# 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 A-level Maths or equivalent.

Useful background: This module is a broad-based 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:

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, 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.
• 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 non-uniqueness 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
• 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