# Learn Discrete Mathematics with Veerarajan PDF: A Comprehensive Guide for Beginners

## Veerarajan Discrete Mathematics PDF Download

Are you looking for a comprehensive and accessible book on discrete mathematics? Do you want to learn the concepts and applications of this fascinating branch of mathematics? If yes, then you might be interested in downloading Veerarajan Discrete Mathematics PDF, a book written by T. Veerarajan, a renowned author and professor of mathematics. In this article, we will tell you what discrete mathematics is, why you should study it, who T. Veerarajan is, what his book offers, how to download it, and how to use it effectively. Let's get started!

## 8. veerarajan discrete mathematics pdf download

## What is Discrete Mathematics?

Discrete mathematics is the study of finite or discrete structures, such as integers, graphs, sets, logic, algorithms, codes, cryptography, etc. Unlike continuous mathematics, which deals with smooth and continuous phenomena, such as real numbers, functions, calculus, etc., discrete mathematics focuses on discrete and distinct objects that can be counted or enumerated. For example, the number of vertices in a graph, the truth value of a logical statement, the encryption key of a message, etc., are all discrete quantities.

Discrete mathematics has many applications in computer science, engineering, cryptography, combinatorics, logic, artificial intelligence, coding theory, etc. It also helps to develop abstract thinking, problem-solving skills, and mathematical reasoning skills. Some examples of discrete mathematics topics are:

Set theory: The study of collections of objects that share some common properties.

Graph theory: The study of networks of nodes and edges that represent relationships or connections.

Combinatorics: The study of counting and arranging discrete objects.

Group theory: The study of algebraic structures that have certain properties under operations.

Coding theory: The study of methods for transmitting and storing information efficiently and securely.

## Why Study Discrete Mathematics?

Discrete mathematics is not only interesting and fun to learn, but also very useful and relevant for many fields and disciplines. Here are some reasons why you should study discrete mathematics:

It helps you to understand the foundations of computer science and programming languages.

It enables you to design and analyze algorithms and data structures for various problems.

It allows you to model and solve real-world problems using discrete structures and methods.

It prepares you to deal with complex and abstract concepts in mathematics and logic.

It enhances your creativity and imagination by exploring different possibilities and patterns.

## Who is T. Veerarajan?

T. Veerarajan is a distinguished author and professor of mathematics at S.A. Engineering College in Chennai, India. He has over 40 years of teaching experience and has written several books on engineering mathematics, probability and statistics, numerical methods, discrete mathematics, etc. He has also received many awards and honors for his contributions to the field of mathematics and education. Some of his achievements are:

He is a Fellow of the Institution of Engineers (India) and a Life Member of the Indian Mathematical Society.

He has published more than 50 research papers in national and international journals.

He has authored more than 20 books on various topics of mathematics, which have been widely used by students and teachers across India and abroad.

He has received the Best Teacher Award from the Government of Tamil Nadu in 2004 and the National Award for Outstanding Academician from the Indian Society for Technical Education in 2007.

He has been a member of various academic bodies and committees, such as the Board of Studies, the Academic Council, the Curriculum Development Committee, etc.

## What is Veerarajan Discrete Mathematics Book?

Veerarajan Discrete Mathematics is a book written by T. Veerarajan that covers the basic concepts and applications of discrete mathematics. It is intended for the beginners who are taking the discrete mathematics course as part of their engineering or computer science curriculum. The book has the following features:

It provides an exhaustive coverage of set theory, which is the foundation of discrete mathematics.

It offers a comprehensive coverage of graph theory and combinatorics, which are essential for solving many problems in computer science and engineering.

It gives an excellent discussion of group theory and its applications in coding, which are important for cryptography and information theory.

It explains the solution procedure of the worked examples in detail, which helps the students to understand the concepts and methods clearly.

It includes a pedagogy that consists of 341 solved examples, 566 short answer questions, 556 descriptive questions, and over 500 figures and tables, which enhance the learning experience and test the understanding of the students.

### Set Theory

Set theory is the branch of mathematics that deals with collections of objects that share some common properties. A set is defined as a well-defined collection of distinct objects, which are called elements or members of the set. For example, 1, 2, 3 is a set that contains three elements: 1, 2, and 3. Some basic concepts and operations in set theory are:

Subset: A set A is said to be a subset of another set B if every element of A is also an element of B. For example, 1, 2 is a subset of 1, 2, 3, but 1, 4 is not. We write A B to denote that A is a subset of B.

Equality: Two sets A and B are said to be equal if they have exactly the same elements. For example, 1, 2, 3 and 3, 2, 1 are equal sets, but 1, 2 and 2, 3 are not. We write A = B to denote that A and B are equal sets.

Union: The union of two sets A and B is the set that contains all the elements that belong to either A or B or both. For example, the union of 1, 2 and 2, 3 is 1, 2, 3. We write A B to denote the union of A and B.

Intersection: The intersection of two sets A and B is the set that contains all the elements that belong to both A and B. For example, the intersection of 1, 2 and 2, 3 is 2. We write A B to denote the intersection of A and B.

Difference: The difference of two sets A and B is the set that contains all the elements that belong to A but not to B. For example, the difference of 1, 2 and 2, 3 is 1. We write A - B or A \ B to denote the difference of A and B.

Complement: The complement of a set A with respect to a universal set U is the set that contains all the elements that belong to U but not to A. For example, if U = 1, 2, 3, then the complement of 1 with respect to U is 2, 3. We write Ac or U - A to denote the complement of A with respect to U.

#### Solved Example

### }, A = a,b,c, B = c,d,e, and C = b,d,f. Find the following sets: - A B - A C - B - C - Ac - (A B)c Answer: We can use the definitions and properties of the set operations to find the required sets. Here are the steps: - A B = x = a,b,c,d,e - A C = x = b - B - C = x B and x C = c,e - Ac = x U and x A = d,e,f - (A B)c = x = f Graph Theory and Combinatorics

#### Graph theory is the branch of mathematics that deals with networks of nodes and edges that represent relationships or connections. A graph is defined as a set of vertices (or nodes) and a set of edges (or links) that connect some pairs of vertices. For example, the following figure shows a graph with six vertices and seven edges: ![graph](https://i.imgur.com/8Q0Zw9y.png) Some basic concepts and terms in graph theory are: - Degree: The degree of a vertex is the number of edges that are incident to it. For example, in the above graph, the degree of vertex a is 3, and the degree of vertex f is 1. - Path: A path is a sequence of vertices that are connected by edges. For example, in the above graph, a-b-c-d is a path of length 3. - Cycle: A cycle is a path that starts and ends at the same vertex. For example, in the above graph, a-b-c-a is a cycle of length 3. - Connected: A graph is said to be connected if there is a path between any two vertices. For example, the above graph is connected. - Tree: A tree is a connected graph that has no cycles. For example, the following figure shows a tree with six vertices and five edges: ![tree](https://i.imgur.com/5r9z6Ym.png) Combinatorics is the branch of mathematics that deals with counting and arranging discrete objects. Some basic concepts and techniques in combinatorics are: - Factorial: The factorial of a positive integer n, denoted by n!, is the product of all positive integers from 1 to n. For example, 5! = 5 4 3 2 1 = 120. - Permutation: A permutation of n distinct objects is an ordered arrangement of those objects. For example, if we have three objects a, b, and c, then there are six possible permutations: abc, acb, bac, bca, cab, cba. The number of permutations of n distinct objects is given by n!. - Combination: A combination of n distinct objects taken r at a time is an unordered selection of r objects from those n objects. For example, if we have three objects a, b, and c, then there are three possible combinations of two objects: ab, ac, bc. The number of combinations of n distinct objects taken r at a time is given by nCr = n! / (r! (n - r)!). - Binomial theorem: The binomial theorem states that for any positive integer n and any two numbers x and y, (x + y)n = Î£nCr xn-r yr where Î£ denotes summation from r = 0 to r = n. Solved Example

Question: How many different ways can five people sit in a row?

Answer: We can use the concept of permutation to solve this problem. We have five distinct people and five seats in a row. The order in which they sit matters, so we need to find the number of permutations of five people. This is given by 5!, which is equal to 120. Therefore, there are 120 different ways for five people to sit in a row.

### Group Theory and Coding

#### Group theory is the branch of mathematics that deals with algebraic structures that have certain properties under operations. A group is defined as a set of elements and an operation that satisfies the following four axioms: - Closure: For any two elements a and b in the group, the result of the operation on a and b is also in the group. - Associativity: For any three elements a, b, and c in the group, the operation is associative, that is, (a * b) * c = a * (b * c), where * denotes the operation. - Identity: There exists an element e in the group such that for any element a in the group, e * a = a * e = a. The element e is called the identity element of the group. - Inverse: For any element a in the group, there exists an element b in the group such that a * b = b * a = e. The element b is called the inverse of a in the group. For example, the set of integers Z under addition (+) forms a group, where the identity element is 0 and the inverse of any integer n is -n. Coding theory is the branch of mathematics that deals with methods for transmitting and storing information efficiently and securely. A code is defined as a set of symbols and rules for encoding and decoding messages. For example, Morse code is a code that uses dots and dashes to represent letters and numbers. Some basic concepts and terms in coding theory are: - Alphabet: An alphabet is a finite set of symbols that are used to form messages. For example, the English alphabet consists of 26 letters. - Message: A message is a sequence of symbols from an alphabet that conveys some information. For example, "HELLO" is a message composed of five symbols from the English alphabet. - Code word: A code word is a sequence of symbols from an alphabet that represents a message or part of a message. For example, ". . . _ _ _ . . ." is a code word in Morse code that represents the message "SOS". - Encoding: Encoding is the process of transforming a message into a code word or a sequence of code words. For example, encoding the message "HELLO" in Morse code results in ". . . . _ . _ . _ _ _ . . _ _ .". - Decoding: Decoding is the process of recovering the original message from a code word or a sequence of code words. For example, decoding the code word ". . . . _ . _ . _ _ _ . . _ _ ." in Morse code results in "HELLO". - Redundancy: Redundancy is the amount of extra information or symbols that are added to a code word or a sequence of code words to make it more reliable or secure. For example, adding parity bits or checksums to code words are ways of introducing redundancy. - Error detection: Error detection is the ability to identify if there are any errors or mistakes in a code word or a sequence of code words. For example, using parity bits or checksums can help to detect errors in code words. - Error correction: Error correction is the ability to correct or fix any errors or mistakes in a code word or a sequence of code words. For example, using Hamming codes or Reed-Solomon codes can help to correct errors in code words. Solved Example

Question: What is the Hamming distance between two binary code words 10101 and 11011?

Answer: The Hamming distance between two binary code words is defined as the number of positions where they differ. For example, the Hamming distance between 10101 and 11011 is 2, because they differ at the second and fourth positions.

## How to Download Veerarajan Discrete Mathematics PDF?

If you are interested in downloading Veerarajan Discrete Mathematics PDF, you have several options to choose from. Here are some sources where you can find and download the book:

### Google Books

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## ![free download books](https://i.imgur.com/9Xl6y9K.png) How to Use Veerarajan Discrete Mathematics PDF?

Now that you have downloaded Veerarajan Discrete Mathematics PDF, you may wonder how to use it effectively. Here are some tips and suggestions that can help you make the most of this book:

Read the preface and introduction of the book to get an overview of the contents and objectives of the book.

Follow the order of chapters and sections as suggested by the author, unless you have a specific reason to skip or rearrange them.

Pay attention to the definitions, examples, and exercises in each section. Try to understand and apply them before moving on to the next section.

Solve as many problems as possible from each section. Check your answers with the solutions given at the end of the book or online.

Review and revise the concepts and methods regularly. Use flashcards, notes, summaries, etc., to help you remember and recall them.

Use additional resources, such as online videos, lectures, tutorials, etc., to supplement your learning and clear your doubts.

Form a study group with other students who are using this book. Discuss and compare your solutions and approaches with them.

Seek feedback and guidance from your teachers or tutors if you have any questions or difficulties with the book.

## Conclusion

In this article, we have discussed what discrete mathematics is, why you should study it, who T. Veerarajan is, what his book on discrete mathematics offers, how to download it, and how to use it effectively. We hope that this article has helped you to learn more about this fascinating and useful subject and inspired you to read Veerarajan Discrete Mathematics PDF. If you have any comments or suggestions, please feel free to share them with us. Happy reading and learning!

## FAQs

Here are some frequently asked questions and answers about Veerarajan Discrete Mathematics PDF:

Q: Is Veerarajan Discrete Mathematics PDF suitable for beginners?

A: Yes, Veerarajan Discrete Mathematics PDF is suitable for beginners who are taking the discrete mathematics course as part of their engineering or computer science curriculum. The book provides a judicious mix of concepts and solved examples that make it ideal for the beginners.

Q: How can I get a hard copy of Veerarajan Discrete Mathematics book?

A: You can get a hard copy of Veerarajan Discrete Mathematics book from online or offline bookstores. You can also order it from the publisher's website: https://www.mheducation.co.in/discrete-mathematics-9780070616783-india.

Q: What are the prerequisites for reading Veerarajan Discrete Mathematics PDF?

A: The prerequisites for reading Veerarajan D