MA138-12 Sets and Numbers
Mathematics can be described as the science of logical deduction - if we assume such and such as given, what can we deduce with absolute certainty? Consequently mathematics has a very high standard of truth - the only way to establish a mathematical claim is to give a complete, rigorous proof. Sets and Numbers aims to show students what can be achieved through abstract mathematical reasoning.
University mathematics introduces progressively more and more abstract ideas and structures, and demands more and more in the way of proof, until by the end of a mathematics degree most of the student's time is occupied with understanding proofs and creating his or her own. This is not because university mathematicians are more pedantic than schoolteachers, but because proof is how one knows things in mathematics, and it is in its proofs that the strength and richness of mathematics is to be found. But learning to deal with abstraction and with proofs takes time. This module aims to bridge the gap between school and university mathematics, by beginning with some rather concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
Number systems: Natural numbers, integers, rationals and real numbers. Existence of irrational numbers.
Euclidean algorithm; greatest common divisor and least common multiple.
Prime numbers, existence and uniqueness of prime factorisation (and non-uniqueness in other “number systems”, e.g. even integers, Gaussian integers).
Properties of commutativity, associativity and distributivity.
Infinity of the primes.
Summing series of integers; proofs by induction.
Basic set theory: Intersection, Union, Venn diagrams and de Morgan’s Laws.
Logical connectives and, or, implies and their relation with intersection and union
Multiplication and long division of polynomials.
Euclidean algorithm for polynomials.
Remainder theorem; a degree n polynomial has at most n roots.
Rational functions and partial fractions.
Incompleteness of the real numbers, completeness of the complex numbers (sketch).
Elementary combinatorics as practice in bijections, injections and surjections.
Cardinality of the set of subsets of a set X is greater than cardinality of X.
Definition of Cartesian product.
Countability of the rational numbers, uncountability of the reals.
Transcendental numbers exist!
The second (and smaller) part of the module explores the elementary properties of
a fundamental algebraic structure called a group. Groups arise in an extraordinary range
of contexts in mathematics and beyond (for example, in elementary particle physics and in
card tricks), and can be used to analyse the symmetry of geometric objects or physical systems.
Modular arithmetic: 3 hours:
Addition, multiplication and division in the integers modulo n.
Some theorems of modular arithmetic.
Permutations and the symmetric group:
Multiplying (composing) permutations.
Cycles and disjoint cycle representation.
The sign of a permutation.
Basic Group Theory
By the end of the module, students should be able to:
- Work with number systems and develop fluency with their properties
- Learn the language of sets and quantifiers, of functions and relations
- Become familiar with various methods and styles of proof
Indicative reading list
None of these is the course text, but each would be useful, especially the first:
A.F.Beardon, Algebra and Geometry, CUP, 2005.
I.N. Stewart and D.O. Tall, Foundations of Mathematics, OUP, 1977.
J. A. Green, Sets and Groups; First Course in Algebra, Chapman and Hall, 1995.
Subject specific skills
Sets and Numbers will provide students with an introduction both to the language and to the methodology of university level mathematics. By studying familiar objects, but from a deeper and more rigorous perspective, they will become accustomed to constructing and evaluating logical arguments, as well as learn how to communicate these arguments precisely. These skills will be used throughout the rest of their degree.
Critical thinking, problem solving and analytical skills, group work.
|Lectures||30 sessions of 1 hour (25%)|
|Tutorials||8 sessions of 30 minutes (3%)|
|Private study||86 hours (72%)|
Private study description
Review lectured material and work on set exercises.
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D2
|Four fortnightly multiple choice tests||15%|
multiple choice tests
|1.5 hr examination (January)||85%|
Assessment group R
Feedback on assessment
Verbal feedback in supervisions, fortnightly multiple choice tests.
This module is Core for:
- Year 1 of USTA-G302 Undergraduate Data Science
- Year 1 of USTA-G304 Undergraduate Data Science (MSci)
- Year 1 of UCSA-G4G1 Undergraduate Discrete Mathematics
- Year 1 of UCSA-G4G3 Undergraduate Discrete Mathematics
- Year 1 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
- Year 1 of UMAA-GV17 Undergraduate Mathematics and Philosophy
- Year 1 of UPXA-FG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
- Year 1 of UPXA-GF13 Undergraduate Mathematics and Physics (BSc)
- Year 1 of UPXA-FG31 Undergraduate Mathematics and Physics (MMathPhys)
- Year 1 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
- Year 1 of USTA-GG14 Undergraduate Mathematics and Statistics (BSc)
- Year 1 of USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics