Foundations of Pure Mathematics B


Unit code: MATH10111
Credit Rating: 15
Unit level: Level 1
Teaching period(s): Semester 1
Offered by School of Mathematics
Available as a free choice unit?: N

Requisites

Co-Requisite

Aims

The aims of this course are to provide a basic introduction to fundamental mathematical concepts of sets, numbers, functions and proof. 

Overview

This lecture course is intended to introduce students to the concept of "proof". The objects of study, sets, numbers and functions, are basic to almost all Mathematics.

Learning outcomes

On successful completion of this module students will be able to:

  • Analyse the meaning of mathematical statements involving quantifiers and logical connectives, and construct the negation of a given statement.
  • Construct truth tables of simple mathematical statements and use these to determine whether two given statements are equivalent.
  • Construct elementary proofs of mathematical statements using a range of fundamental proof techniques (direct argumentation, induction, contradiction, use of contrapositive).
  • Use basic set theoretic language and constructions, and be able to determine whether two given sets are equal.
  • Use elementary counting arguments (pigeonhole principle, inclusion-exclusion, binomial theorem) to compute cardinalities of finite sets.
  • Describe and apply basic number theoretic concepts to compute greatest common divisors and to solve linear congruences.
  • Recall formal definitions and apply these to give examples and non-examples of bijections, equivalence relations, binary operations and (abelian) groups.
  • Compose and invert given permutations, expressing the result in two-line notation and in cycle notation.­­­­

 

Assessment methods

  • Other - 25%
  • Written exam - 75%

Assessment Further Information

  • Supervision attendance and participation; Weighting within unit 10%
  • Coursework; In class test, weighting within unit 15%
  • Two and a half hours end of semester examination; Weighting within unit 75%

Syllabus

The language of mathematics. Mathematical statements, quantifiers, truth tables, proof.

Number theory I. Prime numbers, proof by contradiction

Proof by induction. Method and examples.

Set Theory. Sets, subsets, well known sets such as the integers, rational numbers, real numbers.Set Theoretic constructions such as unions, intersections, power sets, Cartesian products.

Functions. Definition of functions, examples, injective and surjective functions, bijective functions, composition of functions, inverse functions.

Cardinality of sets. Counting of (mostly) finite sets, inclusion-exclusion principle, pigeonhole principle, binomial theorem.

Euclidean Algorithm. Greatest common divisor, proof of the Euclidean Algorithm and some consequences, using the Algorithm.

Congruence of Integers. Arithmetic properties of congruences, solving certain equations in integers.

Relations. Examples of various relations,reflexive, symmetric and transitive relations. Equivalence relations and equivalence classes. Partitions.

Number Theory II. Fundamental theorem of Arithmetic, Fermat's little theorem.

Binary Operations. Definition and examples of binary operations. Definition of groups and fields with examples. Proving that integers mod p ( p a prime) give a finite field. 

Recommended reading

The course is based on the following text:

P.J. Eccles, An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, Cambridge University Press, 1997.

Feedback methods

Feedback seminars will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

  • Lectures - 33 hours
  • Tutorials - 11 hours
  • Independent study hours - 106 hours

Teaching staff

Marianne Johnson - Unit coordinator

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