1. To bridge the gap in language and philosophy between A-level and University mathematics.

2. To train students to think clearly and logically, and to appreciate the nature of definitions, theorems, and proofs.

3. To give an appreciation of the richness and importance of the structures of the integer, rational, real and complex number systems.

After completing the module students should be able to:

1. Use mathematical language and symbols accurately;

Understand the nature of a definition, and show that simple definitions are or are not satisfied by given examples;

 Use theorems to draw logical conclusions from given information

Understand the logic of direct proofs and proofs by contradiction, and construct very simple proofs, including proofs by induction;

Interpret statements involving quantifiers, and negate statements with one or two quantifiers

Use the language of naive set theory

Understand the integer, rational, real and complex number systems and the relationship between them.

The concepts of definitions, theorems (hypothesis and conclusion, converse and contrapositive), proofs (direct proof and proof by contradiction), examples and counter examples, quantifiers and negation will be introduced gradually via examples throughout the module.

[3] Basic propositional logic (exemplified by ''real world'' logic puzzles).

[8] Natural numbers, Peano axioms. Integers. Principle of mathematical induction and proof by induction. Greatest common divisors. Euclid''s algorithm. Fundamental Theorem of Arithmetic (unique factorisation into primes), and applications: greatest common divisors and least common multiples, numbers of divisors.

[5] Sets and maps. Injectivity, surjectivity and bijectivity. Intersection and union, principle of inclusion-exclusion.

[2] Equivalence relations and quotients. Simple examples from ''real life'' and arithmetic.

[9] Definition of rational numbers via integer numbers. Construction of real numbers via Dedekind cuts. Irrational numbers; proof of existence, examples. Sequences. Completeness of the real numbers as their key property. (Connect with Math101.)

[3] Countability. Cantor''s diagonal argument. Existence of transcendental numbers.

[3] Construction of complex numbers from the reals. Basic properties (connect with Math103). Statement and sketch proof of Fundamental Theorem of Algebra.

[3] Revision