A book which will help you with much of this course, and is widely used at British Universities, is Introduction to University Mathematics: Algebra and Analysis, by Geoff C Smith, published by Springer.
If you need help with this course, 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.
Also Wolfram Mathworld is useful mathematical resource. If you want to play and learn mathematics, not necessarily about this course, try cut-the-knot. If you are a truly sad person looking for something to do in those long winter evenings, then you might wonder how many centres a triangle has. You could even visit the Encyclopedia of Triangle Centers.
Here is the course timetable.
For this course, Lectures are Monday 11:15 East Building, Wednesdays 09:15 University Hall (changing to Tuesday 09:15 University Hall from week 2 onwards) and Friday 12:15 University Hall. Sometimes one of these events will be a Problems Class, where solutions to a problem sheet will be discussed.
The Mathematics Department has set up drop-in Workshops to advise students on this course (and others). These are the Monday Workshop 1W 4.40, 14:15-16:05; the Tuesday Workshop 1W 4.40, 10:15-12:05 and the Thursday Workshop 3WN 3.7, 16:15-18:05. These workshops are run by the maths department, and 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.
Here is the syllabus.
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.
Lecture 1 Boole's rules are here. Problem Sheet 1. We introduced some famous sets, and explored the notation for operations on sets. Quantifiers for all (inverted A) was introduced, and we just had time to define a function (map, mapping). 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 2 We introduced the quantifier there is, there are, there exist, there exists (inverted E). We defined constant maps, surjective maps, injective maps and bijective maps. Proposition 3 showed that the composition of two surjections is a surjection, the composition of two injections is an injection, and that the composition of two bijections is a bijection. Proposition 4 showed that, where defined, composition of maps is associative. We empasized that two maps are equal iff three things are true: same domains, same codomains and same action. We also observed that a bijective map gives rise to a map `going backwards'.
Problem Sheet 1 typos. Two minor typos were reported and have been corrected in problem sheet 1 v2 (version 2). The links on this website now point to the corrected version, but you may wish to amend your paper version by hand. Here was the error report from tutor Carolyn Ashurst: In Q4, you have f(x) = x/2 rather than g(x) = x/2. In Q8a, one of your R's is not blackboard bold. Thanks.
Minor Timetable Change From week 2 onwards, our Wednesday 9:15 Algebra 1A lecture will move to Tuesdays, with the exception of Week 3, when it will revert to Wednesday again for one week only (I have a prior engagement on October 18). The location, the University Hall, is unchanged.
Lecture 3 Suppose that f: A → B is a map between non-empty sets. We proved that f is injective if, and only if, it has a left-inverse g, i.e. a map g: B → A such that g o f = IdA. In the same spirit, we proved that f is surjective if and only if it has a right-inverse, i.e. a map h: B → A such that f o h = IdB.
It then follows that f is bijective if and only if it has both a left inverse and a right inverse, but in these circumstances we showed that there is only one possible left-inverse, only one possible right-inverse, and that they are the same map. It therefore makes sense to talk of the (unique) two-sided inverse of a bijective map. We write it as f -1. We noted that f -1 was also a bijection, with inverse f. Finally we showed that there is no bijection between a set and its power set. To clarify the definition of a power set: let X be a set. The collection of all possible subsets of X is called the power set of X. Thus if X = {1,2}, then the power set P(X) is { ∅, {1}, {2}, {1,2} }, a set which has just four elements: ∅, {1}, {2} and {1,2}. It follows that there are infinite sets between which there is no bijection. A final remark on the power set of the empty set. This is P(∅) = { ∅ }. Note that { ∅ } ≠ ∅ because the left-hand set is not a subset of the right-hand set. This is because the left-hand set contains an element ∅ which is not an element of the right-hand set ∅. Clear? Here is Problem Sheet 2.
Lecture 4 Here are the Sheet 1 Solutions. We showed that the natural numbers and the integers are in bijective correspondence. We used Cantor's diagonal argument to show that the natural numbers and the real numbers are not in bijective correspondence. We defined the notion of being equinumerous.
Lecture 5 We defined countably infinite and countable. We introduced the term cardinality and the notation |A| for the size (or cardinality) of a set. We introduced the sensible notation Im f for the image (or range) of a function f: A → B, and the bizarre notation f -1(Y) for the preimage of the subset Y of B (bizarre because f need not be bijective, so that it need not be the case that a function f -1 exists). We made reference to fibres (preimages of singleton sets). Finally we defined a partition Xi (for i ∈ I) of a set X. Each Xi must be non-empty, their union must be X, and if Xi ∩ Xi ≠ ∅, then i = j. We gave several examples of partitions, and observed that if f : A → B is a map, and for each y ∈ Im f we let Ay = { a | f(a) = y }, then the sets Ay, for y ∈ Im f, form a partition of A. We mentioned Srinivasa Ramanujan, and his work on the partition function.
Lecture 6 Transversals, relations and equivalence relations. A relation R on a set X is, by definition, a fixed subset R of X × X. We often use a infix notation for `being related' such as ~. If ~ is an Equivalence Relation and x ∈ X, then we write [x] = { y | y ∈ X, x ~ y } for the equivalence class of y. The equivalence classes form a partition of X. A transversal for a partition, or for an equivalence relation, is a set T which intersects each element of the relevant partition in a singleton set.
Lecture 7 On the other hand, a partition gives rise to an ER, the equivalence classes of which are the sets in the partition. Convesely, the ER defined by the equivalence classes of an ER is the original ER. We started § 2, Number Theory. We explained the notions of an irreducible integer and a prime positive integer. We proved that each integer, other than 1 and -1, is divisible by at least one prime number. We proved, as Euclid did, that there are infinitely many prime numbers.
Here are Problem Sheet 3 and Sheet 2 Solutions
Lecture 8 We proved that if m, n are integers, not both 0, then the greatest common divisor g of m and n is the smallest positive integer of the form λm + μ n. The proof shows that the common divisors of m and n are precisely the divisors of g. From this we deduces that if p is a prime number and it divides a product of two integers, then it must divide one of them. In fact, by induction, we could have proved that if p divides a product of finitely many integers, then it divides (at least) one of them. We stated and proved Gauss's Fundamental Theorem of Arithmetic which states that every positive integer n bigger than one has an essentially unique factroization into prime numbers. The only wriggle room is the order in which the factors are multiplied. We observed that 109 has exactly 100 divisors.
Lecture 9 We discussed Euclid's algorithm, and why it terminates (stops) and also why it outputs the gcd of the inputs. We also observed that unpicking Euclid's Algorithm gives you a way of expressing the gcd of a and b as an integral linear combination a and b. Someone asked about the abbreviation wlog meaning `without loss of generality'. We had used it when making the assumption that b > 0 when analyzing Euclid's Algorithm. This looks like an unjustified assumption, because b might be 0 or perhaps b < 0. However, if b = 0, then a is not 0, so we swap the roles of a and b. If b is still being awkward, we just change its sign. This does not change the gcd. Therefore although we seem to be making an unjustified assumption about b being positive, we are actually dealing with the general case. We emphasize this with the phrase `without loss of generality'. Mathematicians sometimes write wma wlog to mean `we may assume, without loss of generality'. There is also wma wotlog standing for `we may assume, with only temporary loss of generality'. This indicates that you are only dealing with a special case of the problem now, but that later on this will unlock the general problem.
Here is an extra document on counting and choosing which elaborates on the solution to Problem 3 of Sheet 3.
Lecture 10 We discussed algebraic properties of Z_n. We observed that the natural maps from Z to Z_n respect algebraic structure. We defined a Group. We gave many examples of groups.
Lecture 11 We discussed rings, integral domains and fields. We observed that if p is prime, then Z_p is a field.
I promised a handout on Groups, Rings and Fields at the Problems Class on Friday Nov 4th.
Lecture 12 We proved the Chinese Remainder Theorem in two versions. We have a detailed proof in the case that there are just two coprime moduli. We then indicated how the proof could be modified to provide a version of the theorem when there are t pairwise coprime moduli.
Here is an extra document on Abstract Algebra which concerns groups, rings and fields.
Lecture 13 We showed that the remainder theorem holds for polynomials with coefficients in a field, and introduced the notion of an irreducible polynomial.
Lecture 14 We discussed how to prove an analogue of the Fundamental Theorem of Arithmetic for polynomials over a field. We discussied a polynomial version of the Chinese Remainder Theorem, and build a field of size 4.
Problem Sheet 8 (This is the improved Sheet 8, which contains the accidentally omitted problem 1(d)). You may need to refresh your browser to get the new version.
Lecture 15 We introduced matrices, and linear maps from Rn to Rm.
Lecture 16 We discussed the algebra of matrices, and how an m by n matrix gives rise to a linear map from Rn to Rm.
Problem Sheet 9 and a href = "http://people.bath.ac.uk/masgcs/s8sol.pdf">Sheet 8 Solutions
Lecture 17Recap of the definition of a group. Examples. Problem Sheet 10
Lecture 18This was an extra bit of lecture, caused by a short problem class. Group homomorphisms. Images and Kernels are subgroups.
Lecture 19Monomorphisms, epimorphisms, isomorphisms, automorphisms. Fibres of a homomorphism, and a homomorphism is injective if and only if its kernel is trivial. Sheet 9 Solutions
Problem Sheet 11, the vacation sheet. Here are the Sheet 11 Solutions for the vacation problems.
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, IMOAB member.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:
Whereabouts:
