### Elements of Set Theory and Mathematical Logic (MATH-5102)

DESCRIPTION AND OBJECTIVES

The main aim of this course is the study of set theory and the concept of mathematical logic. 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, ordinal numbers, relations, functions, Boolean algebra and fundamentals of propositional and predicate logics.

INTENDED LEARNING OUTCOMES

To learn basic concepts of set theory.
2. To 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. To 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. To define infinite sets, finite sets, countable sets, properties and examples, operations with cardinal numbers.
5. To prove Cantor-Bernstein theorem with some applications of the gamma, beta function and hypergeometric functions.
6. To 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.
7. To learn mathematical logic including propositional calculus, truth tables and predicate calculus.

COURSE CONTENTS

1. Set theory: Sets, subsets,
2. Operations with sets: union, intersection,
3. Difference, symmetric difference,
4. Cartesian product and disjoint union.Functions: graph of a function,
5. Composition; injections, surjections, bijections, inverse function,
6. Computing cardinals: Cardinality of Cartesian product, union.,
7. Cardinality of all functions from a set to another set,
8. Cardinality of all injective, surjective and bijective functions from a set to another set,
9. Infinite sets, finite sets. Countable sets, properties and examples,
10. Operations with cardinal numbers. Cantor-Bernstein theorem,
11. Relations: Equivalence relations, partitions, quotient set; examples,
Parallelism, similarity of triangles,
12. Order relations, min, max, inf, sup; linear order,
13. Examples: N, Z, R, P(A). Well ordered sets and induction,
14. Inductively ordered sets and Zorn’s lemma,
15. Mathematical logic: Propositional Calculus. Truth tables. Predicate Calculus.

Halmos, P. R., Naive Set Theory, emended edition, (Bow Wow Press, (2019)).
2. Lipschutz, S., Schaum's Outline of Set Theory and Related Topics, 2nd edition, (McGraw-Hill Education (1998)).
3. Pinter, C. C., A Book of Set Theory, (Dover Publication, (2014)).
4. O'Leary, M. L., A First Course in Mathematical Logic and Set Theory, 1st edition, (Wiely, (2015)).
5. Smith, D. Eggen, M. and Andre, R.S., A Transition to Advanced Mathematics, 8th edition (Brooks/Cole, (2014))

Assessment Criteria:

Mid-Term Marks:  30

Final-Term Marks: 50

Sessional Marks: 20

Commencement of Classes:

Wednesday 11. 00AM to 12.30 PM

Thursday 11.00AM to 12.30PM

October 12, 2020 (Monday )

Mid-Term Examination:

December 28, 2020 to January 01, 2021 (Monday to Friday)

Final-Term Examination:

March 01 to 05, 2021 (Monday to Friday)

Declaration of Result: March 12, 2021 (Friday)

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