This course is a prerequisite for all theory courses as well as Introduction to Security (Part IB), Artificial Intelligence I and II, Information Theory and Coding (Part II), Security (Part II).
Aims
This part of the course will develop the theory of sets and their uses
in Computer Science. The Lent Term half of the Discrete Mathematics
course will include a series of mini-seminars involving the active
participation of students.
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 define set is the least set
closed under the rules; induction on derivations.
Simple applications.
Tarski's fixed point theorem for monotonic functions on a powerset.
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 books
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.
