# Introduction to University Mathematics - Material for the year 2019-2020

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

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 language and notation of university mathematics, and to introduce the elements of naïve set theory and the nature of formal proof.

Students should:

(i) have the ability to describe, manipulate, and prove results about

sets and functions using standard mathematical notation;

(ii) know and be able to use simple relations;

(iii) develop sound reasoning skills;

(iv) have the ability to follow and to construct simple proofs, including proofs by mathematical induction (including strong induction, minimal counterexample) and proofs by contradiction;

(v) learn how to write clear and rigorous mathematics.

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 $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$, and intervals in $\mathbb{R}$. Inclusion, union, intersection, power set, ordered pairs and cartesian product of sets. Relations. Definition of an equivalence relation. Examples.

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

Writing mathematics. The language of mathematical reasoning; quantifiers: "for all", "there exists". Formulation of mathematical statements with examples.

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

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

1) C. J. K. Batty, *How do undergraduates do Mathematics?*, (Mathematical Institute Study Guide, 1994) https://www.maths.ox.ac.uk/system/files/attachments/study_public_1.pdf

2) K. Houston, *How to think like a mathematician*, (CUP, 2009)

3) L. Alcock, *How to study for a mathematics degree*, (OUP, 2012)

1) G. Pólya. *How to solve it: a new aspect of mathematical method*, (1945, New edition 2014 with a foreword by John Conway, Princeton University Press).

2) G. C. Smith, *Introductory Mathematics: Algebra and Analysis*, (Springer-Verlag, London, 1998), Chapters 1 and 2.

3) Robert G. Bartle, Donald R. Sherbert, *Introduction to Real Analysis*, (Wiley, New York, Fourth Edition, 2011), Chapter 1 and Appendices A and B.

4) C. Plumpton, E. Shipton, R. L. Perry, *Proof*, (MacMillan, London, 1984).

5) R. B. J. T. Allenby, *Numbers and Proofs*, (Butterworth-Heinemann, London, 1997).

6) R. A. Earl, *Bridging Material on Induction*, (Mathematics Department website).