Foundations of Pure Mathematics A
|Unit level:||Level 1|
|Teaching period(s):||Semester 1|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
The course unit unit aims to:
- address the problem of enumerating sets - both finite and infinite!
- to supply evidence that such problems are both intriguing and provocative, and require rigorous proof;
- to explain the fundamental ideas of sets, numbers and functions;
- to compare and contrast language and logic;
- to introduce a detailed study of the integers, including prime numbers and modular arithmetic;
- to show how mathematicians generalize ideas, so unique factorization of integers is shown to hold for permutations;
- to introduce the ideas of algebraic structures and so show, by examples throughout the course, how the same structure can arise in many different situations.
This course introduces students to the concept of proof, by studying sets, numbers and functions from a rigorous viewpoint. These topics underlie most areas of modern mathematics, and recur regularly in years 2, 3, and 4. The logical content of the material is more sophisticated than that of many A-level courses, and the aim of the lectures is to enhance students' understanding and enjoyment by providing a sequence of interesting short-term goals, and encouraging class participation.
On completion of this unit successful students will be able to:
- Analyse the meaning of mathematical statements involving quantifiers and logical connectives, and construct the negation of a given statement.
- Construct truth tables of simple mathematical statements and use these to determine whether two given statements are equivalent.
- Construct elementary proofs of mathematical statements using a range of fundamental proof techniques (direct argumentation, induction, contradiction, use of contrapositive).
- Use basic set theoretic language and constructions to prove results about finite, denumerable and uncountable sets.
- Use elementary counting arguments, such as the pigeonhole principle, the inclusion-exclusion formula and the binomial theorem to compute cardinalities of finite sets and simplify expressions involving binomial coefficients.
- Recall formal definitions and apply these to give examples and non-examples of functions, bijections, equivalence relations, binary operations and groups.
- Recall and justify basic number-theoretic methods, including the Euclidean algorithm, and use them to solve simple arithmetic problems such as linear Diophantine equations.
- Use modular arithmetic to solve linear and simple non-linear congruences.
- Recall the fundamental properties of prime numbers, prove their infinitude and solve elementary problems on primes and prime factorisation.
- State and prove Fermat's Little Theorem and Euler’s theorem and apply them to solving simple questions involving primality testing and Euler’s phi-function.
- Recognise the two-line notation and the cycle notation for permutations and use them to compose, invert and find the order of given permutations.
- Other - 25%
- Written exam - 75%
Assessment Further Information
Supervision attendance and participation: Weighting within unit 10%
Coursework: Weighting within unit 15%
Three hours end of semester examination: Weighting within unit 75%
This covers chapters 1 to 14 of the course text omitting 5.4, 9.4, 11.3, 12.1, 12.2 and 12.3.
Lecture 0: Provocative Examples. First thoughts on infinite sets; examples versus proof; an extended version of Hilbert's Hotel; Cantor's fate (and redemption!).
2. Basic Logic
Lecture 1: The Language of Mathematics. Mathematical statements, propositions; or, and, not; truth tables.
Lecture 2: Implication. Implication; necessary and sufficient, iff; rules of arithmetic; where proofs begin.
Lecture 3: Proof. Direct proof; case-by-case; working backwards.
Lecture 4: Contradiction. Negative statements; proof by contradiction; contrapositives; statements with 'or'.
Lecture 5: Induction. The induction principle and proof by induction; changing the base case; inductive definitions.
Lecture 6: History and Language. Historical origins, natural numbers to complex numbers; notation, belongs to, definitions (by listing, by conditions, by construction); subsets, equality.
Lecture 7: Operations on Sets. Union , intersection, empty set (with class discussion!); identities, Venn diagrams; power set.
Lecture 8: Quantifiers. Universal and existential statements, proof and negation of statements with quantifiers; induction revisited; two free variables.
Lecture 9: Further operations. Cartesian products; power sets.
Lecture 10: Definitions and Examples. Notation; domain, codomain; formulae and examples (including empty set); equality; restriction.
Lecture 11: More Definitions and Examples. Composition; image; sequences and indexing; restriction, graphs.
Lecture 12: Jections. Injections, surjections, bijections; their compositions.
Lecture 13: Inverse functions. Bijections and inverse functions; inverse images; transpositions.
6. Counting Finite Sets.
Lecture 14: Cardinality. Role of the integers, cardinality, is it well-defined? reduction to pigeon-hole principle.
Lecture 15: Principles of Counting. Principles of addition, multiplication, inclusion-exclusion.
Lecture 16: The Pigeon-hole Principle. Proof of the pigeon-hole principle, finite sets of real numbers.
7. Counting Infinite Sets.
Lecture 17: Rationals and Reals. Definition of rationals, and of reals as infinite decimals; Ã' 2.
Lecture 18: Countability. Denumerability, Ã€ 0 , proper subsets; Cartesian products, integer pairs, rationals.
Lecture 19: Uncountability. The reals; power sets and their cardinality.
Lecture 20: Other Types of Real Number. Algebraic and transcendental numbers.
1. Counting Collections of Functions and Subsets.
Numbers of injections and bijections. Numbers of subsets and Binomial Numbers, Pascal's Triangle, Binomial Theorem.
2. Arithmetic - the study of the integers.
Division Theorem, greatest common divisor, Euclid 's Algorithm, linear Diophantine equations.
3. Congruence and Congruence Classes.
Congruences, modular arithmetic, solving linear congruences, Chinese remainder theorem, congruence classes, multiplication tables*, invertible elements, reduced systems of classes*.
4. Partitions and Relations.
Partitions, relations, generalizing congruence classes.
Bijections, two row notation, composition, cycles, the permutation group Sn, composition tables, factorization, order.
6. Prime Numbers.
Sieve of Erastosthenes, infinitude of primes, conjectures about primes, Fermat's Little Theorem, Euler's Theorem*.
7. Groups, Rings and Fields*.
Definitions and very simple properties. Examples from earlier in the course
Students will acquire a copy of
P.J. Eccles, An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, Cambridge University Press, 1997
which will be available as an e-book in Blackboard.
Feedback seminars will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 44 hours
- Tutorials - 11 hours
- Independent study hours - 145 hours