This module introduces students to the concepts and techniques of proof in mathematics, to mathematical probabilty theory and its applications to statistics, and to the ideas and techniques of mathematical experimentation.

By the end of the module a student should be able to:

• Recognise and construct simple proofs in set theory and number theory by induction, contradiction, contrapositive and construction.
• Apply basic results in probabilty, distribution theory and statistical inference to straightforward problems.
• Use the statistical package R to perform statisitical calculations and to produce numerical and graphical data summaries.
• Use computer algebra packages for symbolic manipulation, numerical calculation and graphical visualisation.
• Formulate simple mathematical models and investigate their behaviour using appropriate computational methods.
• Explore, through computer-aided experimentation, unfamiliar problems and topics from various areas of mathematics and its applications.

Proof  

• Axiomatic systems and the need for proof;
• Implication and negation;
• Concept of proof, proofs by induction, contradiction, contrapositive and construction;
• Order;
• Subsets; union and intersections;
• Injections, surjections and bijections between sets
• Cartesian products
• Binary operations
• The power set
• Countability
• Revision of integers, rational and real numbers
• Countability of the reals via Cantor's diagnonal argument
• Divisibilty; LCM and HCF;
• Primes; Euclid's proof of infinite number of primes.

Probability

• Probability theory, including Bayes Theorem
• Some standard discrete distributions: bionomial, Poisson and hypergeometric and continous distributions: normal, exponential.
• Expectation and moments
• Moment generating functions
• Sums of random variables
• Statement of the central limit theorem
• Sampling distributions
• Inference for means and proportions
• Estimation: urbiased estimators; maximum likelihood and moment estimators
• Hypothesis testing: Type I and II errors, power
• Chi-squared Goodness of fit.
• An Introductuction to R

Experiment

• Mathematical Modelling - Principles and tools for constructing simple models. Techniques for computing, visualising and interpreting solutions to problems formulated as mathematical models.
• Computational Packages - Using packages such as MAPLE and/or MATLAB to perform algebraic and numerical computational tasks need to develop and understand mathematical models.
• Experimental Mathematics - A selection of topics from pure and applied mathematics will be explored. Some of these should be familiar from previous modules; others may be preparatory for work to be covered in later modules. The emphasis throughout will be on an investigative approach: exploring various systems by developing mathematical models of them and discovering their properties via computer-aided experimentation.
  

Teaching is by lectures, tutorials, computer lab sessions and tests. Learning takes place through lectures, tutorials, computer lab sessions, tests and exercise sheets.

Proof:  20 hours of lectures and tutorials over 10 weeks in the Autumn semester.

Probability: 40 hours of lectures and tutorials over 12 weeks in the Spring semester.

Recommended Reading : 

Proof  

•   D.L.Johnson, Elements of Logic via Numbers and Sets, Springer, 1998.  

Probability

•  G.M. Clarke and D. Cooke, A Basic Course in Statistics, Arnold

 Experiment    

•  Frank Garvan: The MAPLE Book (Chapman & Hall 2001)
•  André Heck: Introduction to Maple (Springer, 2003)
•  D.J. Higham, N.J. Higham: Matlab Guide (SIAM)
•  B.R. Hunt, R.L. Lipsman, J.M. Rosenberg, K.R. Coombes, John E. Osborn, G.J. Stuck: A Guide to MATLAB: For Beginners and Experienced Users (Cambridge University Press) [available as e-book in  our Library]
• David L Schwartz: Introduction to Maple 8 (Prentice-Hall 2003)