In our day to day life, we use the word set in so many references. But, what are sets? Well, nobody could successfully define the term set. Who ever tried, could not stop from using synonyms as "class'', "collection'', and "aggregate''. Often sets are defined intuitively as an unordered collection of objects. But this definition does not specify "what an object is?'' Such definition leads to parad ......

Whenever we are not certain about occurrence of a particular situation, we use the word probable. Examples include situation like whether a particular deal would materialise; whether a particular theoretical prediction be confirmed by experiment; or something more simpler like whether it would rain today or whether my girlfriend would come with me for an outing. All the above-mentioned situations ......

Logic is the science and art that so directs the mind in the process of reasoning and subsidiary process as to enable it to attain clearness, consistency and validity in those processes. The aim of logic is to secure clearness in the definition and arrangement of our ideas and other mental images, consistency in our judgement and validity in our process of inference. So, logic is "A formal way of ......

Let us consider the following statement: "whatever numbers a and b may be, (a+b)2 = a2+2ab+ b2''. The above statement is a proposition; but the bare formula "(a + b)2 = a2 + 2ab + b2'' alone is not, since it asserts nothing definite unless it was further told, or led to suppose, that a and b are to have all possible values, or are to have such-and-such values. The former of these is tacitly assume ......

One of the most familiar branch of knowledge is that of numbers. We all seem to know about numbers. To the average educated person of the present day, the obvious starting-point of mathematics would be the series of natural numbers: 1, 2, 3, 4, . . . . In this chapter, we shall study the classification of numbers in general. We will navigate through the properties of integers (whole numbers), rat ......

A function whose domain is a set N of natural numbers and range a set R of real numbers is called a real sequence. Thus a real sequence is denoted symbolically as s : N → R. Here, we shall be dealing with real sequences only, so we shall use the term 'sequence' to denote a real sequence. Also, since the domain for a sequence is always N, a sequence is specified by the values sn, n ∈ N. ......

The study of nature of R is all that we are concerned in this chapter. Further, when we define R, we actually try to establish a mapping from which the correlating relation R goes and to which it goes. Russel suggested that we use the word referent for the term from which the relation goes, and the term relatum for the term to which it goes. That is to say, it is characteristic of a relation of tw ......

In this chapter, we shall discuss about a special type of relation called a function. It is one of the important concepts in the field of mathematics and computer science. Let us assume that two sets of objects are given. Let us designate one of them as the first set and other as the second set. Next let us assume that with each element of the first set is associated a particular element of the se ......

In this chapter we will study the properties of relations, their representation as a matrix or a directed graph. Further we will also discuss about a special kind of relation called partial order.

In this chapter we shall study a particular type of mathematical structure called groups. Let us first discuss, what do we actually mean by a mathematical structure? A collection of objects with operations defined on them and the accompanying properties form a mathematical structure or system. In studying groups, we would be dealing with a concept called binary operation.

Codes are a form of logical construct. Coding permits us to reduce the study of some object to other. The coordinate method made it possible to encode geometric objects by analytic expressions, which turned out to be quite important in the field of mathematics. The set theory also made it possible to encode probability concepts. However, the means of coding were auxiliary, and not a subject of inv ......

In this chapter we introduce the notion of rings and study their basic properties. Just as groups, rings are algebraic systems. While the abstract notion of a group is founded on the set of bijections of a set onto itself, the notion of a ring will be seen to have arisen as generalizations from the set of integers and the additive and multiplicative properties of integers. We know that groups are ......

Among the concepts which have profound influence in computer science, one of them is congruence. This falls under the category of Modular Arithmetic. Carl Friedrich Gauss was the inventor of this concept and he was the first to study it in detail and to publish a monumental work titled Disquisitiones Arithmeticae in 1801. We will first state one basic theorem on which the concept of congruence res ......

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