Autumn

Unit(s) of Assessment	Weighting Towards Module Mark( %)
One class test   Exam	40    60
Qualifying Condition(s)    A weighted aggregate mark of 40% is required to pass the module.

This module introduces students to the main types of proof, within the context of some simple but fundamental mathematical ideas.

By the end of the unit, students should be able to recognise and construct simple proofs in set theory and number theory by induction, contradiction, contraposition and construction.

Logic axiomatic systems and the need for proof; implication and negation.

Concept of proof; proof by contradiction, contraposition and induction.

Revision of integers, rational and real numbers.

Sets: definition of a set; order; subsets; union and intersection;

injections, surjections and bijections between finite sets;

cartesian products.

Introduction to number theory: divisibility, LCM and HCF; primes;

Euclid 's proof of infinite number of primes.

Fundamental Theorem of Arithmetic; modular arithmetic; Fermat's theorem;

Wilson 's theorem; solution of x^2+y^2=z^2 in integers.  Continued fractions.

Some basic notions of elementary Euclidean geometry (time permitting):

Pythagoras’ theorem, Cauchy-Schwarz inequality, triangle inequality.

 Introduction to plane geometry.  

Teaching will be by lectures and problem classes. In addition to reading

the lecture notes, students will learn by tackling a range of assessed and unassessed problems.

Recommended

1) D.L.Johnson, Elements of Logic via Numbers and Sets, Springer, 1998;

2) Tom Apostol: Introduction to Analytic Number Theory, Springer, 1998.

Further Reading

1) H. Davenport, The Higher Arithmetic, 7th ed., CUP, 1999;

2) A. Cupillari: The Nuts and Bolts of Proofs, Academic Press, 2001;

3) K. Houston: How to Think Like a Mathematician, CUP, 2009.

October 10