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Department of Mathematics



Next: MA1102 Algebraic Structures and Number Systems Up: ModuleGuide03-04 Previous: MA1061 Probability

MA1101 Proof and Logical Structures

MA1101 Proof and Logical Structures

Credits: 10 Convenor: Prof. J.R. Hunton Semester: 1 (weeks 1 to 6)
Prerequisites:
Assessment: Coursework: 20% Examination: 80%
Lectures: 18 Problem Classes: 6
Tutorials: none Private Study: 45
Labs: none Seminars: none
Project: none Other: none
Surgeries: 6 Total: 75

Subject Knowledge

Aims

This module aims to introduce the student to rigorous university level mathematics. It discusses the need for and nature of Proof and the types of logical argument used in a Mathematics degree. In the process a number of basic terms, symbols and ideas are developed.

Learning Outcomes

Students will know the standard types of mathematical proof - direct proofs, proofs by cases, proof by contradiction and by contrapositive, proof by induction. Students will also know some simple and some classical examples of proofs, such as the irrationality of $\sqrt 2$ and the existence of an infinite number of primes, and be able to present and construct some simple proofs themselves. Students will be familar also with the need for careful definition of concepts and the careful statement of mathematical results. They will have some facility at the formation of the negation of statements. They will also be familar with a number of basic pieces of mathematical language and symbol, including that of functions and equivalence relations.

Methods

Lectures, supervisions, example sheets and class tests, surgeries.

Assessment

Marked problem sheets, class tests, written examination.

Subject Skills

Aims

To develop students' critical understanding of mathematical method and written communication skills.

Learning Outcomes

Students will have familiarity with the stardard types of mathematical argument, be able to reproduce and manipulate definitions, statements and arguments with precision, recognising the need for such precision and be able to present written arguments in a coherent and logical form.

Methods

Lectures, supervisions, example sheets and class tests, surgeries.

Assessment

Marked problem sheets, class tests, written examination.

Explanation of Pre-requisites

This module is a first mathematics module and makes no use of other modules. Some familiarity with higher level school mathematics will be assumed.

Course Description

This module is intended to help students understand what consitutes a proper mathematical statement and proof, and why one needs these concepts and level of precision. It aims to introduce the need for mathematical argument through the unfamilar world of counting infinite numbers and proceeds to look at the standard types of mathematical argument and language used. In the process a number of very simple and familar ideas are worked through, as are a couple of classical arguments from antiquity - the irrationality of $\sqrt 2$ and the infinite number of primes. This module also allows the development of a number of pieces of mathemtical terminology used throughout the degree - the ideas of function and equivalence relation, for example. A small amount of set theoretic language is introduced and used.

Syllabus

The idea of and need for mathematical statements and proofs. Direct proofs, proofs by cases, proof by contradiction, proof by contrapositive, proof by induction. The irrationality of $\sqrt 2$, the infinite number of primes. Discussion of negations and logical connectives (and, or, not, implies, iff). Discussion of familar examples such as $n$ even if and only if $n^2$ even. Functions and the properties of being 1-1, onto or bijective; equivalence relations.

Reading list

Recommended:

P.J. Eccles, An Introduction to Mathematical Reasoning, Cambridge University Press.

Background:

N. L. Biggs, Discrete Mathematics, Oxford University Press.

P. M. Cohn, Algebra I, John Wiley.

D. Driscoll Schwartz, Conjecture and proof, Harcourt Brace.

S. Galovich, Doing Mathematics, Harcourt Brace.

R. Garnier and J. Taylor, 100% Mathematical Proof, John Wiley.

A. G. Hamilton, Numbers, Sets and Axioms, Cambridge University Press.

G. Pólya, How to Solve It, Penguin.

S. Singh, Fermat's Last Theorem, Fourth Estate.

D. Solow, How to Read and Do Proofs

I. Stewart and D. Tall, The Foundations of Mathematics, Oxford University Press.

D. J. Velleman, How to Prove It, Cambridge University Press.

Resources

Problem sheets, lecture rooms, surgeries and supervision rooms. Human resources.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA1102 Algebraic Structures and Number Systems Up: ModuleGuide03-04 Previous: MA1061 Probability

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Last updated: 2004-02-21
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