Department of Computer Science and Technology

Part IA CST

# Discrete Mathematics

Principal lecturers: Prof Marcelo Fiore, Prof Frank Stajano
Taken by: Part IA CST
Term: Michaelmas (continuing in Lent)
Hours: 24
Format: In-person lectures
Suggested hours of supervisions: 6
Prerequisites: This course is a prerequisite for all theory courses.
This course is a prerequisite for: Category Theory, Compiler Construction, Computation Theory, Cryptography, Formal Models of Language, Introduction to Probability, Machine Learning and Bayesian Inference, Multicore Semantics and Programming
Exam: Paper 2 Question 7, 8, 9, 10
Past exam questions, Moodle, timetable

## 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.