# Department of Computer Science and Technology

Course pages 2019–20

Subsections

## Paper 2: Discrete Mathematics

This course is not taken by NST students.

Lecturers: Professor M. Fiore and Professor F. Stajano

No.of lectures: 24 (continued into Lent term)

Suggested hours of supervisions: 6

This course is a prerequisite for all theory courses as well as: Probability, Security, Artificial Intelligence, Compiler Construction and the following Part II courses: Machine Learning and Bayesian Inference and Cryptography

### Aims

The course aims to introduce the mathematics of discrete structures, showing it as an essential tool for computer science that can be clever and beautiful.

### Lectures

• Proof [5 lectures].

Proofs in practice and mathematical jargon. Mathematical statements: implication, bi-implication, universal quantification, conjunction, existential quantification, disjunction, negation. Logical deduction: proof strategies and patterns, scratch work, logical equivalences. Proof by contradiction. Divisibility and congruences. Fermat’s Little Theorem.

• Numbers [5 lectures].

Number systems: natural numbers, integers, rationals, modular integers. The Division Theorem and Algorithm. Modular arithmetic. Sets: membership and comprehension. The greatest common divisor, and Euclid’s Algorithm and Theorem. The Extended Euclid’s Algorithm and multiplicative inverses in modular arithmetic. The Diffie-Hellman cryptographic method. Mathematical induction: Binomial Theorem, Pascal’s Triangle, Fundamental Theorem of Arithmetic, Euclid’s infinity of primes.

• Sets [9 lectures].

Extensionality Axiom: subsets and supersets. Separation Principle: Russell’s Paradox, the empty set. Powerset Axiom: the powerset Boolean algebra, Venn and Hasse diagrams. Pairing Axiom: singletons, ordered pairs, products. Union axiom: big unions, big intersections, disjoint unions. Relations: composition, matrices, directed graphs, preorders and partial orders. Partial and (total) functions. Bijections: sections and retractions. Equivalence relations and set partitions. Calculus of bijections: characteristic (or indicator) functions. Finite cardinality and counting. Infinity axiom. Surjections. Enumerable and countable sets. Axiom of choice. Injections. Images: direct and inverse images. Replacement Axiom: set-indexed constructions. Set cardinality: Cantor-Schoeder-Bernstein Theorem, unbounded cardinality, diagonalisation, fixed-points. Foundation Axiom.

• Formal languages and automata [5 lectures].

Introduction to inductive definitions using rules and proof by rule induction. Abstract syntax trees.

Regular expressions and their algebra.

Finite automata and regular languages: Kleene’s theorem and the Pumping Lemma.

### Objectives

On completing the course, students should be able to

• prove and disprove mathematical statements using a variety of techniques;

• apply the mathematical principle of induction;

• know the basics of modular arithmetic and appreciate its role in cryptography;

• understand and use the language of set theory in applications to computer science;

• define sets inductively using rules and prove properties about them;

• convert between regular expressions and finite automata;

• use the Pumping Lemma to prove that a language is not regular.

### Recommended reading

Biggs, N.L. (2002). Discrete mathematics. Oxford University Press (Second Edition).
Davenport, H. (2008). The higher arithmetic: an introduction to the theory of numbers. Cambridge University Press.
Hammack, R. (2013). Book of proof. Privately published (Second edition). Available at:
http://www.people.vcu.edu/ rhammack/BookOfProof/index.html
Houston, K. (2009). How to think like a mathematician: a companion to undergraduate mathematics. Cambridge University Press.
Kozen, D.C. (1997). Automata and computability. Springer.
Lehman, E.; Leighton, F.T.; Meyer, A.R. (2014). Mathematics for computer science. Available on-line.
Velleman, D.J. (2006). How to prove it: a structured approach. Cambridge University Press (Second Edition).

75=75