**3.1 The study units**

The study units are the core course materials. They are grouped into five blocks. You are expected to complete one unit over a two week period including the associated assessment questions. You should refer to the course schedule on the OLE for details as to when each unit is to be studied.

Each block is accompanied by a *Problem Booklet* for extra practice. The blocks are:

__Block I - Foundations in discrete mathematics__

*Unit 1 Sets*

- Basic properties of sets
- Subsets
- Operations on sets
- Set identities

*Unit 2 Logic*

- Introduction to p-and-q game
- Developing an algebra for p-and-q game
- Logical operations
- Equivalence of logical statements
- Quantifiers

*Unit 3 Relations*

- Properties of relations
- Equivalence relations
- Partial orders
- Operations in relations

__Block II - Functions __

*Unit 1 Introduction to functions*

- Domain, codomain, image
- Graphs of functions
- Linear, quadratic, exponential function
- Trigonometric function
- Flooring and ceiling functions

*Unit 2 Properties of functions*

- Bijective, injective and surjective functions
- Inverse function
- Composition of functions
- Sequences
- Sum and product notations

__Block III - Proofs and combinatorics__

*Unit 1 Proofs*

- Direct proof
- Proof by contradiction
- Proof by exhaustion
- Mathematical induction

*Unit 2 Counting*

- Fundamental principles of counting
- Permutations and combinations
- Elements of probability
- Conditional probability

*Unit 3 Binomial expansions*

- Binomial coefficients and combinatorial identities
- Binomial theorem
- Properties of binomial coefficients
- Binomial expansion and approximations

__Block IV - Recursion and iteration__

*Unit 1 Homogeneous recurrence relations with constant coefficients*

- Mathematical representations of recurrence relations
- Solving recurrence relations by arithmetic and geometric progression
- Second-order linear homogeneous recurrence relations with constant coefficients

*Unit 2 Non-homogeneous recurrence relations with constant coefficients*

- Non-homogeneous linear recurrence relations with constant coefficients
- Second-order non-homogeneous linear recurrence relations

*Unit 3 Iterating functions*

- Iteration sequences
- Exact and approximate solutions
- Bisection method
- Secant method

__Block V - Algebraic structure__

*Unit 1 Matrices*

- Matrix property
- Matrices addition and subtraction
- Matrix multiplication and inverse
- Transformation and matrices
- Eigenvalues and eigenvectors

*Unit 2 Complex numbers*

- Describe the complex number system
- Manipulate calculations on complex numbers
- Power of complex numbers
- olving equations with complex numbers

All study units will be posted into the OLE.

**3.2 Working through the study units**

Each unit contains guidelines indicating what items you need to study the unit. Objectives at the beginning of each topic set out in detail what you should be able to do after completing it.

When studying a unit, you should remember to work through each exercise and problem within the unit, as well as those in the textbook as indicated in the unit. It is important to try your very best to solve problems on your own before looking at the solutions; don't just turn to the solution as the easy way out. On the other hand, don't ignore the solution — do use it if you are stuck, and read through the solution to compare it with your own.

You must not skip the sections that direct you to listen to audio recordings or watch the video clips; if you do, you'll miss some useful teaching points and explanations.

In the units you will come across examples, exercises and activities. At the end of your study of a unit, you should complete the corresponding assignment questions. Although it will be tempting to refer to the units while you do so, we think that you will gain more by trying to get as far as possible at your first attempt with any help other than referring to the handbook. You can, of course, refer back to the unit before you finalize your answers.

**3.3 Scientific Notebook**

In the same way that electronic calculators have replaced the abacus, computer mathematics software is increasingly being used by mathematicians to relieve them from tedious and lengthy calculations. It has now become essential for anyone using mathematics professionally to be able to make use of such software. To meet this aim we will be introducing you to the use of *Scientific Notebook* and at the same time using it to enable you to experiment with different mathematical techniques. This encourages discovery as well as allowing the analysis of more realistic problems without the burden of extensive calculations/manipulations.

The intention here is not to automate all calculations. The course material will require you to first develop mastery of all techniques through the use of pen-paper-calculator. Only when you are competent in handling this approach will you be allowed to switch to the software. This is important as you will not have access to the software in the examination.

*Scientific Notebook* combines the ease of entering text and mathematics in natural notation with the power of symbolic computation. This package, in fact, allows you to enter a mathematical expression and then to have the program manipulate it in a number of different ways. You can add explanatory text to your work and print it out for submission to your tutor.

**3.4 Audio programmes***

Some units include teaching delivered on audio recordings. These audio programmes (as well as the video programmes) should provide you with some variety in your study. In most programmes you will be led by the speaker through an example or exercise displayed in the unit.

Each unit will indicate exactly when you should listen to the audio section. You will need to have your unit, a pencil and some paper to hand. You should pause the recording whenever you are instructed to do so, and work on the exercises as suggested. Of course, you may want to pause the recording at other times to gain a better understanding of some of the material.

A special feature of the audio programmes is that both English and Chinese versions are provided. You can choose the language of presentation with which you think you'll learn the most.

* Please note that all the audio and video recordings for this course have been uploaded to the OLE for easy access and no physical CDs will be provided. In the course materials, all references to the 'audio/video CD' should be taken to mean 'audio/video recordings on the OLE'.

**3.5 Video programmes**

Some study units include sections based on videos. These are used when a visual presentation better suits the topic being studied. You should try your best to watch these programmes when directed by the unit. The content of each one will be briefly described in the unit. The programmes are in English with Chinese subtitles.

**3.6 Other materials**

**3.6.1 ***MATH S121* Handbook

Please read carefully the handbook regulation of this presentation as given below:

You will not be allowed to bring *MATH S121* Course Handbook to the exam. Another copy of *MATH S121* Handbook will be given to you together with the exam paper.

**3.6.2 Course schedule**

This sets out an overall timetable for the study weeks for each unit as well as when each assignment should be completed. It is available on the OLE.

Experience shows that spreading your study evenly over the presentation is very important. The submission dates of the assignments are designed to help you to do this. You are expected to have completed your study of the unit in time to be able to complete the assignment questions so that you can submit them before the associated due date.

**3.6.3 Problem Booklet**

Each block includes one of these booklets. Each one provides extra exercises that you can work through to help reinforce your understanding of the material. The *Problem Booklet* is divided into sections to match those in the units. You can pause your study at any point to work through relevant exercises.

You are not required to attempt every exercise in these booklets — only enough to give you confidence that you understand the associated technique/topic in the unit. You will also find these booklets a useful source of revision exercises for the examination.

**3.6.4 Stop press**

These act as a sort of course newsletter containing useful information or errata. They will be made available only through the OLE.