20 credits at level HE4
This module lays the foundation for all of the material on the pathway concerning algebra and discrete mathematics. It begins by introducing the notion of a set, together with the allied notions of subsets and union, intersection and Cartesian product of sets. This leads into some work on logic, approached using truth tables, and a survey of some of the more usual methods of mathematical proof. Relations and mappings are introduced, building further on the work on sets, and this paves the way for the introduction of abstract notion of a group. Groups are studied in some depth, including work on permutation groups and modular arithmetic groups. Finally, rings and fields are introduced, and it is shown how the familiar field of real numbers may be extended to give the field of complex numbers.
1. Set Theory Sets, subsets, power set, union, intersection, set difference, De Morganís Laws, Cartesian product, sets of numbers, Euclidís Algorithm.
2. Logic and Proof Statements and predicates, negation, conjunction, disjunction, truth tables, implication connectives, universal and existential quantifiers, methods of proof.
3. Relations and Mappings Relations and their properties, mappings, injectivity, surjectivity, bijectivity, composite and inverse mappings, binary operations, Cayley tables.
4. Groups and their Properties semigroups, groups, Abelian groups, subgroups, centraliser and centre, cyclic groups, homomorphisms of groups, isomorphisms of groups.
5. Symmetric groups and Modular Arithmetic Groups Permutations, symmetry groups, congruence of integers.
6. Rings, Integral Domains and Fields Rings, polynomial rings, complex numbers, the Argand diagram, arithmetic of complex numbers, polar coordinates, De Moivreís theorem.
On completion of the module the student should gain familiarity with the various abstract notions introduced, skills in problem solving for specific examples, and the ability to write mathematical arguments and proofs in a clear and lucid style.
Approximately two-thirds of the available time will be devoted to lectures based on printed notes. Class discussion and participation will be encouraged. The remainder of the time will be devoted to attempting and discussing the structured exercises which appear at the end of each chapter of the notes.
when you have successfully completed this module you will:
to demonstrate that you have achieved the learning outcome you will:
|1.||have an understanding of sets.||find the union, intersection and Catesian product of a pair of sets, and verify laws about these using Venn diagrams.|
|2.||have an understanding of propositional logic,||use truth tables to demonstrate logical equivalence.|
|3.||have an understanding of mappings and their properties.||verify whether or not a mapping is surjective, injective or bijective.|
|4.||be familiar with the concept of a group.||verify the group axioms for a given algebraic structure, and verify that a mapping is a homomorphism.|
|5.||have familiarity with symmetric, dihedral and modular arithmetic groups.||complete Cayley tables for a variety of small finite groups, and perform algebra of permuations and modular arithmetic.|
|6.||have fluency with the arithmetic of complex numbers.||add, subtract, multiply and divide complex numbers.|
Your achievement of the learning outcomes for this module will be tested as follows:
|Description||One in class test and one piece of work to be complted outside of class time.||Unseen writtten examination.|
There are no prerequisites for this module.
No restrictions apply.
Burn, R.P. Groups: A Path to Geometry, CUP (1985).
Barnard, Tony. Mathematical Groups, Hodder (1996).
Cartwright, Mark. Groups, Macmillan (1993).
Dean, Neville. The Essence of Discrete Mathematics, Prentice-Hall (1997).
Devlin, Keith. Sets, Functions and Logic, 2nd ed., Chapman & Hall (1992).
Devlin, Keith. The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd ed., Springer (1993).
Eccles, Peter J. An Introduction to Mathematical Reasoning: Numbers, Sets & Functions, CUP (1997).
Humphreys, J. F. & Prest, M. Y. Numbers, Groups and Codes, CUP (2004).
Lederman, Walter. Introduction to Group Theory,
Addison-Wesley Longman (1996).
Mendelson, Elliott. Introduction to Mathematical Logic,
Chapman & Hall (1987).
Ross, Kenneth. Discrete Mathematics, 3rd ed., Prentice-Hall, (1992).
Whitehead, C.. Guide to Abstract Algebra, Macmillan (2003).
Whitelaw, T. A. An Introduction to Abstract Algebra, Chapman & Hall (1995).
|Host Subject Group:||Mathematics|
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