Sciences, Technology & Medicine

## Discrete Mathematics

Discrete Mathematics, the study of finite mathematical systems, is a hybrid subject. The topics covered in this book have been chosen keeping in view the knowledge required to understand the functioning of the digital computer because many of its properties can be understood and interpreted within the frame work of finite mathematical systems. This book provides reasonably rigorous and compact treatment of major mathematical techniques available for study in theoretical computer science. An attempt has been made to cover elementary to advance concepts in each chapter to take care of the needs of students who may earlier not have knowledge of these topics. Each chapter contains sufficiently large number of solved examples and illustrations to explain definitions, principles, theorems and other descriptive material. This is followed by sets of Self Practice Problems with Hints and Answers to stimulate further learning and interest. The book is designed to be self-contained and comprises of 13 chapters. It can be used by students of BSc (Maths), BCA/BIS/BIT and BE/BTech as an introduction to the fundamental concepts of discrete mathematics and by students of MSc, MCA, MTech for the development of more advanced mathematical concepts required in computer science applications.
Additional Info
• Publisher: Laxmi Publications
• Language: English
• ISBN : 978-93-5138-143-3
Total Price: 44.85
• Chapter 1

### Set Theory Price 2.99  |  2.99 Rewards Points

This chapter introduces the notations and terminology of set theory which was originated in the 1895 by the German mathematician G. Cantor. In our daily lives, we often use phrases or words such as a bunch of keys, a pack of cards, a class of students, a team of players, etc. The words bunch, pack, class and team all denote a collection of several discrete objects. Also, the dictionary meaning of set, is a group or a collection of distinct, definite and distinguishable objects selected by means of some rules or description. In this chapter, we shall elaborate and discuss the importance of this word in the application of mathematics, computer science and engineering to real-life problems. Venn diagram representation of sets can be applied to understand relationship between set theory and logical arguments in Boolean algebra. Moreover, this concept simplify, clarify and unify many other mathematical concepts
• Chapter 2

### Number Theory Price 2.99  |  2.99 Rewards Points

Number theory helps us to understand properties of integers, specially the properties of positive integers. Integer numbers are used in solving problems associated with the transmissions, coding and manipulation of numerical data. The development of electronic computers has enabled us to perform number-theoretic calculations much easily on personal computers or programmable calculators. In this chapter, we shall discuss some of the basic properties of integers particularly divisibility and congruence relation with their applications.
• Chapter 3

### Relations And Digraphs Price 2.99  |  2.99 Rewards Points

The relations such as, less than, perpendicular to, not equal to, subset of, etc. are often used in mathematics and computer science problems concerning discrete objects. These relations help establishing a relation between pair of objects taken in a particular order. A relation between two sets can be defined by listing their elements as an ordered pairs. In this chapter, we will discuss relations defined on sets and various ways of representing finite relations along with their properties.
• Chapter 4

### Functions Price 2.99  |  2.99 Rewards Points

In this chapter, we will discuss a special kind of relation called a function. The knowledge of functions is very useful both in mathematics and computer science. Functions can also be used for counting and knowing the cardinality of sets. The terms such as mapping ,transformation, etc., are also used for functions to depict a relation between two discrete objects
• Chapter 5

### Mathematical Induction Price 2.99  |  2.99 Rewards Points

One of the most important techniques to prove many mathematical statements or formulae which cannot be easily derived by direct methods is sometimes derived by using the principle of mathematical induction. In this principle, the word induction is associated with the inductive reasoning, by which a conclusion is drawn from a large number of special cases. Actually mathematical induction is deductive in nature, for it leads to a definite conclusion. It is usually employed in proving the validity of a statement involving all positive integer values of the number n.
• Chapter 6

### Combinatorics Price 2.99  |  2.99 Rewards Points

In this chapter counting techniques which are useful in the development of several important concepts in mathematics and computer science particularly permutations and combinations will be discussed. The concept of permutations and combinations supports pigeonhole principle and is useful in the analysis of probability of an event.
• Chapter 7

### Fundamentals Of Probability Price 2.99  |  2.99 Rewards Points

The decision-makers always face some degree of risk while selecting a particular decision (course of action or strategy) to solve a decision problem. It is because each strategy can lead to a number of different possible outcomes (or results). Thus, it is necessary for the decision-makers to enhance their capability of grasping the probabilistic situation so as to gain a deeper understanding of the decision problem and base their decisions on rational considerations. For this the knowledge of the concept of probability, probability distributions and various related statistical terms is needed. The knowledge of probability and its various types of distributions helps the development of probabilistic decision models. This chapter is devoted to
• Chapter 8

### Recurrence Relations And Generating Functions Price 2.99  |  2.99 Rewards Points

In this chapter, we will discuss finite order linear recurrence relations with constant coefficients. There is no general algorithm (or method) for solving all recurrence relations, however, generating functions will be introduce to solve recurrence relations.
• Chapter 9

### Graph Theory Price 2.99  |  2.99 Rewards Points

Graph theory is used to analyse problems of combinatorial nature that arise in computer science, operations research, physical sciences and economics. The term graph is familiar to you because it has been used in the context of straight lines and linear inequalities. The graphs are used to represent a problem involving discrete arrangements of objects that are related to each other, but without taking into consideration their internal properties. In this chapter, first we will combine the concepts of graph theory with digraph of a relation to define a more general type of graph that has more than one edge between a pair of vertices. Second, we will identify basic components of a graph, its features and many applications of graphs.
• Chapter 10

### Trees Price 2.99  |  2.99 Rewards Points

A tree is a connected graph without cycles (also called paths). There are several different form of a tree sand can be applied for solving a wide variety of problems in computer science applications, such as organising and sorting of data in a database, coding theory, language compliers, etc. A graph without self loop parallel edges is also a tree. In this chapter, we will discuss various types of undirected trees, that too, without certain paths called cycles and their properties. Obviously, then all multigraphs cannot be considered as trees.
• Chapter 11

### Algebraic Structures Price 2.99  |  2.99 Rewards Points

A set of objects with mathematical operation or operations defined on it and the properties associated with it form an algebraic system or structures.
• Chapter 12

### Mathematical Logic Price 2.99  |  2.99 Rewards Points

In this chapter, we will introduce certain rules and techniques of mathematics that deal with the method of logical reasoning to recognise valid logical arguments. A Greek philosopher and scientist Aristotle (381–322 DC) is said to be the first person to have studied logical reasoning. Logical reasoning, for example is applied by a software engineer to verify correctness of programs; by a lawer in the courtroom to defend his client; by a scientist to draw conclusions from experiments and so on. In this chapter, few of the basic ideas about the logic – the algebra of propositions (also called proposition calculus that deals statements with true and false and their analysis) and conclusions (also called predicate calculus that deals with the propositions containing variables) that are useful in designing circuits of digital computers, automata theory and computability, artificial intelligence, etc. will be discussed.
• Chapter 13

### Boolean Algebra Price 2.99  |  2.99 Rewards Points

The concept of Boolean algebra was first developed by an English mathematician, George Boole, in 1947, to simplify logical statements and solve logical problems. Claude Shannon, later on, in his paper ‘A Symbolic Analysis of Relay and Switching Circuits’ pointed out the use of Boolean algebra in the analysis and design of logical circuits. It is, therefore, used in designing of computers based on binary numbers 1 and 0. Numbers 1 and 0, correspond to two possible states of an electric circuit – current flowing or not, switch on or off.
• Chapter 14

### Posets And Lattices Price 2.99  |  2.99 Rewards Points

Recall from Chapter 1 that a non-empty set A on which a partial ordering relation, (generally denoted by ≤ , a relation that is reflexive, anti-symmetric and transitive) is defined is called a partially ordered set or poset and is written as (A, ≤ ). In this chapter, we will discuss those posets in which every subset { a , b } consisting of only two elements – greatest lower bound (glb) and least upper bound (lub)
• Chapter 15

### Language, Grammar And Automata Price 2.99  |  2.99 Rewards Points

In this chapter we shall discuss three important concepts useful in the study of computer science: Language, Grammar and Finite Automata. To communicate in any language of the world such as Hindi, English, German, French, etc. (also called natural languages), we need to know the letters of alphabet and then know how the letters are combined in a meaningful sentence. The set of rules of syntax that lead to the automatic translation of one natural language to another is referred as formal language. One simple method of developing a formal language is a phase structure grammar or simple grammar.

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