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Semester 2
Proofs and Problem Solving (MATH08059)
Subject
Mathematics
College
SCE
Credits
20
Normal Year Taken
1
Delivery Session Year
2023/2024
Pre-requisites
Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.
Course Summary
This course is designed to introduce and develop the fundamental skills needed for advanced study in Pure Mathematics. The precise language of professional mathematicians is introduced and the skills needed to read, interpret and use it are developed. The 'Axiomatic Method' will be developed along with its principal ingredients of 'Definition' (a statement of what a term is to mean), 'Theorem' (something that inevitably follows from the definitions) and 'Proof' (a logical argument that establishes the truth of a theorem). Constructing proofs, and much other mathematical practice relies on the difficult art of 'Problem Solving' which is the other main theme of the course. Facility comes only with practice, and students will be expected to engage with many problems during the course. The principal areas of study which are both essential foundations to Mathematics and which serve to develop the skills mentioned above are sets and functions, and number systems and their fundamental properties.
Course Description
This syllabus is for guidance purposes only: 1. Sets, proofs quantifiers, real numbers, rationals and irrationals. 2. Inequalities, Mathematical induction.3. Upper Bounds, Least Upper Bounds and Limits.4. Monotone Convergence. Decimals, Series.5. Complex numbers, roots of unity, polynomial equations, fundamental theorem of algebra. 6. Euclidean algorithm, prime factorization, prime numbers.7. Congruence, primality testing.8. Equivalence relations, functions.9. Counting and choosing, binominal coefficients, more set theory. 10. Permutations.
Assessment Information
Written Exam 50%, Coursework 50%, Practical Exam 0%
Additional Assessment Information
Coursework 50%, Examination 50%
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