Dear Algebra Diary by Geoff Smith (MA10209, 2012)

Downloadable material

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 talk to fellow students, 5 ask the person in charge of your tutorial group, 6 go to a workshop drop-in session, 7 go to a MASH drop-in session, 8 Ask nRich (see below), 9 ask your personal tutor (unless you have one of the odd ones who wants to discuss your life) and if all else fails, 10 make an appointment with me by e-mail.

The best bulletin board that supports undergraduate mathematicians is nRich. You register at that site, and then click on the Ask nRich link. There are bulletin boards at various levels. The one appropriate to undergraduates is Higher Dimension. You may find some early undergraduate topics being discussed in Onwards and Upwards. The latter is really for people aged 16-18, but smart school students often discuss university mathematics.

Discussions worth reading Over at Cambridge, Tim Gowers is introducing a new feature this year, his Weblog. While it will be tailored to his first year lectures at his university, I expect we will 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).

Timetable Here is the course timetable. For this course, Lectures (and Problem Classes) are at Tuesday 9:15 University Hall, Thursday 16:15 University Hall and Friday 12:15 (Problem Class) University Hall.

The Mathematics Department has set up drop-in Workshops to advise students on this course (and others). These are 14:15 -- 16:05 Monday 1W4.40, 10:15 -- 12:05 Tuesday 1W4.40 and 17:15 -- 19:05 1W4.40. You can drop in to any of them for as long, or as short, as you wish. The workshops will not run in Week 1 but our Algebra 1A tutorials will run in week 1. There are also drop-in help sessions organized by MASH. These are on Tuesday 14:15 -- 16:05 in 1W2.5, Wednesday 13:15 -- 15:05 in 1W3.15 and Thursday 13:15 -- 15:05 in 4W1.7. Please let me know if any of this information is broken.

Syllabus: Here is the syllabus.

Feedback The best way to get feedback on your progress is via your tutor. Hand-in work by the specified time and date in the relevant 4W level 1 slot. Your tutor will comment on your work, which should be returned to you about a week later. It is a crime to go to a tutorial without having tried the problems on the current sheet. It is also a crime to go to a tutorial without knowing the meaning of every word used on the problem sheet (since if you do not understand the words, you will not understand the questions). It is most definitely not a crime to be stuck, confused or bamboozled, and your tutor should be able to help you there. Do not ask your tutor to show you how to do the problems. Rather your tutor should help you understand the problems with sufficient insight that you can do some of the problems yourself. Problems come in varying levels of difficulty, and it would be an extraordinary student who could regularly do them all. What do you do if you think I have made an error in lectures, on a question sheet or on a solution sheet? Well, first check if someone else agrees with you. Then if so, send me an e-mail with the relevant information.

Lecturer's Lecture Notes These are in my head, and therefore cannot be borrowed. If you miss a lecture or lectures, then borrow a set of notes from a reliable scribe, and copy them up by hand or photocopying machine.

Problem Sheet Solutions Written solutions will be put up at this site after the work has been handed in.

Lecture 1 Boole's rules are here. Problem Sheet 1. In this lecture we made three definitions and introduced some notation. We defined the notion of a set, and wrote x ∈ A to mean that x is an element of the set A. We defined the notion of a subset, and wrote A ⊆ B. We decided to write A = B when both A ⊆ B and B ⊆ A. It follows that there is a unique set with no elements, so we can apply the definite article to the empty set ø. The set of natural numbers is ℕ, the set of integers is ℤ, the set of rational numbers is ℚ, the set of real numbers is ℝ and the set of complex numbers is ℂ. We introduced interval notation (a,b), [a,b], (a, b] and [a,b]. Thus, for example, (1,2) = { x | x ∈ ℝ, 1 < x, x < 2}. Finally I asked if there was a collection of open intervals with the property that the intersection of each pair is not the empty set, but the intersection of the whole collection is the empty set. Incidentally, I spent last Sunday marking the 2012 UK Mathematical Olympiad for Girls.

Lecture 2 We introduced more notation, including that for intersection and union. We discussed Boole's rules at some length, including De Morgan's laws. We began to discuss maps. A dark warning was issued against allowing the set of all sets to be a set, for Bertrand Russell is waiting for the unwary, and will hit you with a paradox if you do that. Note the quality of the moustache.

Lecture 3 We introduced the identity map on a set, and constant maps. We defined composition of maps, and showed that composition of maps, where defined, is an associative process. We defined injective, surjective and bijective maps. We proved that each of these last three types of map is closed under map composition (Proposition 10). Here is Problem Sheet 2, due in on Monday October 17.

Here are Solutions to sheet 1. Note the use of the "maps to" symbol, a right arrow with a short vertical tail. This is used to describe how a map acts. Thus f: x |-> x^2 means the same as f(x) = x^2. Sorry about the home made symbols, I am still looking for the way to display the "maps to" symbol in html. This is not the same symbol as the right arrow which sits between the domain and codomain of a map.

Lecture 4 We characterized injections, surjections and bijections between non-empty sets in terms of the existence of left, respectively right, respectively (unique) two sided inverses. We introduced the notion of the power set of a set. We showed that if A is a finite set, then |P(A)| = 2ⁿ. We began the proof that for any set B, there is no bijection between B and P(B).

Lecture 5. We finished the proof mentioned above. We proved that there is a bijection between the natural numbers and the integers. We proved that if S is an infinite subset of the natural numbers, then there is a bijection between S and the natural numbers. We proved that there is a bijection between the set of ordered pairs of natural numbers and the natural numbers. We stated (but will not prove) that if A and B are sets, then either there is an injection from A to B, or there is an injection from B to A (or possibly both). We also stated, but will not prove in lectures, the Schroeder-Bernstein Theorem, that if A and B are sets, and there are injections both from A to B, and from B to A, then there is a bijection between A and B. Problem Sheet 3 and Sheet 2 Solutions.

Lecture 6 We defined countability: a set S is countable if (and only if) there is an injection f : S → ℕ We proved that the set ℚ of rational numbers is (infinite) countable. We used Cantor's diagonal argument to show that the real interval [0,1/2] is not countable, and so ℝ is not countable. Note that there was a typo (chalko?) in the diagonal argument. At one point I wrote y_i when I should have written y_ii.

Lecture 7 Here are notes v2 on partitions and equivalence relations. here are the changes to version 1 in case you have it. First line, R is a subset of S X S, not R is a subset of S. In "Properties of Relations", the erroneous spelling reflextive was eliminated. In Examples of Partitions (iii), change "for" to "form". In Examples of Equivalence Relations (ii), insert the missing comma after the word "sets". In the discussion after teh Examples of Transversals, correct the mangled spelling of "equivalence". In teh final part on scary notation, change [3] to [2].

At the Problems Class on Friday October 19 2012, a student called Miles proposed a better solution to Sheet 2, problem 8(d), than the one which I had suggested in the solutions sheet.

Here is Problem Sheet 4

Lecture 8 We started number theory. We defined prime numbers, and proved that there are infinitely many of them. We defined coprimality, and we showed that if m and n are integers and not both 0, then the smallest positive integer g expressible as rm + sn (with integers r and s) is the greatest common divisor of m and n, and moreover that every common divisor of m and n will divide g. We stated the Fundamental Theorem of Arithmetic, that every positive integer greater than 1 is the product of prime numbers in a (more or less) unique way. We got as far as proving the existence of such a factorization, and will address uniqueness in the lecture on Thursday. This note addresses the chalko in Euclid's proof, and the matter of good housekeeping (induction) as promised in the lecture. Here are Sheet 3 Solutions.

Lecture 9 We completed the proof of Gauss's Fundamental Theorem of Arithmetic. We discussed how to count the number of divisors of a natural number by looking at its factorization into prime numbers. We compared the prime factorization of positive integers m and n with their prime factorizations. We discussed Euclid's algorithm, why it terminates, and why it gives the gcd of two positive integers as the output. Here is a question which has a nice answer: "what is the sum of the divisors of 1000?". You can do it in your head (provided that you are relaxed about multiplying a three-digit number by a two-digit number, and who isn't?).

Sheet 4 Solutions and Problem Sheet 5.

Lecture 10 We endowed the integers mod n with well-defined addition and multiplication. We defined the notion of a group, and gave several examples. Here are some notes which will assist you with Sheet 5, Questions 1 and 2 in particular, and life in general.

Lecture 11 We discussed Groups and Rings. Here is a handout on these algebraic structures.

Sheet 5 Solutions and Problem Sheet 6.

Nov 6 Lecture 12 We proved the Chinese Remainder Theorem, and gave lay interpretations.

Thu Nov 8 Lecture 13 We introduced Euler's φ-function, and proved that it was multiplicative with respect to coprime arguments (by exploiting the Chinese Remainder Theorem). We also proved that of f, g are polynomials in K[X] where K is a field, and g is not the zero polynomial, then there are q, r in K[X] such that f = qg + r and deg r < deg g. There were a couple of glitches. (i) In the proof that deg(f + g) ≤ max {deg f, deg g}, I wrote (a_l + b_l)X^k+l but the exponent should be l rather than k + l. (ii) Also in the calculation of φ(p^k), the set being subtracted should have been { tp | 0 ≤ t < p^k-1} (and not { tp | 0 ≤ t < p}).

Here is Problem Sheet 7 and Sheet 6 Solutions.

Tue Nov 13 Lecture 14 The theory of K[X] is astonishingly similar to that of the integers.

Thu Nov 15 Lecture 15 We began the theory of linear maps and matrices. Here is Sheet 7 solutions and Problem Sheet 8.

Tue Nov 20 Lecture 16 More on linear maps and matrices. Small glitch in one proof. Fix to follow. How to invert a 2 by 2 real matrix (or indeed, any 2 by 2 matrix with entries in a field). There was a glitch in the lecture, in (iii) where we were establishing that if the determinant of a real 2 by 2 matrix is 0, then it has no inverse. Here is a fix. Here is Problem Sheet 9 Sheet 8 solutions.

Thu Nov 22 Lecture 17 The area/volume interpretation of determinants for 2 by 2 and 3 by 3 matrices with real entries.

Tue Nov 27 Lecture 18 Inverting an invertible square matrix. Sheet 9 solutions. Problem Sheet 10.

Tue Dec 4 Lecture 19 Development of Group Theory. Cosets, Lagrange;s Theorem. Fermat's Theorem and the Fermat-Euler Theorem as examples of Lagrange's theorem. Homomorphisms, kernels and images.

Thu Dec 6 Lecture 20 Each element of S_n is either an even permutation or an odd permutation, and no permutation is in both categories. Therefore S_n is the union of two cosets of A_n, and A_n has size n!/2. Sheet 10 solutions. Vacation Problem Sheet 11.

Tue Dec 11 Lecture 21 The use of the sign of a permutation to define a determinant of a square matrix as an alternating sum of monomials.

Thu Dec 13 Lecture 22 The orbit stabilizer theorem, and Platonic solids. Vacation Problem Sheet 11 Solutions.

Contact Information: My home page.

Dr Geoff Smith                         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 advisory board. 
                                       EGMO advisory board.

Physical Whereabouts:

• In Bath.

Mental Whereabouts:

• ZF + AC + CH.