Dear Algebra Diary by Geoff Smith (MA10209, 2017)


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 (2/10) Boole's rules are here. Problem Sheet 1. We introduced set notation. Lecture 1.

Lecture 2 (2/10 and 3/10) 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).

Here are pre-sessional problem sheet solutions. You can read these solutions if you have a University of Bath account. These solutions are not available to the wider community from here. Please do not share them.

Lecture 4 (9/10) 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.

Problem Sheet 2. Here are some notes on bijections. Here are Problem Sheet 1 solutions. Problem 5(d) of Sheet 1 inspired these notes on compositional square roots of the doubling map. See Sheet 1 Problem 5(d). Also several students have asked me questions about maps, so here is a document which might clarify a few things: playing with maps.

Lecture 5 (9/10 and 10/10) Lecture 5. This concerned countability and uncountability.

Problem Sheet 3. Here are Problem Sheet 2 solutions.

Lecture 6 (16/10) Lecture 6.

Lecture 7 (16/10 and 17/10) Lecture 7.

Lecture 8 (23/10) Lecture 8.

Lecture 9 (16/10 and 17/10) Lecture 9.


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. (The marking schemes are approximate, and from memory). 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.

Contact Information: My home page.

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, 
                                       EGMO advisory board.
				       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.

Physical Whereabouts:

Mental Whereabouts: