Introductory description

University mathematics introduces progressively more and more abstract ideas and structures, and demands more and more in the way of proof, until by the end of a mathematics degree most of the student's time is occupied with understanding proofs and creating his or her own. This is not because university mathematicians are more pedantic than schoolteachers, but because proof is how one knows things in mathematics, and it is in its proofs that the strength and richness of mathematics is to be found.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Indicative Content:

Naive Set Theory, Counting and Lists:
Sets and functions, injections, surjections and bijections, permutations.
Lists, sublists, lists as functions, strings.
Subsets, power sets, partition, infinite versus finite, Cantor's Theorem.

Operations on Sets, Lists, Functions:
Ordered pairs, cartesian products, functions and graphs, functions and lookup tables.
Union, intersection, set difference, list concatenation.
Composition, iteration, orbits, Cantor-Schroeder-Bernstein, cardinalities.
Relations:
Reflexive, symmetric, transitive.
Orders, equivalence classes and relations: integers, rational numbers, partitions.
Kernels and co-kernels, well-definedness, modular arithmetic.

Logic:
Variables, booleans, negation, operations.
Operators and formulas via truth tables.
Quantifiers, tautologies, deduction rules.

Proof:
What is proof? False proofs, examples, subtle issues (diagrams, hand-waving)
Kinds of proof: direct, contraposition, contradiction, construction, cases.
Recursion, induction, pigeonhole principle, counting.

Algorithms in Algebra and Cryptography:
What is algorithm? Euclid's algorithm, operational complexity, P=NP
Discrete Logarithm, introduction to cryptography, Diffie-Hellman key exchange.
Prime factorisation, primality testing, Chinese Remainder Theorem
RSA (Rivest–Shamir–Adleman) public key exchange

Learning outcomes

By the end of the module, students should be able to:

• Students will work with number systems and develop fluency with their properties;
• they will learn the language of sets and quantifiers, of functions and relations, and will become familiar with various methods and styles of proof.

Indicative reading list

None of these is the course text, but each would be useful, especially the first:
A.F.Beardon, Algebra and Geometry, CUP, 2005.
I.N. Stewart and D.O. Tall, Foundations of Mathematics, OUP, 1977.
J. A. Green, Sets and Groups; First Course in Algebra, Chapman and Hall, 1995.

Subject specific skills

After taking this module, students will be familiar with the concept of a rigorous mathematical proof and most of commonly used notations.

Transferable skills

Students will work with number systems and develop fluency with their properties; they will learn the language of sets and quantifiers, of functions and relations, and will become familiar with various methods and styles of proof.