## Dear Algebra Diary by Geoff Smith (MA10209, 2018)

Here is the timetable of lectures and problem classes. In week 1, problem classes will be lectures. You will either be timetabled for Block 1 or Block 2.
```Mon 11:15 8W1.1     Lecture A (Block 2)
Mon 14:15 EB1.1     Lecture A (Block 1)
Mon 16:15 Univ Hall Lecture B (Block 2)
Tue 09:15 Univ Hall Lecture B (Block 1)
Wed 10:15 3WN2.1    Problem Class (Block 2)
Wed 12:15 CB1.10    Problem Class (Blook 1)
```
Book The book which will help you with much of this course, and is widely used at British Universities, is Introductory Mathematics: Algebra and Analysis, by Geoff C Smith, published by Springer.

Aims and Objectives This course is an introduction to pure mathematics. If you follow the course, you will have a grasp of contemporary mathematical notation, and will become familiar with various methods of proof.

Need Help? You have the following options (in order of preference): 1 think, 2 read the lecture notes, 3 read the book, 4 Use a search engine, 5 talk to fellow students, 6 ask the person in charge of your tutorial group, 7 go to a workshop drop-in session MASH, 8 ask your personal tutor (unless you have one of the odd ones who wants to discuss your life) and if all else fails, 9 make an appointment with me by e-mail.

Discussions worth reading Over at Cambridge, Tim Gowers has a Weblog. While it is tailored to his first year lectures at his university, we can all learn a thing or two from Tim. There are Terry Tao's maths quizzes, which seem to be very attractive aids to learning pure maths in general (not just for this course). Both Terry and Tim have Fields Medals (these are like Nobel prizes, but for clever people).

Lecture 1 (1/10) Boole's rules are here. Problem Sheet 1. We introduced set notation. Lecture 1.

Lecture 2 (1/10 and 2/10) Lecture 2. We introduced map notation, and proved that composition of maps, when defined, is an associative process.

Lecture 3 (3/10) In week 1, the Problems Class slots will be used for an extra lecture. Lecture 3. We proved that each of the three classes of injective, surjective and bijective functions are closed under map composition, when it is defined. We defined the power set of a set, and proved that there is no surjective map from a set to its power set. We were exploring some ideas of Georg Cantor (the material in the link on "Philosophy, religion and Cantor’s mathematics" is particularly interesting).

Lecture 4 (8/10) Lecture 4. We stated the Schröder-Bernstein theorem. We also showed that there are infinitely many infinite sets, no two of which are in bijective correspondence. We characterized injective and surjective maps in terms of (not necessarily unique) left and right inverse maps. We showed that bijective maps have a unique left inverse, a unique right inverse, and that these rival inverses are equal.

Lecture 5 (8/10 and 9/10) Lecture 5. This concerned countability and uncountability. Extra Notes On compositional square roots in Problem Sheet 1. Also how to improve maps until they become bijections.

Lecture 6 (15/10) Lecture 6. Cantor's diagonal argument which proves that the real numbers are not a countable set. Also partitions and equivalence relations.

Please treasure these examples of notational abuse and diversity and examples of equivalence relations.

Lecture 7 (15/10 and 16/10) Lecture 7. More on relations, and the start of Number Theory.

Lecture 8 (22/10) Lecture 8. The theorem of Bezout and its consequences.

Lecture 9 (22/10 and 23/10) Lecture 9. This concerned the subtle part of Gauss's Fundamental Theorem of Arithmetic which concerns uniqueness, and the proof of correctness of Euclid's algorithm to calculate greatest common divisors.

The Fundamental Theorem of Arithmetic can go wrong if you try to generalize it. For example, consider Evenland

Lecture 10 (22/10) Lecture 10. Applications of Euclid's algorithm, and Euler's phi-function.

Lecture 11 (29/10 and 30/10) Lecture 11. The theorems of Fermat and Euler-Fermat, and the Chinese Remainder Theorem.

Here is a helpful sheet on the algebra of the integers modulo n.

Lecture 12 (5/11) Lecture 12.

Lecture 13 (5/11 and 6/11) Lecture 13.

Here is an elaboration on Euler's formula.

Lecture 14 (12/11) Lecture 14.

Lecture 15 (12/11 and 13/11) Lecture 15.

Here is a hand calculation of long division of integers and polynomials (UK style), and a polynomial gcd calculatioe

Lecture 16 (19/11) Lecture 16.

Lecture 17 (19/11 and 20/11) Lecture 17.

Lecture 18 (26/11) Lecture 18.

Lecture 19 (26/11 and 27/11) Lecture 19.

Here are some sheets to support your understanding of groups. There is stuff on cycle notation, on even and odd permutations and on cyclic groups.

Lecture 20 (03/12) Lecture 20.

Lecture 21 (03/12 and 04/12) Lecture 22.

Lecture 22 (10/12) Lecture 22.

The Vacation Problem Sheet with a Christmas theme. Here are Problem Sheet 10 solutions

Lecture 23 (10/12 and 11/12) Lecture 23.

Lecture 23 (10/12 and 11/12) Lecture 23.

For reasons beyond my understanding, some solutions are missing from the library's repository of past exams. Here is the exam sat in Jan 2013 and the associated solutions. The library has the exam paper for 2014, but not the solutions. Here they are: solutions for the exam in Jan 2014. Here is the exam sat in Jan 2016 and associated solutions. Here are the 2017 solutions and I left off the solution to question 7 so now you have it. (The marking schemes are sometimes approximate). If you think I have made a mistake in an answer, and you can find another student or staff member who agrees with you, then please let me know.

```Dr Geoff Smith MBE                         Email: G.C.Smith@bath.ac.uk
Department of Mathematical Sciences    Tel:   +44 (0)1225 386182 (direct)
University of Bath                     Fax:   +44 (0)1225 386492
Bath BA2 7AY                           Room:  4W2.16
England                                UKMT volunteer,
BMO chair,
IMO president,
IMO 2019 Ltd chair.
```
The email address G.C.Smith@bath.ac.uk is the official University of Bath format, though once upon a time the shiny new format was masgcs@bath.ac.uk -- the one that people actually use is, of course, entirely different: gcs@maths.bath.ac.uk -- as far as I know, all these incantations work.