy and Procedure
• Student Assessment (Coursework) Policy and Procedure
• Extenuating Circumstances Application
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure
• Code of practice for teaching and learning

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the Academic Skills website. In rare cases where online submission using Turnitin software is not technically possible; or where not using Turnitin software has been justified by the Course Convener and approved by the Associate Dean (Education) on the basis of the teaching model being employed; students shall submit assessment online via ‘Wattle’ outside of Turnitin, or failing that in hard copy, or through a combination of submission methods as approved by the Associate Dean (Education). The submission method is detailed below.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Participation

Although there are no participation requirements (hurdles) for this course please note that:

You must attend the workshop in which you are enrolled to earn your workshop participation score. Workshop participation scores contribute to your final grade. Details of how this is assessed are given in the details of this assessment item stated elsewhere in this document.

Assessable material for the course is specified by lecture content (rather than any text). So it is important to keep up regular monitoring of lectures, preferably by attending but otherwise by accessing the lecture recordings and/or PDF notes on Wattle.

Examination(s)

This course includes a final examination, and an in-class mid-semester examination. The details and mode of delivery for exams will be communicated through the course Wattle site and the ANU examination timetable.

Assessment Task 1

MATH6005 Workshop Participation (best 8 of 10)

Workshops run in teaching weeks 2, 3, 4, 5, 6 and 8, 9, 10, 11, 12. You should enrol in and then attend one workshop, at the same time, each week. You can select from available workshop times as described on the course Wattle page. Each workshop is 90 minutes long; the 30 minutes following the end of the scheduled workshop time is an optional consultation session for students to ask questions after their workshop has ended.

In the week before a workshop, a worksheet will be made available on Wattle. Several problems will be identified as problems that you should solve BEFORE the workshop,; you should bring along your attempts and be ready to talk them at the start of the workshop. During the workshop you will work collaboratively in small groups (two or three students) on all of the worksheet questions, asking for help from the demonstrator as needed. Throughout the workshop, your group may be asked to present a solution(s) to your demonstrator or participate in class discussions. The aims are to sharpen your (already emerging) understanding of the course material by working together with your classmates, to give you many opportunities to practice and improve your skill in verbal communication of mathematics, and to give your demonstrator opportunities to correct any misconceptions that you or other class members may have about the course material and provide feedback about your work.

For each workshop, a participation score will be determined by your demonstrator and recorded in Wattle. You earn your score by diligently preparing to discuss the problems indicated as those you should solve before the workshop (this includes bringing along a record of your attempts), participating throughout the workshop with energy and enthusiasm, and by the quality of understanding demonstrated by you and your group throughout the workshop. A rubric for assessing your workshop participation will be available on the course wattle page. To perform well in the workshop participation, you will be expected to be familiar with the lecture material up to and including that covered in the previous week and to have solved the problems indicated as those you should solve before the workshop,

At the end of the semester, your workshop participation scores will be combined to compute the workshop participation contribution to your final grade. To allow for occasional absences (medical or otherwise), your workshop participation contribution to your final grade will be calculated from your best eight (out of ten) workshop marks. This means that you may earn up 2.5% of your final grade in any given workshop.

Assessment Task 2

MATH6005 Mid-semester examination

The mid-semester examination will be an in-person exam with 70 minutes of writing time. All material covered in the first six weeks of the course will be examinable. The exam will be held during the Wednesday lecture slot in week 7, in the lecture room. Some scaling of marks on the mid-semester exam may occur if the distribution of marks leads to results significantly out of line with previous years. Note that the MATH6005 mid-semester exam will be different to the MATH1005 mid-semester exam, as needed to assess the masters-level learning outcomes, and will be held on a different day.

This exam is redeemable via the final exam: if the percentage score on the mid-semester exam is less than the percentage score on the final exam, only the final exam will count and it will be weighted at 80% instead of 60%.

Assessment Task 3

MATH6005 Final Examination

The final examination will be an in-person exam with 15 minutes of reading time followed by 180 minutes of writing time. The entire course will be examinable. The exam will be held during the ANU final examination period at a time and date determined by the ANU Examinations Office. Please check the course Wattle site and the ANU Examination Timetable to confirm the date, time, and location of the exam. Note that the MATH6005 final exam will have some differences in comparison to the MATH1005 final exam..

Some scaling of marks on the final exam may occur if the distribution of marks leads to results significantly out of line with previous years.

Academic Integrity

Academic integrity is a core part of the ANU culture as a community of scholars. The University’s students are an integral part of that community. The academic integrity principle commits all students to engage in academic work in ways that are consistent with, and actively support, academic integrity, and to uphold this commitment by behaving honestly, responsibly and ethically, and with respect and fairness, in scholarly practice.

The University expects all staff and students to be familiar with the academic integrity principle, the Academic Integrity Rule 2021, the Policy: Student Academic Integrity and Procedure: Student Academic Integrity, and to uphold high standards of academic integrity to ensure the quality and value of our qualifications.

The Academic Integrity Rule 2021 is a legal document that the University uses to promote academic integrity, and manage breaches of the academic integrity principle. The Policy and Procedure support the Rule by outlining overarching principles, responsibilities and processes. The Academic Integrity Rule 2021 commences on 1 December 2021 and applies to courses commencing on or after that date, as well as to research conduct occurring on or after that date. Prior to this, the Academic Misconduct Rule 2015 applies.

 

The University commits to assisting all students to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. All coursework students must complete the online Academic Integrity Module (Epigeum), and Higher Degree Research (HDR) students are required to complete research integrity training. The Academic Integrity website provides information about services available to assist students with their assignments, examinations and other learning activities, as well as understanding and upholding academic integrity.

Online Submission

You will be required to agree to a declaration as part of the submission of your assignments, that will record your understanding of ANU academic integrity principles. The assignment document will state the deadline for submission. You should keep a copy of both your completed document and its pdf file. Note that pdf is the only file format permitted for online submission in MATH6005. MATH6005 does not use Turnitin, having been granted an exemption.

Hardcopy Submission

Except under very special course-wide conditions (e.g. major breakdown of the Wattle system) hardcopy submission of assignments will not be permitted in MATH6005.

Late Submission

Late assignments will incur penalties as per ANU Policy. We will not accept assignment submissions more than 5 business days after the due date. Late submission of assessment tasks without an extension are penalised at the rate of 5% of the possible marks available per working day or part thereof. Late submission is not accepted for examinations.

Referencing Requirements

The Academic Skills website has information to assist you with your writing and assessments. The website includes information about Academic Integrity including referencing requirements for different disciplines. There is also information on Plagiarism and different ways to use source material. Any use of artificial intelligence must be properly referenced. Failure to properly cite use of Generative AI will be considered a breach of academic integrity.

Returning Assignments

Except in exceptional circumstances, you will be able to view your marked assignment via Wattle within a week of the due date. Exceptions include, but are not restricted to, unexpected unavailability of the marker and disruptions to the timetable resulting from public holidays.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure. Extensions may be granted for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

Assignments cannot be resubmitted.

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information.
In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service – including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy.
If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes.

Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Accessibility for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents all ANU students