This is an archived syllabus from 2013-2014

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COMP11120 Mathematical Techniques for Computer Science

Level 1
Credits: 20
Enrolled students: 214

Course leader: Aravind Vijayaraghavan

Additional staff: view all staff

Assessment methods

• 85% Written exam
• 15% Coursework

Timetable

Semester	Event	Location	Day	Time	Group
Sem 1 w1-5,7-11	Lecture	1.1	Mon	12:00 - 12:00	-
Sem 1 w1-5	Lecture	St Peters House CHAPLAINCY	Tue	12:00 - 12:00	-
Sem 1 w2-5,7-11	Examples	LF15	Fri	10:00 - 10:00	W
Sem 1 w2-5,7-11	Examples	IT407	Fri	12:00 - 12:00	Y
Sem 1 w2-5,7-11	Examples	IT407	Fri	14:00 - 14:00	B+X
Sem 1 w3+	Examples	LF15	Tue	14:00 - 14:00	A+Z
Sem 1 w7-12	Lecture	Chemistry G.51	Tue	12:00 - 12:00	-
Sem 1 w12	Lecture	Roscoe TH B	Mon	12:00 - 12:00	-
Sem 2	Lecture	1.1	Fri	09:00 - 09:00	-
Sem 2	Lecture	1.1	Thu	11:00 - 11:00	-
Sem 2 w2+	Examples	LF15	Thu	13:00 - 13:00	Y
Sem 2 w2+	Examples	LF15	Tue	14:00 - 14:00	B+X
Sem 2 w2+	Examples	LF15	Tue	15:00 - 15:00	A+Z
Sem 2 w2+	Examples	LF15	Mon	15:00 - 15:00	W
Sem 2 w22	Lecture	1.1	Mon	09:00 - 09:00	-
Sem 2 w23	Lecture	1.1	Mon	09:00 - 09:00	-
Sem 2 w27	Lecture	1.1	Mon	09:00 - 09:00	-

Overview

Aims

This full-year course unit focuses on the use of mathematics as a tool to model and analyse real-world problems arising in computer science. Four principal topics, drawn from the traditional areas of discrete mathematics as well as some continuous mathematics, will be introduced: symbolic logic, probability, discrete structures, and vectors and matrices. Each topic will be motivated by experts introducing relevant real-world problems arising in their own specialism.

Abstraction is fundamental to computer science. Hence, a fundamental emphasis of this course unit is to introduce mathematical techniques and skills to enable the student to design and manipulate tractable and innovative abstract models of chunks of the real-world. These techniques and skills include appropriate mathematical notations and concepts. These range over the four principal areas mentioned above. Formalisation in mathematics has, in general, significant cost. Therefore, to be of practical use, the benefits arising from formalisation, such as succinctness, unambiguity, provability, transformability and mechanisability, must outweigh the costs. A key aim of the course is for the student to appreciate this issue and know how and when to use particular techniques.

The specific aims of the course unit are:

• To demonstrate the relevance of mathematics to computer science.
• To introduce fundamental mathematical techniques of abstraction.
• To demonstrate applicability of particular mathematical techniques and skills for particular types of computer science problem.
• To appreciate the costs and benefits of mathematical modelling.

The delivery style will place more emphasis on students undertaking appropriate background reading, i.e. being more independent learners, and use the lectures more to demonstrate examples and solutions and not working through every detail of a particular or concept.

The course unit is delivered by staff from both the School of Computer Science and the School of Mathematics .

Syllabus

Semester 1

Introduction to Discrete Structures - 4 lectures

Standard number systems and arithmetical operations on them; introduction to syntax. A brief introduction to sets and set theoretic operations.

Logic - 7 lectures

About propositions, truth and falsity, logical connectives and their meaning (via truth tables); propositional formulas --- the logical language, truth table semantics.

Validity, contingency, unsatisfiability; logical equivalences (laws);

Simple logical manipulation and reasoning, normal forms, satisfiability algorithms;

Predicates and individuals, universal and existential quantifiers, first order logic in use;

Probability - 8 lectures

Random experiments. Finite or countable discrete sample spaces. Events, disjoint events. Algebra of events. Definition of probability of an event when outcomes are equally likely. General definition of probability of an event. Axioms of probability. Addition Rule of two events

Conditional probability. The product rule. The total probability of two events. Definition of a partition of a sample space (exhaustive events). The total probability rule for a partition set. Bayes Theorem. Independent events and the product rule for independent events.

Definition of a random variable. Discrete random variables and their distributions. The probability mass function and the cumulative distribution function for a discrete random variable. The Binomial and Poisson distributions.

Semester 2

Discrete Structures - 9 lectures

Sets, functions and maps, use in specification, mathematical induction.

Binary relations, domain and range, properties of relations, relational composition, closure properties, equivalence relations, partitions.

Graphs and networks, directed/undirected, acyclic/cyclic, paths, networks in action, structural induction.

Motivational Guest Lecture

The importance of vectors and matrices in graphics.

Vectors and Matrices - 12 lectures

Vectors:

Reminder of 2-D and 3-D coordinate systems and right-angle trig; Vectors (Equality, Parallel, Addition / Subtraction, Multiplication by a scalar); Vectors in a cartesian system; Scalar and Vector Products; Vector Equations of lines and planes.

Matrices:

Concept, Equality, Addition, Subtraction and Multiplication by a scalar; Matrix Multiplication; Associative Law; Multiplication of matrix times vector; Matrices and systems of equations.

Geometrical Transformations:

Homogeneous Coordinates; Affine Transformations; The transformation matrix; Translation, Rotation, Reflection, Scaling, Shear; Transformations and inverse transformations; Combined Transformations and the Associative Law.

Teaching methods

Lectures

44 in total, 2 per week

Examples classes

22 in total, 1 per week

Feedback methods

Study hours

• Assessment written exam (4 hours)
• Lectures (46 hours)
• Practical classes & workshops (20 hours)

Employability skills

• Analytical skills
• Problem solving

Learning outcomes

On successful completion of this unit, a student will be able to:

Learning outcomes are detailed on the COMP11120 course unit syllabus page on the School of Computer Science's website for current students.

Reading list

Additional notes

Course unit materials

Links to course unit teaching materials can be found on the School of Computer Science website for current students.