Aims

To present the notion and techniques of proof, illustrated by results in set theory and basic combinatorics. To stimulate logical thinking and to develop students' skills at constructing mathematical arguments.

Module summary

This module introduces the concept of proof in mathematics. Proof sets mathematics apart from other subjects: within mathematics we can prove statements are unambiguously true, rather than simply collecting evidence in support of statements as in other areas of science. The objects studied in this module are sets and functions between sets. Important techniques of proof presented will include proof by induction and proof by contradiction. The module will conclude with a variety of methods for counting the number of elements in sets.

Outline Of Syllabus

Mathematical terminology. Logic. Techniques of proof. Set theory. Mathematical induction. Functions: composition; injective and surjective functions, inverse functions.
Counting arguments for the number of elements in a set: sequences, permutations and combinations.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.