Demonstration Course
Course information | Demonstration Course | SOMAS
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1.
Chapter 1 — All sixteen environments
This chapter contains every environment type. It includes titled and untitled blocks, inline and display mathematics, lists, a figure, and multiple reveal panels. Colour, layout shape, and numbering preferences come from the Moodle environment-type settings; they are not set by this source file.
- 1.1. Learning and orientation
- 1.2. Definitions and reasoning
- 1.3. Practice, media, and consolidation
- 2. Chapter 2 — Numbering restarts in a new chapter
- 2.1. Experiment
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Week 1: Relations and Binary Operations
LEARNING OUTCOMES By the end of this week, you will be able to: * Compute cardinalities and domain/range for Cartesian products and relations. * Test relations for reflexivity, symmetry, and transitivity. * Construct and inspect Cayley tables for algebraic properties. * Prove the Partition Theorem, uniqueness of inverses, and the Reverse Order Law. 🔗 CONNECTION TO PRIOR LEARNING: FROM SETS TO ALGEBRAIC OPERATIONS In elementary set theory, we group isolated mathematical objects into sets. But how do we model interactions, rules, comparisons, and mappings between distinct objects? We package them as ordered pairs. In Chapter 1, we study subsets of Cartesian products — the binary relations that form the foundational language of modern abstract algebra. René Descartes (1596 1650) Each problem that I solved became a rule which served afterwards to solve other problems.
- 1.1. Cartesian Products and Binary Relations
- 1.2. Equivalence Relations and Partitions
- 1.3. Binary Operations and Cayley Tables
- 1.4. Properties of Operations and Inverses
- 1.5. Reference and Perspective
- Week 2: The Integers and Factorisation
- 2.1. The Integers as a Ring &The Division Algorithm
- 2.2. Divisibility &Greatest Common Divisors
- 2.3. The Euclidean &Extended Euclidean Algorithms
- 2.4. Prime Numbers, Euclid's Lemma &The Fundamental Theorem of Arithmetic
- Week 3: Modular Arithmetic and Cryptography
- 3.1. Modular Arithmetic &Congruence Relations
- 3.2. Computing with Congruence Classes ( \mathbb{Z}_{n} )
- 3.3. Applications in Coding Theory &Check Digits
- 3.4. Modular Cryptography &Public-Key Foundations
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