Course Name :Discrete Mathematics
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To understand mathematical reasoning in order to read, comprehend and construct Mathematical arguments as well as to solve problems, occurred in the development of programming languages.
To work with discrete structures such as graphs to study the structure of the world wide web, to model a computer network and to find the shortest path between two places in a transportation network.
Upon successful completion of this course, the student will be able to:
Apply the logical structure of proofs and work symbolically with connectives and quantifiers to produce logically valid, correct and clear arguments.
Evaluate elementary mathematical arguments and identify fallacious reasoning
Reformulate statements from common language to formal logic. Apply truth tables and the rules of propositional and predicate calculus.
Model and solve real-world problems using graphs, both quantitatively and qualitatively.
MODULE-I (4Hours)
Propositional Logic, Connectives, Truth tables of compound propositions, Propositional Equivalence.
Project 1: Given the truth values of the propositions p and q, find the truth values of the conjunction, disjunction, implication, bi-implication, converse, contrapositive and inverse.
MODULE-II (3Hours)
Theory of inference, Predicates and Quantifiers, Rules of Inference.
Project 2:Build valid arguments of a given set of propositional logics and quantified statements using rules of inferences.
MODULE-III (3 Hours)
Relations and its properties, Partial Ordering, POSET, Totally Ordered Set.
Project 3:Define the properties of a relation on a set using the matrix representation of that relation with examples.
MODULE-IV (3Hours)
Hasse Diagram, Maximal & Minimal Elements of a Poset, Greatest & Least Elements of a Poset, Supremum & Infimum of a Poset, Lattice.
Project 4: Find a Topological Sort of a Poset.
MODULE-V (3 Hours)
Introduction to Graph Theory, Graph Terminology and Special types of Graphs, Representation of Graphs.
Project 5:Describe how some special types of graphs such as bipartite, complete bipartite graphs are used in Job Assignment, Model, Local Area Networks and Parallel Processing.
MODULE-VI (3 Hours)
Graph Isomorphism, Connectivity, Euler and Hamiltonian Graphs, Planar Graphs, Graph Coloring.
Project 6(i):Describe the scheduling of semester examination at a University and Frequency Assignments using Graph Coloring with examples. Find also their Chromatic numbers.
Project 6(ii):List out 10 pairs of Non-isomorphic graphs and explain the reason behind it.
Project 6(iii):List out all features ofEuler and Hamiltonian Graphs. Justify whether the given set of graphs are Euler and Hamiltonian. Construct a Gray Code where the code words are bit strings of length three.
MODULE-VII (4 Hours)
Trees and their Properties, Spanning Trees, Minimum Spanning Trees, Kruskal’s Algorithm.
Project 7:Find a minimum spanning tree in a given weighted graph using Kruskal’s Algorithm.
Text Books:
Logic and Proofs (Mathematical Logic)
Propositional Logic:
CONVERSE, INVERSE, AND CONTRA POSITIVE
SOME MORE CONNECTIVES
Tautology and Contradiction, Logical Equivalence:
Equivalence Formulas: Other Connectives
Assignment 1 on Module 1: Construction of Truth Tables and Proving Logical Equivalences.
Project 1: Given the truth values of the propositions p and q, find the truth values of the compound propositions.
Theory of Inference, Inference rules.
Rules of Inference, formulae
https://youtu.be/HcS4lqXxrV4
Theory of Interference, deriving conclusions, from the given premises.Theory of Inference-converted
Assignment II : Deriving the valid conclusions from the given set of premises.
Project 2:Build valid arguments of a given set of propositional logics and quantified statements using rules of inferences.
Representation of Relations, Poset, totally ordered set.
Assignment III.
Project 3:Define the properties of a relation on a set using the matrix representation of that relation with examples.
Assignment : Construction of Hasse diagrams with different posets.
Maximal & Minimal Elements of a Poset, Supremum & Infimum of a Poset.
Project 4: Find a Topological Sort of a Poset.
Introduction to Graph Theory, Graph Terminology.
https://youtu.be/g9U_kSjmmTI
Special types of Graphs
Project 5:Describe how some special types of graphs such as bipartite, complete bipartite graphs are used in Job Assignment, Model, Local Area Networks and Parallel Processing.
Connectivity, Euler , Hamilton paths Graph Coloring.
Isomprphic graphs, Euler, planner graphs, and Graph coloring
Project 6(i):Describe the scheduling of semester examination at a University and Frequency Assignments using Graph Coloring with examples. Find also their Chromatic numbers.
Project 6(ii):List out 10 pairs of Non-isomorphic graphs and explain the reason behind it.
Project 6(iii):List out all features ofEuler and Hamiltonian Graphs. Justify whether the given set of graphs are Euler and Hamiltonian. Construct a Gray Code where the code words are bit strings of length three.
Kruskal’s Algorithm.
Assignment.
Project 7:Find a minimum spanning tree in a given weighted graph using Kruskal’s Algorithm.
Presentation by the students based on the assignment works with different Applications.
Working as Associate Professor of Mathematics CUTMAP Vizianagaram from July 2017 to Till Date. Awarded Ph.D in 2016 from Andhra University. Having an experience of 19 years in teaching and Research. Published 15 Papers in the area of Fixed point theory in National & International Journals.