This course is taken by Part IA (50% Option) students.

Lecturer: Dr M.P. Fiore

No. of lectures and mini-seminars: 12 + 4

Prerequisite course: Discrete Mathematics I

This course is a prerequisite for all theory courses as well as Security (Part IB and Part II), Artificial Intelligence (Part IB and Part II), Information Theory and Coding (Part II).

Aims

This part of the course will develop the theory of sets and their uses
in Computer Science.

Associated with this course is a series of four mini-seminars
involving the active participation of students. These are tickable, so
every 50% Option student must attend all four
seminars, and deliver a mini-seminar at one of them.

Lectures

Sets and constructions on sets.
Basic sets, comprehension, indexed sets, unions, intersections,
products, disjoint unions, powersets. Russell's paradox.

Relations, functions and partial functions.
Composition and identity relations. Lambda notation for functions.
Direct and inverse image of a relation.
Equivalence relations and partitions.
Cardinality especially countability, Cantor's diagonal argument.

Sets and logic.
Powersets as boolean algebras.
Characteristic functions.
Venn diagrams.

Well-founded induction.
Well-founded relations and well-founded induction. Other induction
principles as instances of well-founded induction.
Applications.

Introduction to inductive definitions.
Using rules to define sets.
Reasoning principles: the inductively defined set is the least set
closed under the rules; induction on derivations.
Simple applications.

Objectives

On completing this part of the course, students should be able to

understand and use the language of set theory; prove and
disprove assertions using a variety of techniques

understand boolean operations as operations on sets and
formulate statements using boolean logic

apply the principle of well-founded induction

define sets inductively using rules, and prove properties about
them

Recommended reading

Comprehensive notes will be provided.

Devlin, K. (2003). Sets, functions, and logic: an introduction to abstract mathematics. Chapman and Hall/CRC Mathematics (3rd ed.).
Biggs, N.L. (1989). Discrete mathematics. Oxford University Press.
Mattson, H.F. Jr (1993). Discrete mathematics. Wiley.
Nissanke, N. (1999). Introductory logic and sets for computer scientists. Addison-Wesley.
Pólya, G. (1980). How to solve it. Penguin.