Aims: 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.

Learning to deal with abstraction and with proofs takes time. This module aims to bridge the gap between school and university mathematics, by beginning with some rather concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.

Indicative Content (please refer to the lecture notes for the definitive 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.
  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.
• Algebra:
  
  Groups - definitions, examples, applications.
  Group homomorphisms. The symmetric group, the parity of a permutation.
  Fields - definitions, examples, applications.

Objectives: 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.

Books:
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