Course Tittle: Elements of Set Theory and Mathematical Logic

Course Code: MATH-5102

Credit Hours: 03

DESCRIPTION & OBJECTIVES

Everything mathematicians do can be reduced to statements about sets, equality and membership which are basics of set theory. This course introduces these basic concepts. The course aims at familiarizing the students with cardinals, relations and fundamentals of propositional and predicate logics.

Recommended Books:

1. Liebeck M. A Concise introduction to pure Mathematics, CRC Press, 2011.
2. Biggs N. L., Discrete Mathematics, Oxford University Press, 2002.
3. Garnier R., Taylor J., Discrete Mathematics, Chapters 1,3,4,5, CRC Press, 2010.
4. Fraenkal A.A., Abstract Set Theory, North-Holland Publishing Company, 1966.
5. Suppes P., Axiomatic Set Theory, Dover Publication, 1972.
6. Halmos P.R., Naive Set Theory, New York, Van Nostrand, 1950.
7. Rotman B., Kneebone G.T., The Theory of sets and Transfinite Numbers,  Oldbourne London, 1968.
8. Smith D., Eggen M., Andre R.S., A Transition to Advanced Mathematics, Brooks/Cole 2001

CONTENTS

Set theory: Sets, subsets, operations with sets: union, intersection, difference, symmetric difference, Cartesian product and disjoint union. Functions: graph of a function. Composition; injections,  surjections, bijections, inverse function. Computing cardinals: 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. Infinite sets, finite  sets. Countable sets, properties and examples. Operations  with cardinal numbers.  Cantor- Bernstein  theorem. Relations: Equivalence relations, partitions, quotient set; examples, parallelism, similarity of triangles. Order relations, min, max, inf, sup; linear order. Examples: N, Z, R, P(A). Well-ordered sets and induction. Inductively ordered  sets and Zorn’s lemma. Mathematical logic: Propositional Calculus. Truth tables. Predicate Calculus.

RESEARCH PROJECT

N/A

ASSESSMENT CRITERIA

Mid exam:       30

Sessional:        20

Project:            --

Assignments:   10

Presentation:   10

Final exam:      50

Total:               100

RULES AND REGULATIONS

75% attendance is compulsory to appear in Final Term exam.