module specification

This module develops a range of mathematical techniques including set theory, logic, relations
and functions. The techniques are particularly relevant to students doing computing and
mathematics.
 

None

The aim of this module is to provide a firm foundation for study in further mathematics and computing. It will also enable students to apply the methods to a range of practical problems (A2). In addition, students will evaluate their mathematical strengths and weaknesses and implement an action plan (A1).

Logic: representation of simple verbal arguments; truth-tables; logical equivalence, validity and consequence; resolution proof method; logic circuits and Karnaugh maps. Predicate logic.

Sets: introduction to notation; set operations; Venn diagrams; universal, empty and subsets; set identities (De Morgan etc); duality; power sets, ordered pairs and Cartesian products.

Relations: representations of relations; equivalence relations; partitions; partial orderings.
Functions: ways of defining functions; composition; inverse functions.
Proof methods: proofs by contradiction and induction.
Natural numbers: number bases; twos compliment representation.

This module will be delivered through a mixture of lectures (24 hours including revision) and tutorials (24 hours including a class test). The lectures will develop theory, explain the methods and techniques and demonstrate them by going through examples. The tutorials will provide students with the opportunity of reviewing their lecture notes and working through the problems designed for their practice, which will underpin the skills and techniques demonstrated in the lectures. Students will be encouraged to construct valid and precise mathematical arguments and will be expected to produce solutions using appropriate notational and stylistic conventions. Self-study exercises (48 hours) will enable students to monitor their own progress. Students will be expected to produce solutions to the problems in the lecture notes. This will form the basis for the preparation for assessments that will also include the revision of basic concepts, revisiting problem sheets and working through past examination papers in a guided and structured way. Students are also expected to spend further 54 hours on directed reading.
A booklet of lecture notes will be provided to students at the beginning of the semester and answers for exercise questions will be put on the Web.

After successful completion of this module students should be able to,
1. Understand the meaning of mathematical definitions of sets and perform set operations.
[A2]
2. Symbolise simple verbal arguments and test their validity by truth-tables and by resolution.
[A2]
3. Use truth-tables to test for tautologies, contradictions and for logical equivalence.
4. Work with simple predicate logic language and appreciate its uses. [A2]
5. Understand the meaning of mathematical definition of relations and to determine what
relations are equivalence relations. [A2]
6. Understand the meaning of mathematical definition of functions, to use it to construct
functions and to determine which functions are one-to-one. [A2]

There will be two elements of the assessment that reflect all the learning outcomes. One of them will be coursework, which consists of solutions of selected exercises together with notes on reading done outside the module sessions (L1-L6, A1). The other element is the exam, which will provide students with an opportunity to monitor their progress and to boost their confidence of success (L1-L6, A1 and A2). In order to pass the module, a student must attempt both elements and achieve at least 40% of marks in aggregate.

In order to pass the module, a student must seriously attempt all three elements and achieve at least 40% of marks in aggregate.

1. Lecture Notes: “Discrete Mathematics”. They are also available on the student’s drive.
2. A. Chetwynd and P. Diggle, Discrete Mathematics, Arnold, 1994.
3. P. Grossman, Discrete Mathematics for Computing (2nd edition), Macmillan, 2002.
4. K. H. Rosen, Discrete Mathematics and Its Applications, McGraw Hill, 1994.
5. A. Simpson, Discrete Mathematics by Examples, McGraw Hill, 2002.
6. T. Feil and J. Krone, Essential Discrete Mathematics for Computer Science, Pearson Education, 2003.