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2010/1 Module Catalogue
 Module Code: MAT1026 Module Title: PROOF
Module Provider: Mathematics Short Name: MAT1026
Level: HE1 Module Co-ordinator: BARTUCCELLI M Dr (Maths)
Number of credits: 10 Number of ECTS credits: 5
Module Availability


Assessment Pattern

Unit(s) of Assessment


Weighting Towards Module Mark( %)


One class test










Qualifying Condition(s) 


A weighted aggregate mark of 40% is required to pass the module.




Module Overview
Module Aims

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.

Learning Outcomes
Module Content

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.  

Methods of Teaching/Learning

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.

Selected Texts/Journals



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.

Last Updated

October 10