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)

**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.

Problem Sheet 4. Here are some notes on equivalence relations.

**Lecture 8 (23/10)** Lecture 8.

Here are Problem Sheet 3 solutions.

**Lecture 9 (23/10 and 24/10)** Lecture 9.

**Lecture 10 (30/10)** Lecture 10.

Notes on the algebra of the integers modulo n.

**Lecture 11 (30/10 and 31/10)** Lecture 11.

Here are Problem Sheet 4 solutions.

Here are Problem Sheet 5 solutions.

**Lecture 12 (6/11)** Lecture 12.

**Lecture 13 (6/11 and 7/11)** Lecture 13.

**Lecture 14 (13/11)** Lecture 14.

**Lecture 15 (13/11 and 14/11)** Lecture 15.

**Lecture 16 (20/11)**
Lecture 16.

**Lecture 17 (20/11 and 21/11)** Lecture 17.

Here are some notes on cyclic groups and also some notes on cycle notation for permutations.

**Lecture 18 (27/11)** Lecture 18.

**Lecture 19 (27/11 and 28/11)** Lecture 19.

**Lecture 20 (4/12)**
Lecture 20.

Problem Sheet 10. Problem Sheet 9 solutions.

**Lecture 21 (4/12 and 5/12)**
Lecture 21.

**Lecture 22 (11/12)**
Lecture 22.

Christmas Vacation Problem Sheet. Problem Sheet 10 solutions.

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

Geoff's highly recommended important Christmas present to the students following this course is some essential reading.

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.

**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:**

- In Bath.

** Mental Whereabouts:**