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Module MA2C03: Discrete Mathematics

Credit weighting (ECTS)

10 credits

Semester/term taught

Michaelmas and Hilary terms 2016-17

Contact Hours

22 weeks, 3 lectures including tutorials per week

Lecturer

Prof Andreea Nicoara

Learning Outcomes

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

• Construct reasoned logical arguments to identify and justify basic properties of mathematical objects that are specified as sets, relations on sets, functions between sets, and/or monoids.
• Identify formal languages generated by simple context-free grammars, and construct specifications of context-free grammars and finite state machines that generate and/or determine formal languages, given specifications of such formal languages.
• Recognize and identify properties of undirected graphs that are networks consisting of vertices together with edges joining pairs of vertices, and find examples of isomorphisms between such graphs satisfying given criteria.
• Find solutions to certain types of homogeneous and inhomogeneous linear ordinary differential equations of degree at least two, using methods based on the use of power series, and also methods based on the identification of particular integrals and complementary functions, where the coefficients of the differential equation are constants and the forcing function is typically constructed from polynomial, exponential and trigonometric functions.
• Expound and apply basic properties of exponential and trigonometric functions, where the arguments of those functions are complex numbers and variables, and thereby obtain results that are relevant to the basic implementation of the Discrete Fourier Transform.
• Perform calculations within the algebra of vectors in three-dimensional space, and the algebra of quaternions, and apply the results of such calculations to the solution of simple geometrical problems.
• Perform calculations in basic number theory, justified on the basis of theorems explicitly presented and proved within the module, that have relevance to the implementation of public key cryptographic systems such as the Rivest-Shamir-Adelman (RSA) public key cryptosystem.

Module Content

Specific topics addressed in this module include the following:

• The Principle of Mathematical Induction
• Sets, Relations and Functions
• Introduction to Abstract Algebra
• Introduction to Formal Languages and Context-Free Grammars
• Introduction to Graph Theory
• Ordinary Differential Equations
• Trigonometric Identities, Complex Exponentials and Periodic Sequences
• Vectors and Quaternions
• Introduction to Number Theory and Cryptography

Lecture notes, assignments, worked solutions to problems from previous years and further information relevant to the module are available from the module webpage at http://www.maths.tcd.ie/~dwilkins/Courses/MA2C03/.

Module Prerequisite

Module CS1001 (Mathematics I), or an equivalent module developing the necessary mathematical skills in areas such as calculus and linear algebra.

Assessment Detail

This module will be examined in a 3 hour examination in Trinity term. Also students should complete a small number of assignments during the academic year. The final grade at the annual examination session will be a weighted average over the examination mark (90%) and the continuous assessment mark (10%). The final grade at the supplemental examination session will be wholly determined by the supplemental examination paper.