Introductory description

Discrete mathematics forms the mathematical foundation of computer science and cyber security. It forms the basis of how computers work, allows us to prove system correctness and security, and underlies modern cryptography. This course introduces the discrete structures used by computers, as well as how to use them to solve problems in cyber security.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Numbers and sets: Basic algebra: Types of numbers and their properties. Sets and their operations, set countability and power sets.
Logic and proof: Proposition and predicates. Common proof techniques including direct proof, contrapositive, exhaustion, induction and more. Common mistakes made in proofs.
Functions and relations: Functions and their inverses. Injections, surjections and bijections. Properties of relations including (anti-)symmetry, reflexivity and transitivity. Partial orderings, total orderings and equivalence relations.
Number theory: Divisibility, modular arithmetic and Fermat's little theorem. The fundamental theorem of arithmetic. The GCD and LCM, and Euclid's algorithm.
Combinatorics: The multiplication and addition principles. Permutations.
Probability: Uniform distribution. Independent and mutually exclusive events.
Graph theory: Adjacency, distance and incidence matrices. Social network theory.

Learning outcomes

By the end of the module, students should be able to:

• Use a variety of techniques to prove and disprove mathematical statements
• Recognise common mistakes made in mathematical proofs
• Analyse the probabilities of random events, and use their findings to make informed recommendations
• Understand how people form social networks from a graph-theoretic perspective
• Use number-theoretic techniques to develop simple cryptographic systems
• Use formal verification techniques to analyse the security of simple programs

Indicative reading list

Johnsonbaugh, Richard, "Discrete mathematics", 8 Ed, Pearson Education Limited (2019)

Balakrishnan, V. K., “Schaum's Outline of Combinatorics”, McGraw-Hill (1995)

Karumanchi, Narasimha, “Data Structures and Algorithms Made Easy: Data Structure and
Algorithmic Puzzles”, 2 Ed, CareerMonk (2011)

View reading list on Talis Aspire

Subject specific skills

This course equips students with the foundational mathematical skills necessary in computer science and cyber security, including logic and proof, functions and their inverses, graphs, and probability, and applies these skills in a cyber context.

Transferable skills

Numeracy, logical reasoning, problem solving, written communication skills, and increased numerical confidence