MA138 Sets and Numbers
Lecturer: Saul Schleimer and Robert Kropholler
Term(s): Term 1
THIS MODULE IS NOT AVAILABLE TO MATHS STUDENTS
Commitment: 30 lectures, assignments with weekly online quizzes based on them
Assessment: 15% from online quizzes and 85% from April exam
Formal registration prerequisites: None
Assumed knowledge: Grade A in Alevel Maths or equivalent.
Useful background: This module is a broadbased introduction to the language of mathematics. Learning a language takes time, so it would be very useful to learn or review the various number systems (integer, rational, real, complex), modular arithmetic, set theory notation, and the basic techniques of proofs (direct, by induction, by contradiction, by contrapositive, by cases, by construction).
Synergies: This module is core; it is relevant to, and required for, all other maths modules at Warwick. Most later pure mathematics modules specifically:
 MA140/MA152 Mathematical Analysis 1 and 2 or MA142/MA143 Calculus 1 and 2
 MA149 Linear Algebra or MA148 Vectors and Matrices
Leads to: The following modules have this module listed as assumed knowledge or useful background:
 MA222 Metric Spaces
 MA260 Norms, Metrics and Topologies
 MA257 Introduction to Number Theory
 MA249 Algebra II: Groups and Rings
 MA3E1 Groups and Representations
 MA3A6 Algebraic Number Theory
 MA3D5 Galois Theory
 MA3H3 Set Theory
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:

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, CantorSchroederBernstein, cardinalities.  Relations:
Reflexive, symmetric, transitive.
Orders, equivalence classes and relations: integers, rational numbers, partitions.
Kernels and cokernels, welldefinedness, 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, handwaving)
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.Rings  definitions, examples, applications.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.
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. But 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.
Content:
Numbers
Number systems:
 Natural numbers, integers, rationals and real numbers. Existence of irrational numbers
 Euclidean algorithm; greatest common divisor and least common multiple
 Prime numbers, existence and uniqueness of prime factorisation (and nonuniqueness in other “number systems”, e.g. even integers, Gaussian integers)
 Properties of commutativity, associativity and distributivity
 Infinity of the primes
 Summing series of integers; proofs by induction.
Language
Basic set theory:
 Intersection, Union, Venn diagrams and de Morgan’s Laws
 Logical connectives and, or, implies and their relation with intersection and union
Polynomials
 Multiplication and long division of polynomials
 Binomial theorem
 Euclidean algorithm for polynomials
 Remainder theorem; a degree n polynomial has at most n roots
 Rational functions and partial fractions
 Incompleteness of the real numbers, completeness of the complex numbers (sketch)
Counting
 Elementary combinatorics as practice in bijections, injections and surjections
 Cardinality of the set of subsets of a set X is greater than cardinality of X
 Russell’s paradox
 Definition of Cartesian product
 Countability of the rational numbers, uncountability of the reals
 Transcendental numbers exist!
The second (and smaller) part of the module explores the elementary properties of a fundamental algebraic structure called a group. Groups arise in an extraordinary range of contexts in mathematics and beyond (for example, in elementary particle physics and in card tricks), and can be used to analyse the symmetry of geometric objects or physical systems.
Modular arithmetic: 3 hours:
 Addition, multiplication and division in the integers modulo n
 Some theorems of modular arithmetic
 Equivalence relations.
Permutations and the symmetric group:
 Multiplying (composing) permutations
 Cycles and disjoint cycle representation
 The sign of a permutation
 Basic Group Theory
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