Discrete Mathematics BSc

at University of Warwick

Your first year will establish the foundations of Discrete Mathematics and its applications, covering proof, formal arguments, rigour and calculations, as well as mathematical reasoning, combinatorial analysis and discrete structures.

In your second year, you will develop a rigorous understanding of the subject's theoretical basis, which will prepare you for later specialisation.

In your third year, you will work alongside academics on an individual project as well as focusing on applications of Discrete Mathematics to Computer Science, and completing advanced modules on algorithms and computation.

In each year of your course, you are expected to study a core group of modules and make up the required normal load for the year by choosing a set of optional modules. There is a choice of optional modules available and there may be requirements to be satisfied by the choices: that a minimum number be chosen from a specific list.

Year One
Programming for Computer Scientists
On this module, whatever your starting point, you will begin your professional understanding of computer programming through problem-solving, and fundamental structured and object-oriented programming. You will learn the Java programming language, through practical work centred on the Warwick Robot Maze environment, which will take you from specification to implementation and testing. Through practical work in object-oriented concepts such as classes, encapsulation, arrays and inheritance, you will end the course knowing how to write programs in Java, and, through your ability to analyse errors and testing procedures, be able to produce well-designed and well-encapsulated and abstracted code.

Design of Information Structures
Following on from Programming for Computer Scientists, on the fundamentals of programming, this module will teach you all about data structures and how to program them. We will look at how we can represent data structures efficiently and how we can apply formal reasoning to them. You will also study algorithms that use data structures. Successful completion will see you able to understand the structures and concepts underpinning object-oriented programming, and able to write programs that operate on large data sets.

Discrete Mathematics and its Applications 1
On this foundation module, you’ll learn the basic language, concepts and methods of discrete mathematics, while developing your appreciation of how these are used in algorithms and data structures. By the end, you should be able to appreciate the role of formal definitions, mathematical proofs and underlying algorithmic thinking in practical problem-solving. You’ll acquire knowledge of logic, sets, relations and functions, and learn summation techniques (manipulations and finite calculus) and concepts including asymptotics and the big-O notation to prepare you for more advanced techniques in computer science.

Discrete Mathematics and its Applications 2
During this module, you will build on your foundations in discrete mathematics through the study of concepts such as discrete probability and number theory; learning how to apply these methods in problem-solving. By the end of your course, you’ll be able to use algebraic techniques (including linear and matrix algebra) to analyse basic discrete structures and algorithms, and understand the importance of asymptotic notation, and be able to use it to analyse asymptotic performance for some basic algorithmic examples. Also, you will study the properties of graphs and related discrete structures, and be able to relate these to practical examples.

Linear Algebra
Linear algebra addresses simultaneous linear equations. You will learn about the properties of vector spaces, linear mappings and their representation by matrices. Applications include solving simultaneous linear equations, properties of vectors and matrices, properties of determinants and ways of calculating them. You will learn to define and calculate eigenvalues and eigenvectors of a linear map or matrix. You will have an understanding of matrices and vector spaces for later modules to build on.

Calculus 1/2
Calculus is the mathematical study of continuous change. In this module there will be considerable emphasis throughout on the need to argue with much greater precision and care than you had to at school. With the support of your fellow students, lecturers and other helpers, you will be encouraged to move on from the situation where the teacher shows you how to solve each kind of problem, to the point where you can develop your own methods for solving problems. By the end of the year you will be able to answer interesting questions like, what do we mean by `infinity’?

Sets and Numbers
It is in its proofs that the strength and richness of mathematics is to be found. University mathematics introduces progressively more abstract ideas and structures, and demands more in the way of proof, until most of your time is occupied with understanding proofs and creating your own. Learning to deal with abstraction and with proofs takes time. This module will bridge the gap between school and university mathematics, taking you from concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.

Introduction to Probability
This module takes you further in your exploration of probability and random outcomes. Starting with examples of discrete and continuous probability spaces, you will learn methods of counting (inclusion-exclusion formula and multinomial coefficients), and examine theoretical topics including independence of events and conditional probabilities. You will study random variables and their probability distribution functions. Finally, you will study variance and co-variance, including Chebyshev’s and Cauchy-Schwarz inequalities. The module ends with a discussion of the celebrated Central Limit Theorem.

Year Two
Combinatorics
In this module you learn the basics about discrete structures that lie at the heart of many real-world problems. A key notion is that of a graph, which is an abstract mathematical model for a network, such as a street network, a computer network, or a network of friendships. You learn to argue about these structures formally, and to prove interesting theorems about them. This will train your ability to think outside of the box.

Algorithmic Graph Theory
This project-based module will provide you with experience of designing, developing and implementing a significant project, under supervision. From submission of the outline and detailed specification, you will produce regular progress reports, until presenting your final results. This is an excellent opportunity to develop important professional business skills, including independent learning, self-discipline, organisation and time management. Providing you with experience of undertaking a significant individual design and development exercise from conception through to design, implementation and delivery.

Formal Languages
You will gain a fundamental understanding of formal languages and how the Chomsky hierarchy classifies them. You’ll study techniques for exploring the regularity of languages using closure properties and pumping lemmas, whilst also considering automata models, alongside the notion of computability. These concepts are central to computer science, and completion will see you able to specify between, and translate, various forms of formal language descriptions. You’ll learn methods of lexical analysis and parsing, and be able to argue whether a formal language is regular or context free. The teachings will discuss Turing machines and philosophical concepts such as decidability, reducibility and the halting problem.

Algorithms
Data structures and algorithms are fundamental to programming and to understanding computation. On this module, you will be using sophisticated tools to apply algorithmic techniques to computational problems. By the close of the course, you’ll have studied a variety of data structures and will be using them for the design and implementation of algorithms, including testing and proofing, and analysing their efficiency. This is a practical course, so expect to be working on real-life problems using elementary graph, greedy, and divide-and-conquer algorithms, as well as gaining knowledge on dynamic programming and network flows.

Year Three
Discrete Mathematics Project
Through this practical module, you’ll gain experience in undertaking a significant individual design and development exercise in discrete mathematics, from conception through to design, implementation and delivery. Starting with the selection of a topic and location of a suitable supervisor, you’ll be responsible for regular progress reports, and a presentation of your final results alongside a detailed written report. In addition to enhancing your technical knowledge, this process will help you develop important skills such as self-discipline, time management, organisation and professional communications.

Complexity of Algorithms
Are you ready for a challenge? In this module, you’ll learn to analyse the intrinsic difficulty of various computational challenges, and to specify variations that may be more tractable. This will require you to learn notions of the complexity of algorithms, and what makes some computational problems harder than others. You’ll undertake a close study of what makes an algorithm efficient, and study various models of computation, in particular, models of classical deterministic and non-deterministic computations.

Approximation and Randomised Algorithms
On this module, you will gain an introductory understanding of approximation and randomised algorithms, which often provide a simple, viable alternative to standard algorithms. You’ll learn the mathematical foundations underpinning the design and analysis of such algorithms. Whilst gaining experience of using suitable mathematical tools to design approximation algorithms and analyse their performance. You’ll also learn techniques for designing faster but weaker algorithms for particular situations, such as large running times. You can expect to cover important concepts, including linearity of expectation, Chernoff bounds, and deterministic and randomised rounding of linear programs.

Optional modules
Optional modules can vary from year to year. Example optional modules may include:

• Professional Skills
• Functional Programming
• Visualisation
• Computer Security
• Logic and Verification
• Groups and Rings
• Combinatorial Optimisation
• Introduction to Number Theory
• Metric Spaces
• Stochastic Processes
• Introduction to Mathematical Statistics

The precise modules available to students may depend on module prerequisites (i.e. for some module choices it is necessary for you to have taken a particular module in a previous year).

All applicants, regardless of their visa requirements, are required to show that their ability to understand and express themselves in both written and spoken English is sufficiently high for them to derive full benefit from their degree course. The level of competence required depends on what type of degree you wish to study. Many applicants can demonstrate this through a satisfactory level of achievement in, for example, GCSE English Language or the English courses within the International Baccalaureate.

Applicants whose academic qualifications meet the University’s admissions requirements, but whose English language qualifications are not accepted as equivalent, may be offered a place at Warwick conditional on an acceptable English language qualification being achieved before they join the University. A number of English language tests are accepted; some of the most frequently offered ones are detailed below.

Please note that certain English tests (including IELTS and PTE) must be taken within two years and one month prior to the start of your course at Warwick. Applicants whose tests fall outside this timescale will need to retake a suitable English language test.

Applicants who require a visa to study in the UK may, depending on the circumstances, be asked to provide evidence of a secure English language test (e.g. IELTS for UKVI (Academic)), rather than an examination (such as International Baccalaureate English) due to the visa requirements set by the UK government.

ENGLISH LANGUAGE REQUIREMENTS FOR UNDERGRADUATE STUDENTS (2022 Entry)

If you do not meet the English Language Requirements for your course in full, you may still be eligible for the course on condition of:

• Meeting the English Language Requirements for our Pre-sessional English course, and
• Successfully completing the Pre-sessional English course

Please use the table below to find the relevant band for the course(s) in which you are interested. You can sort the table by clicking on the heading of the column you wish to sort by.

If you are an overseas or EU student enrolling in 2022, your annual tuition fees will be as follows:

• Band 1 – £22,280 per year (classroom-based courses, including Humanities and most Social Science courses)
• Band 2 – £28,410 per year (laboratory-based courses, plus Theatre and Performance Studies, Economics, and courses provided by Warwick Business School, with exceptions)

Fees for 2023 entry have not been set. We will publish updated information here as soon as it becomes available, so please check back for updates about 2023 fee rates before you apply.

Warwick Undergraduate Global Excellence Scholarship 2022

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