# Discrete Mathematics

By Minirani S   |   MPSTME, NMIMS Deemed University, Vile Parle West, Mumbai.
Learners enrolled: 989
Discrete mathematics involves the study of objects which are separated or spaced apart from each other. For example, finite sets and the set of integers are discrete sets, while the set of real numbers would be considered to be a continuous, or non-discrete, set of objects. The difference between discreteness and continuity can also be seen in distinguishing between digital signals (discrete) and analog signals (continuous). As these examples suggest, discrete mathematics forms a conceptual complement to the continuous processes which underlie the study of calculus. Discrete sets often carry additional structures such as an operation (addition, multiplication, concatenation, union or intersection, for example) or an inequality relationship, and, when present, these structures are instrumental in developing deeper theories. With both the subject itself, as well as the experience of working with mathematical arguments, the course will provide a foundation for moving into higher level mathematics courses such as real analysis, abstract algebra, math modelling, geometry and topology.
Summary
 Course Status : Ongoing Course Type : Not Applicable Duration : 12 weeks Start Date : 16 Jan 2023 End Date : 10 Apr 2023 Exam Date : Enrollment Ends : 15 Mar 2023 Category : Mathematics Credit Points : 2 Level : Undergraduate

### Course layout

WEEK -1

Module - 01
Sets and subsets-Representations-Cardinality-Set operations-Set identities- principle of mathematical induction- principle of inclusion-exclusion
Assignment - 1
Deadline – last day of the week

Module - 2
Ordered sets-languages-phase structure grammars-types of grammars and languages
Assignment - 2
Deadline – last day of the week

Module - 3
Interaction/Discussion on Module 1 & 2 and Assignments

WEEK - 2

Module - 4
Basics of counting-permutations-combinations. Pigeon hole principle
Assignment - 3
Deadline – last day of the week

Module - 5
Elements of probability-sample spaces-asssigning probabilities to events-equally likely outcomes
Assignment - 4
Deadline – last day of the week

Module - 6
Interaction\Discussion on Module 4 & 5 and Assignments

WEEK - 3

Module - 7
Relations and their properties-binary relations and their applications-representing relations-equivalence relations
Assignment - 5
Deadline – last day of the week

Module - 8
Product sets and partitions-partial orderings-lattices-chains and antichains-functions
Assignment - 6
Deadline – last day of the week

Module - 9
Interaction/Discussion on Module 7 & 8 and Assignments

WEEK - 4

Module - 10
Basic definitions and terminology-multigraphs-weghted graphs-paths and circuits
Assignment - 7
Deadline – last day of the week

Module -11
Shortest paths- Eulerian paths and circuits-Travelling Salesman problem-planar graphs
Assignment - 8
Deadline – last day of the week

Module - 12
Interaction/Discussion on Module 10 & 11 and Assignments

WEEK - 5

Module -13
Definitions-properties-applications of trees
Assignment - 9
Deadline – last day of the week

Module - 14
Spanning trees-minimum spanning trees-applications-Prims and Kruskals algorithm
Assignment - 10
Deadline – last day of the week

Module -15
Interaction/Discussion on Module 13 & 14 and Assignments

WEEK-6

Module - 16
Finite state machines with output- finite state machines with no output-finite state automata- language recognition by finite state machines-designing finite state automata
Assignment - 11
Deadline – last day of the week

Module - 17
Complexity of algorithms-comparing algorithms-Discrete numeric functions-manipulation of numeric functions-generating functions-operations on generating functions-
Assignment - 12
Deadline – last day of the week

Module - 18
Interaction/Discussion on Module 16 & 17 and Assignments

WEEK - 7

Module - 19
Linear recurrence relations- homogeneous and non-homogeneous and solutions
Assignment  - 13
Deadline – last day of the  week

Module - 20
Solutions of non-homogeneous type-particular solution and total solution-solution by the method of generating functions
Assignment  - 14
Deadline – last day of the  week

Module - 21
Interaction/Discussion on Module 19 & 20 and Assignments

WEEK - 8

Module - 22
Binary operations-semigroups-products and quotient of semigroups
Assignment  - 15
Deadline – last day of the  week

Module - 23
Groups- products and quotient groups-other mathematical structures
Assignment  - 16
Deadline – last day of the  week

Module - 24
Interaction/Discussion on Module 22 & 23 and Assignments

WEEK -  9

Module - 25
Lattices and algebraic structures-sub lattices-isomorphic lattices-Principle of duality-special types of lattices-
Assignment  - 17
Deadline – last day of the  week

Module - 26
Boolean Algebra-Boolean expressions and Boolean functions-identities of Boolean algebra-
Assignment  - 18
Deadline – last day of the  week

Module - 27
Interaction/Discussion on Module 25 & 26 and Assignments

WEEK - 10

Module - 28
Functional completeness-logic gates-exmples of circuits-minimisation of circuits-Karnaugh maps-Quine Mc Clusckey method
Assignment  - 19
Deadline – last day of the  week

Module- 29
Interaction/Discussion on Module 28 & other modules and Assignments

Module- 30
Term end assessment:
Will provide the video answers to the frequently asked questions by the students.
Interactions and Online evaluation process

### Books and references

1. Elements of Discrete Mathematics, C.L. Liu., Tata McGraw-Hill Publishing Company Ltds. 2nd edition
2. Discrete Mathematics and its Applications (SixthEdition), Kenneth.H. Rosen, TMH Education Pvt. Ltd.
3. Set Theory and Related Topics, Seymour Lipschutz Schaum’s Outline Series, Second Edition, TMH Education Pvt. Ltd.
4. Applied Discrete Structures for Computer Science, Alan Doerr & Kenneth Lefvasseur.
5. Introduction to Automata Theory, Languages and Computation, John Hopcroft & Jeffrey D. Ullman, Narosa Publishing Company, 1999.

### Minirani S

MPSTME, NMIMS Deemed University, Vile Parle West, Mumbai.
Dr. Minirani S is currently working as an Associate Professor in the Department of Basic Sciences and Humanities in the Mukesh Patel School of technology, Management and Engineering, NMIMS Deemed University, Vile Parle West, Mumbai. Dr. Minirani S has completed her Ph. D. in Mathematics from National Institute of Technology, Calicut. Her areas of interest include Fractal Geometry, Discrete Mathematics and Fluid Dynamics.

### Course certificate

30 % for in course Assessment &amp; 70 % of end term Proctored Exam