Synopsis for Introduction to Pure Mathematics

There will be 8 introductory lectures in the first two weeks of Michaelmas term.

Overview

Prior to arrival, undergraduates are encouraged to read Professor Batty's study guide "How do undergraduates do Mathematics?"

The purpose of these introductory lectures is to establish some of the basic notation of mathematics, introduce the elements of (na\" ive) set theory and the nature of formal proof.

Learning Outcomes

Students will:

1. have the ability to describe, manipulate, and prove results about sets and functions using standard mathematical notation;
2. know and be able to use simple relations;
3. develop sound reasoning skills;
4. have the ability to follow and to construct simple proofs, including proofs by mathematical induction (including strong induction, minimal counterexample) and proofs by contradiction.

Synopsis

The natural numbers and their ordering. Induction as a method of proof, including a proof of the binomial theorem with non-negative integral coefficients.

Sets: examples including the natural numbers, the integers, the rational numbers, the real numbers. Inclusion, union, intersection, power set, ordered pairs and cartesian product of sets. Relations. Definition of an equivalence relation. Examples.

Maps: composition, restriction; injective (one-to-one), surjective (onto) and invertible maps; images and preimages.

Rules for writing mathematics with examples. Formulation of mathematical statements with examples. Hypotheses, conclusions, "if", "only if", "if and only if", "and", "or". Quantifiers: "for all", "there exists".

Problem solving in mathematics: experimentation, conjecture, confirmation, followed by explaining the solution precisely.

Proofs and refutations: standard techniques for constructing proofs; counter examples. Example of proof by contradiction and more on proof by induction.

Reading

1. C. J. K. Batty, How do undergraduates do Mathematics? (Mathematical Institute Study Guide, 1994).

Further Reading

1. G. C. Smith, Introductory Mathematics: Algebra and Analysis (Springer-Verlag, London, 1998), Chapters 1 and 2.
2. Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis (Wiley, New York, Third Edition, 2000), Chapter 1 and Appendices A and B.
3. C. Plumpton, E. Shipton, R. L. Perry, Proof (MacMillan, London, 1984).
4. R. B. J. T. Allenby, Numbers and Proofs, (Arnold, London, 1997).
5. R. A. Earl, Bridging Material on Induction. (Mathematics Department website.)
6. G. Pólya. How to solve it: a new aspect of mathematical method (Second edition, Penguin, 1990)