MATH A1 Lecture Notes

Discrete Mathematical Models Course Notes

Course Information

  • Institution: Mathematical Sciences Institute

  • Semester: Semester 1, 2026

Acknowledgment of Country

  • Acknowledge and celebrate the First Australians.

  • Pay respects to elders past and present.

  • Acknowledge the Ngunnawal and Ngambri people, Traditional Owners of the University’s Acton campus.

Preface to the Course

  • Credits: Developed by mathematicians over more than a decade.
      - Contributing Authors:
        - Malcolm Brooks
        - Judy-anne Osborne
        - Adam Piggott
        - Pierre Portal
        - Kane Townsend

  • Mathematical concepts developed over millennia.

Learning Outcomes

  • Upon successful completion of MATH1005, students will be able to:
      1. Recall, invent, interpret examples related to mathematical constructs in discrete mathematics as models of processes.
      2. Recognize, define, explain, and use terminology and notation from discrete mathematics.
      3. Identify logical structures of statements and arguments to prove or disprove statements.
      4. Perform mathematical calculations in discrete mathematics using course methods.
      5. Write proofs and construct explicit counterexamples for statements in discrete mathematics.

  • Upon successful completion of MATH6005, in addition to MATH1005 outcomes, students will be able to:
      6. Formulate responses to complex problems using a deep understanding of the material.
      7. Communicate understanding and skills in discrete mathematics with colleagues and non-experts and apply knowledge in occupational situations.

Reference Materials

  • Lecture Notes: Self-contained, available on the Canvas course page.

  • Textbook: Susanna S. Epp, Discrete Mathematics with Applications: Metric Version, 5th Edition, Cengage Learning, 2019. ISBN: 9780357114087.
      - eBook available through ANU library.
      - Discount available for purchase through the publisher's website with coupon code WOW10.

Assessment

  • MATH1005 Assessment: Workshop preparation & participation, mid-semester exam (Week 7), final exam.

  • MATH6005 Assessment: Same as MATH1005.

  • All students must sign up for a workshop & consultation.

  • No workshops in Weeks 1 and 7.

  • Dates, times, and locations of mid-semester and final exams TBD.

  • If workshop falls on a public holiday, rescheduling is required.

  • Students must read the class summary.

Learning Design

  • Lectures: 3 hours per week.

  • Processing: Approximately 4 hours per week for reviewing lectures, working on examples, attending consultations, and preparing for workshops (1.5 hours per week).

  • Workshops: 1.5 hours per week, collaborating with classmates and demonstrator.

  • Exams: Summative assessments testing learning outcomes.

Why Study Discrete Mathematical Models?

  • Definition: Discrete mathematical models are abstract representations indexed by positive integers, avoiding continua.

  • Discrete mathematics is the study of these models.

Examples and Applications:

  • Sudoku Puzzle:
      - Exercise in discrete mathematics, requiring strategies, guarantees of success, and evaluations of the effectiveness of strategies.

  • Web Page Ranking:
      - Problem: Ordering search engine hits effectively.
      - Solution: Google’s PageRank algorithm - combines discrete models with linear algebra.

Relation to Computing

  • Discrete mathematics is foundational to computer science.

  • Memory Models:
      - Describe computer memory abstract models and reflect on time representation (discrete time).

Logic in Mathematics and Computer Science

  • Statements: Basic unit of logic delivering true or false values.
      - Examples:
        - True: "Australia is in the Southern Hemisphere."
        - False: "Canberra is in New Zealand."

Compound Statements

  • Combining Statements: Logical connectives (FF, FF) can combine individual statements into compound statements.

Logical Connectives

  1. NOT (Negation)
        - $
    eg p$ is defined by:
        | p | $
    eg p$ |
        |---|---------|
        | T | F |
        | F | T |

  2. AND (Conjunction)
        - $p ext{ AND } q$ is defined by:
        | p | q | $p ext{ AND } q$ |
        |---|---|--------------|
        | T | T | T |
        | T | F | F |
        | F | T | F |
        | F | F | F |

  3. OR (Disjunction)
        - $p ext{ OR } q$ is defined by:
        | p | q | $p ext{ OR } q$ |
        |---|---|--------------|
        | T | T | T |
        | T | F | T |
        | F | T | T |
        | F | F | F |

  4. IMPLIES/ If-Then (→)
        - $p
    ightarrow q$ is defined by:
        | p | q | $p
    ightarrow q$ |
        |---|---|--------------|
        | T | T | T |
        | T | F | F |
        | F | T | T |
        | F | F | T |

  5. IFF (If and Only If ↔)
        - $p ext{ IFF } q$ is defined by:
        | p | q | $p ext{ IFF } q$ |
        |---|---|--------------|
        | T | T | T |
        | T | F | F |
        | F | T | F |
        | F | F | T |

Order of Precedence Among Connectives

  • Carefully observe the order of logical operations during evaluations, prioritizing parentheses.

Truth Tables

  • Used to define logical connectives and evaluate the truth of compound statements.

Tautologies and Contradictions

  • Tautology: A statement form true for all interpretations.

  • Contradiction: A statement form false for all interpretations.

Quick Activities Suggested

  • Review video materials.

  • Master vocabulary and practice creating truth tables.

  • Explore optional textbook options.

Conclusion

  • Overall, understanding discrete mathematical models, logical formalisms, and the relation to computing creates a potent foundation for advanced studies in mathematics and computer science.