MTH4110 Mathematical Structures

Description

This module is intended to introduce students to the concerns of mathematics, namely clear and accurate exposition and convincing proofs. It will attempt to instil the habit of being "precise but not pedantic". The module covers an informal account of sets, functions and relations, and a sketch of the number systems (natural numbers, integers, rational, real and complex numbers), outlining their construction and main properties.

Details

Organiser: Professor P J Cameron

Level: 4 Credit value: 15 Semester: A

Overlaps:

• MTH4100 Calculus I (if taken in 2011–12 or earlier)
• MTH4101 Calculus II (if taken in 2011–12 or earlier)
• MTH4104 Introduction to Algebra (if taken in 2011–12 or earlier)
• MTH4107 Introduction to Probability (if taken in 2011–12 or earlier)

Prerequisites: A-Level Mathematics or equivalent

Restrictions: Not open to Associate Students.

Assessment: 10% mid-term test, 90% final exam

Organiser's module website: http://qmplus.qmul.ac.uk/

Comment: The overlaps apply only to modules taken in 2011–12 or earlier and are a consequence of small syllabus changes.

Syllabus

1. Definitions, theorems, proofs, counterexamples. Examples of proofs. How to construct proofs, and how to recognise false proofs.
2. Sets, subsets, operations on subsets. Functions; injective, surjective and bijective functions. Relations, including equivalence relations. Counting finite sets (including the number of r-subsets of an n-set). Infinite sets; countable and uncountable.
3. Natural numbers and induction. Integers and rational numbers with a sketch of the construction. Real numbers (treated as infinite decimal and binary expansions) including some completeness properties. Countability of the rationals and uncountability of the reals.
4. Complex numbers. The complex plane with cartesian and polar coordinates; addition and multiplication. Statement of the Fundamental Theorem of Algebra.

Learning outcomes

http://www.maths.qmul.ac.uk/undergraduate/modules/learning-outcomes#MTH4110

Learning resources

Reading list:

• Timothy Gowers, Mathematics: A very short introduction, Oxford University Press, Oxford (2002)
• Kevin Houston, How to think like a mathematician, Cambridge University Press, Cambridge (2009)

Undergraduate Modules for Academic Year 2013–14 (last updated 6 June 2013)