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Course code  :  PMAT 12212

Title               :  Mathematics for Computing II

Pre-requisites :  PMAT 11212

Learning Outcomes:

Upon successful completion of the course unit, the student should be able to:

1. demonstrate knowledge in matrices
2. solve systems of linear equations
3. compute eigenvectors and eigenvalues
4. solve real world problems using counting techniques
5. demonstrate knowledge in basic graph theory.

Course Contents:

Matrices - Matrix notations, Algebra of matrices, Inverse of a matrix, Elementary row operations and
calculation of inverse matrix, Determinant of a matrix, Minor and cofactor, Properties of determinants, Adjoint
of a matrix and calculation of inverse matrix.
Systems of Linear Equations - Matrix form of a system of equations and conditions for unique solutions,
augmented matrix and echelon form, Conditions for existence of a unique solution, infinitely many solutions
and no-solution.
Vector Spaces- Linear Independence, Basis and Rank, Linear Mappings, Eigenvalues and Eigenvectors
Counting Techniques - Factorial notation, Sum rule and product rule, Pigeonhole principle, Permutations and
combinations, Binomial expansion.
Graphs - Graph notation and basic definitions, Complete and bipartite graphs, Euler and Hamiltonian circuits,
Directed and undirected graphs, Planar graphs, Graph colouring, Trees and their basic properties

Method of Teaching and Learning: Lectures, interactive classroom sessions, and case discussions 

Assessment: End of course unit examination, group assignment, mid-term examination, class attendance

Recommended Reading:

1. Johnsonbaugh, R. (8th Ed., 2017). Discrete Mathematics, Pearson.
2. Rosen, K.H. & Krithivasan, K. (7th Ed., 2011). Discrete Mathematics and Its Applications,
McGraw-Hill.
3. Kreyzig, E. (8th Ed., 2006). Advanced Engineering Mathematics, Wiley Student Edition.
4. Susanna, S. E. (2010). Discrete Mathematics with Applications, Cengage Learning.
5. Larson, R. & Falvo, D. (6th Ed., 2009). Elementary Linear Algebra, Houghton Mifflin Harcout
Publishing Compan

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Course code  :  PMAT 12242

Title               :  Discrete Mathematics II

Pre-requisites :  PMAT 11223

Learning Outcomes:

On successful completion of the course, the student should be able to;

1. demonstrate knowledge of topics including divisibility, prime numbers, congruences and Diophantine
equations
2. understand the logic and methods behind the major proofs in Number Theory
3. apply various properties relating to the integers including the Well-Ordering Principle, primes, unique
factorization, the division algorithm, greatest common divisors, and modular arithmetic
4. understand and prove theorems/lemmas and relevant results in graph theory
5. apply the basic concepts of graph theory, including Eulerian trails, Hamiltonian cycles, bipartite graphs,
planar graphs, and Euler characteristics on solving problems
6. apply algorithms and theorems in graph theory in real-world applications
7. apply counting principles to solve problems.

 Course Contents:

Counting: Basic Principles of Counting, Pigeonhole Principle, Permutations and Combinations.
Number Theory: The Well Ordering Principle, Divisibility and Division Algorithm, The Greatest Common
Divisor, The Euclidean Algorithm, Prime Numbers, Infinitude of Primes, The fundamental theorem of arithmetic,
Linear Diophantine Equation, Modular Arithmetic: solving linear congruence.
Graph Theory: Graph Terminology, Special Types of Simple Graphs, Subgraphs, Euler Cycles, Hamiltonian
Cycles, Representations of Graphs, Graph Isomorphism, Planar Graphs, Kuratowski’s Theorem, Graph
Colouring.

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Johnsonbaugh, R., (8^th edition , 2017), Discrete Mathematics, Pearson.
2. Rosen, K.H., Krithivasan, K., (7^th edition, 2013), Discrete Mathematics and Its Applications, McGraw-Hill.
3. Rosen, K.H. (6th Ed., 2010). Elementary Number Theory and Its Applications, Pearson.

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Course code  :  PMAT 12253

Title               :  Theory of Calculus

Pre-requisites :  PMAT 11223

Learning Outcomes:

On successful completion of the course, the student should be able to;

1. classify a real number as a natural, whole, integer, rational, or irrational number and demonstrate knowledge
of the axiomatic description of the field of real numbers and prove theorems from the given set of axioms
2. evaluate limits given analytic, graphical, numerical function information and describe in simple language the
statements of limit laws and use these laws to evaluate limits and state the definition of continuity and use
the definition to ascertain the continuity or dis-continuity of a function at a point
3. state the limit definition of derivative of a function at a point and use the limit definition to calculate a
derivative or identify where the derivative fails to exist at a point
4. apply the Chain Rule to find the derivative of a composition of functions and interpret both continuous and
differentiable functions geometrically and analytically and apply Rolle's Theorem, the Mean Value Theorem
5. explain indeterminant forms and use L’Hopital’s rule to evaluate limits involving indeterminant forms
6. describe upper and lower Darboux sums, Riemann sum and compare them and write the statement of the
Fundamental Theorems of Calculus and explain what the theorems say about definite integrals
7. visualize and sketch the surface generated by revolving a graph of a function about an axis of evolution and
calculate the volume of a solid and volume of a solid revolution by using disk, washer and cylindrical
shells methods
8. classify improper integrals and determine the integrals, identify the types of improper integrals and rewrite
them as proper integrals with a limits and use the words convergent and divergent to describe an
improper integral.

 Course Contents:
Real Numbers: Field Structure, Ordered fields and their properties, Open and Closed sets, Maximum, Minimum,
Supremum and Infimum of a set, Completeness axioms, Archimedean Property, Denseness of subsets of real
numbers
Real Valued Functions of a Real Variable: Review of Polynomial, Rational, Algebraic, Trigonometric,
exponential and Logarithmic Functions, Composite Functions, Piece-wise Functions
Functions and Limits: Limits of Functions, Left and Right-hand limits, Squeeze theorem, Continuous Functions,
Asymptotes and limits involving infinity
Derivative and Applications: Derivative of a function, Chain rule, Logarithmic and Implicit differentiation,
Higher order derivatives, Rolle’s Theorem, Mean Value Theorem
Indeterminate Forms: L’hospital Rule.
Integrals: Darboux and Riemann Integrations, Fundamental Theorem of Calculus, Improper Integrals (First &
second kind)
Applications of Integrals: Volumes and Solid revolutions, Volumes by Cylindrical Shells, Mean Value Theorem
for Integrals, Arc Length, Area of a Surface of Revolution

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Stewart, J. (9th Ed., 2020). Calculus Early Transcendentals, Cengage Learning, Inc.
2. Larson, R. & Edwards, B.H. (11th Ed., 2018). Calculus, Brooks/Cole, Cengage Learning
3. Ross, K.S. (2nd Ed., 2015). Elementary Analysis: The Theory of Calculus, Springer.
4. Hass, J., Heil, C. & Weir, M.D. (14th Ed., 2017). Thomas' Calculus: Early Transcendentals, Single
Variable, Pearson

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Course code  :  PMAT 11203

Title               :  Topics in Basic Mathematics

Learning Outcomes  :

At the end of this course, the student should be able to

1. demonstrate the knowledge in basic discrete mathematical concepts in sets, functions, and relations
2. apply the basic concepts in discrete mathematics to a given problem
3. explain the basics of matrices and its applications
4. find the solution of system of linear equations
5. understand exponential, logarithmic and trigonometric functions
6. identify finite series
7. use binomial theorem in various problems.

Course Contents:

Sets: Set notations, Sets of numbers and intervals, Subsets and equal sets, Power set, Cartesian product of sets,
Algebra of sets

Relations: Relations and their properties, Functions as relations, Relations on a set, Properties of Relations
Functions: Function notation, One-to-one and onto functions, Graphs of functions, Exponential, Trigonometric
and logarithmic functions

Matrices: Algebra of matrices, Types of square matrices, Transpose of a matrix, Inverse of a matrix,
Determinants of order two and three

Linear Equations: Solving linear equations, Solving system of linear equations, Applications of linear systems
in biological stream

Finite series: Types of finite series, Binomial theorem

Method of Teaching and Learning: A combination of lectures and tutorial discussions.

Assessment: Based on the tutorials, tests, and end of course examination.

Recommended Reading: 

1. Johnsonbaugh, R. (8th Ed., 2017). Discrete Mathematics, Pearson.
2. Rosen, K.H. & Krithivasan, K. (8th Ed., 2018). Discrete Mathematics and Its Applications, McGraw-
Hill.
3. Koshy, T. (2004). Discrete mathematics with Applications, Elsevier.
4. Balakrishnan, V.K. (2012). Introductory Discrete Mathematics, Courier Corporation