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Sathyabama Institute of Science and Technology M.Sc - Mathematics SMT5201 Foundations of Mathematics Syllabus

DEPARTMENT OF MATHEMATICS SCHOOL OF SCIENCE AND HUMANITIES

SMT5201
FOUNDATIONS OF MATHEMATICS
L T P CREDITS TOTAL MARKS
3 1 0 4 100

UNIT I 13 Hours
Pre-requisites: Sets, subsets, Set operations and the laws of set theory and Venn diagrams. Examples of finite and infinite
sets. Finite sets and the counting principle. Empty set, properties of empty set. Standard set operations. Classes of sets.
Power set of a set (Quick review). Cartesian product of two and more sets, relations. Difference and Symmetric difference of
two sets. Set identities, Generalized union and intersections (As in section 1.7 of Text book 1).

UNIT II 12 Hours
Relations: Product set, Relations (Directed graph of relations on set is omitted). Composition of relations, Types of relations,
Partitions, Equivalence relations with example of congruence modulo relation, Partial ordering relations, n-ary relations. (As in
Chapter 3 of text book 2 excluding 3.7).

UNIT III 13 Hours
Functions Pre-requisites: Basic ideas such as domain, co-domain and range of functions. Equality of functions, Injection,
Surjection and Bijection (Quick review). Syllabus: Identity function, constant functions, product (composition) of functions,
theorems on one-one and onto functions, Mathematical functions, Recursively defined functions (As in Chapter 4 of text book
2). Indexed collection of sets, Operations on indexed collection of sets (As in 5.1, 5.2 and 5.3 of text book 2). Special kinds of
functions, Associated functions, Algorithms and functions, Complexity of Algorithms (As in Chapter 5.7 of text book 2).
Equipotent sets, Denumerable and countable sets, Cardinal numbers (Definitions and examples only as in 6.1, 6.2, 6.3 and
6.5 of text book 2).

UNIT IV 11 Hours
Basic Logic-1 Introduction, propositions, truth table, negation, conjunction and disjunction. Implications, biconditional
propositions, converse, contra positive and inverse propositions and precedence of logical operators. Propositional
equivalence: Logical equivalences. Predicates and quantifiers: Introduction, Quantifiers, Binding variables and Negations. (As
in Chapter 1 of Text book 1).

UNIT V 11 hours
Basic Logic-2 Methods of proof: Rules of inference, valid arguments, methods of proving theorems; direct proof, proof by contradiction, proof by cases,
proofs by equivalence, existence proofs, uniqueness proofs and counter examples. (As in Chapter 1 of Text book 1).

TEXT BOOKS
1. K.H. Rosen: Discrete Mathematics and its Applications (fifth edition), Tata McGraw Hill Publishing Company, New Delhi.
2. S. Lipschutz: Set Theory and related topics (Second Edition), Schaum Outline Series, Tata McGraw-Hill Publishing
Company, New Delhi.

REFERENCE BOOKS
1. P.R. Halmos: Naive Set Theory, Springer.
2. E. Kamke, Theory of Sets, Dover Publishers.