Discrete Mathematics and Number Theory

Paper Code: 
CMAT111
Credits: 
4
Contact Hours: 
60.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Acquaint the students with the fundamentals of discrete mathematics and number theory and its applications.
  2. Make the students aware about graph theory, theorems related to prime numbers, relations and digraph etc.

Course Outcomes (Cos):

 

 Course

Learning outcomes

 

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

 

 

 

CMAT 111

Discrete Mathematics and Number Theory

 (Theory)

 

The students will be able to –

CO1: Represent a graph using an adjacency list and an adjacency matrix and apply graph theory to application problems such as computer networks.

CO2: Determine if a graph has an Euler or a Hamilton path or circuit.

CO3: Determine whether a graph is a binary tree, N-ary tree, or not a tree; use the properties of trees to classify trees, identify ancestors, descendants, parents, children, and siblings; determine the level of a node, the height of a tree or subtree and apply counting theorems to the edges and vertices of a tree.

CO4: Perform tree traversals using preorder, inorder, and postorder traversals and apply these traversals to application problems; use binary search trees or decision trees to solve problems.

CO5: Find quotients and remainders from integer division and apply Euclid’s algorithm and backwards substitution

CO6: Determine the proof of the Chinese remainder theorem and know its application.

Approach in teaching:

 

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

Unit I: 
I
9.00
Graphs Theory: Basic Terminology, Types of graph, paths and cycles, Euler graph and cycle, Hamiltonian graph and cycle, Shortest path algorithm (Djikstras algorithm), Graph isomorphism, Planar graph, Graph colorings and chromatic number.
 
Unit II: 
II
9.00
Relation and Diagraphs: Product sets and partitions, Paths in relation and diagraphs, Properties of relations, Equivalence relations. Trees: Introduction, m-ary trees, Properties of trees, Spanning trees, Minimal spanning trees, Binary search trees.
 
Unit III: 
III
9.00
Pigeonhole principle, Recurrence relation, Generating functions. Ordered relations and Structures: Partially ordered sets, Extremal elements of partially ordered sets.
 
Unit IV: 
IV
9.00
Elementary divisibility properties, Division algorithm, Greatest common divisor,  Least common multiplier, Euclid’s lemma.
 
Unit V: 
V
9.00
Bezout’s lemma, Prime number, Euclidean Algorithm, Fundamental theorem of arithmetic, Congruence, Chinese remainder theorem.
 
Essential Readings: 
  • Bernard Kolmann, Robert C. Busby and Sharon Ross, Discrete Mathematical Structures, PHI Delhi, 2008.
  • R.C. Choudhary, M.C. Goyal and D.C. Sharma, Discrete Mathematics, Ramesh Book Depot, 2018.
  • C. L. Liu, Elements of Discrete Mathematics, McGraw Hill,2009.
  • Thomas Koshy, Elementary Number Theory with applications, Elsevier Academic Press, 2014.
  • S.K. Pundir and R. Pundir, Theory of Numbers, Pragati Prakashan, Meerut, 2012.
 
References: 
  • Norman Biggs, Discrete Mathematics, Oxford University Press UK, 2003.
  • V. K. Bala krishnan, Introductory Discrete Mathematics, Prentice Hall, 2010.
  • Richard Johnson Baugh, Discrete Mathematics, Pearson Education Asia, New Delhi, 2008.
  • David M. Burton, Elementary Number Theory, McGraw Hill, 2007.
  • J.P.Trembly, and R. Manohar, Discrete Mathematical Structures with Applications to Computer Science, McGraw Hill Pub. Co. Ltd, New Delhi, 2010.
  • Ralph. P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, Pearson Education Asia, Delhi, 2019.
 
Academic Year: 
2022-2023
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