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Discrete structures Notes Unit 1 – 5
Unit 1: Set Theory
“Set theory” is a fundamental branch of mathematics that deals with the study of sets, which are collections of distinct objects, called elements. These objects can be anything—numbers, symbols, or even other sets.
Unit 2: Mathematical Logic and Circuits
“Mathematical logic” is a branch of mathematics that focuses on formal systems, reasoning, and the structure of statements and proofs. It involves the study of propositional logic and predicate logic.
Unit 3: Modern Algebra
“Modern algebra”, also known as abstract algebra, is a branch of mathematics that studies algebraic structures such as groups, rings, fields, and vector spaces. It focuses on the generalization and abstraction of arithmetic operations.
Unit 4: Graph Theory
“Graph theory” is a branch of mathematics that studies graphs, which are structures consisting of vertices and edges. Graphs can represent a wide range of relationships and systems, such as social networks, computer networks, and transportation routes.
Unit 5: Data Analysis
“Data analysis” is the process of inspecting, cleaning, transforming, and modeling data to discover useful information, support decision-making, and draw conclusions. It involves techniques for exploring large datasets to identify patterns, trends, and relationships.
Scope of Discrete structures
- Algorithm Design and Analysis:
- Understanding algorithm efficiency and correctness.
- Proving algorithm properties.
- Data Structures:
- Designing efficient data storage and retrieval mechanisms.
- Analyzing data structure performance.
- Database Systems:
- Modeling and querying data.
- Database design and optimization.
- Computer Networks:
- Routing protocols, network topology, and flow control.
- Cryptography:
- Developing secure communication systems.
- Studying encryption and decryption techniques.
- Artificial Intelligence:
- Knowledge representation, search algorithms, and machine learning.
Objectives of Discrete structures
- Develop logical and analytical thinking: Discrete structures foster the ability to reason logically, analyze problems systematically, and construct rigorous proofs.
- Provide a foundation for computer science: It lays the groundwork for understanding fundamental concepts in computer science, such as data structures, algorithms, database systems, and computer networks.
- Enhance problem-solving skills: By studying various problem-solving techniques and strategies within discrete structures, students can improve their ability to tackle complex challenges.
- Introduce mathematical tools for computer science: The course equips students with essential mathematical concepts and tools that are widely used in computer science applications.
UNIT – 1
1. Set Theory
- Sets, Types of Sets, Basic Operations on Sets, Venn diagram, Cartesian product of two sets, Distributive
law, De Morgan’s Law. - Functions: Interval and sub-intervals. Definition of function and examples, polynomial, rational,
exponential, logarithmic and trigonometric functions.
UNIT – 2
2. Mathematical Logic and circuits
- Basic Concepts, Propositions or Statements, Truth Table, Connectives and Compound Propositions,
Implication, Bi-conditional of Connectives, Converse, Inverse and Contra positive of an Implication,
Tautology, Logical Equivalence, Switching Circuits
UNIT – 3
3. Modern Algebra
- Binary Operations, Properties of Binary Operations, Semi group, Monoid, Groups, Finite and Infinite
Groups, Algebra of Groups, Subgroups and other Groups.
UNIT – 4
4. Graph Theory
- Graph, Multi Graph, Complete Graph, Bi Graph, Degree, Degree Sequence, Matrices of graphs, tree,
spanning trees
UNIT – 5
5. Data Analysis
- Data and Statistical Data, Frequency Distribution, Graphical Representation, Measure of the Central
Tendency, Measures of Dispersion (Mean Deviation and Standard Deviation)
Recommended Books and Resources
- “Discrete Mathematics and Its Applications” by Kenneth H. Rosen: Comprehensive coverage of discrete mathematics topics.
- “Discrete Mathematics” by Seymour Lipschutz and Marc Lipson: Schaum’s Outline series for problem-solving practice.
- “Concrete Mathematics” by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik: Advanced topics in discrete mathematics.
- Online Resources: Khan Academy, Coursera, edX.
Practical Assignments
- Logic and Proofs: Solving problems related to propositional and predicate logic.
- Set Theory and Relations: Exercises involving set operations, relations, and functions.
- Combinatorics and Counting: Problems on permutations, combinations, and the pigeonhole principle.
- Graph Theory and Trees: Implementing and analyzing graph algorithms.
- Boolean Algebra: Simplifying Boolean expressions and designing digital circuits.
- Recurrence Relations: Solving recurrence relations using various methods.