Description and Objective

This course is used to introduce the basics of the theory of sets and some of its applications. The purpose of sets is to house a collection of related objects. They are important everywhere in mathematics because every field of mathematics uses or refers to sets in some way. They are important for building more complex mathematical structure

Prerequisite

 None

Learning Outcomes

On Completion of this module, the learner will be able to:

1. learn basic concepts of set theory.
2. define sets, subsets, operations with sets: union, intersection, difference, symmetric difference, Cartesian product and disjoint union. graph of a function. Composition; injections, surjections, bijections, inverse function.
3. discuss the cardinality of Cartesian product, union, cardinality of all functions from a set to another set, cardinality of all injective, surjective and bijective functions from a set to another set.
4. define infinite sets, finite sets, countable sets, properties and examples, operations with cardinal numbers.
5. prove Cantor-Bernstein theorem with some applications of the gamma, beta function and hypergeometric functions.
6. define equivalence relations, partitions, quotient set; examples, parallelism, similarity of triangles, order relations, min, max, inf, sup; linear order, well ordered sets and induction, inductively ordered sets and Zorn’s lemma.

Contents

1. Set Theory,
2. Equivalent sets,
3. Countable and uncountable sets,
4. The concept of cardinal numbers,
5. Addition and multiplication of cardinals,
6. Cartesian product as sets of functions,
7. Addition and multiplication of ordinals,
8. Partially ordered sets, axiom of choice,
9. The Gamma function,
10. The Beta Function,
11. Solution in series of Bessel, Legendre and Hyper geometric differential equations,
12. Properties and Applications of Bessel function,
13. Properties and Applications of Legendre polynomial,
14. Generating function and Recurrence relations of Bessel function Legendre polynomials,
15. Relations between gamma, beta and hypergeometric function,
16. Properties and Applications of hypergeometric function.

 Recommended Books

1. Lipschutz, S.,Schaum's Outline of Set Theory and Related Topics, 2nd ed. (McGraw-Hill Education 1998).
2. Spiegel M. R., Schaum's Outline of Advanced Mathematics for Engineers and Scientists, (McGraw-Hill Education 2009).

 Suggested Books

1. Suppes P.,  Axiomatic Set Theory, (Dover Publications, 1972).
2. Halmos P.R., Naïve Set Theory, (Springer, 1998).
3. Temme N.M., Special Functions: An Introduction to the Classical Functions of Mathematical Physics, 1st ed. (Wiley-Interscience.1995).

RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS

The projects assigned in this course follow as assignments the exercises related to the topics from the suggested book. 

Assessment Criteria

       Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

       Mid-Term Exam:   30

       Final-Term Exam: 50

Key Dates and Time of Class Meeting

Monday                                                   11:00 PM-12:30 PM 

Thursday                                                  09:30 PM-11:00 PM 

Commencement of Classes                                                   October 26, 2020

Mid Term Examination                                                           December 28, 2020 to January 01, 2021

Final Term Examination                                                         March 01, 2021 to March 05, 2021 

Declaration of Result                                                             March 12, 2021