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 (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.

Problem Sheet 1 solutions and Problem Sheet 2 .

**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.

Problem Sheet 2 solutions and Problem Sheet 3 .

**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.

Problem Sheet 3 solutions and Problem Sheet 4 .

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.

Problem Sheet 4 solutions and Problem Sheet 5 .

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.

Problem Sheet 5 solutions and Problem Sheet 6 .

Here is an elaboration on Euler's formula.

**Lecture 14 (12/11)**
Lecture 14.

Problem Sheet 6 solutions and Problem Sheet 7 .

**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.

Problem Sheet 7 solutions and Problem Sheet 8 .

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

**Lecture 18 (26/11)**
Lecture 18.

Problem Sheet 8 solutions and Problem Sheet 9 .

**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.

Problem Sheet 9 solutions and Problem Sheet 10 .

**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.

Christmas Problem Sheet 11 solutions

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, 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:**