The dictionary meaning of the term "Logic" is the Science of reasoning correctly. The rules of logic is precise meaning to mathematics statements. These rules are used to distinguish between valid and invalid mathematics arguments. Logical reasoning is used in mathematics to prove theorems, in Computer Science to verify the correctness of program and to prove theorems, in natural and physical sci ......

In this unit we describe the process of derivation by which one demonstrates that a particular formula in a valid consequence of given set of premises. The method of derivation involving predicated statement calculus and also certain additional rules which are required to deal with the formulae involving quantifiers. The rules P and T, regarding the introduction of a premise at any stage of deriva ......

The concept of sets in mathematics is the fundamental or basic ideas for the development of higher mathematical concepts. The word set is used in mathematics to mean any well defined collection of items. The items in a set are called the elements of the set. For example, we can refer to the set of all the employees of a particular company, the set of all ASCII character, the set of all integers th ......

Combinatorics, the study of arrangements of objects is an important part of discrete mathematics. This subject was studied as long as the seventeenth century. When combination questions arose in the study of gambling games, Enumeration, the counting of objects with certain properties, is an important part of combinatorics. We must count object to solve many different type of problems for instance. ......

Relationships between elements of sets occur in many contexts everyday we deal with relations such as the relations of father to son, brother to sisters etc. Relationships between elements of sets are represented using the structure called a relation, which is just a subset of the Cartesian product of the sets. Relations can be used to solve problems such as determining which pairs of cities are l ......

Naturally we can now ask: Are there relations that simultaneously manifest all three properties? The answer is yes; For instance the relation is logically equivalent to on the set of propositions has all these properties. Such as a relation is an equivalence relation. For example in some programming languages .the names of variables can contain an unlimited
number of characters that are checked w ......

We are very familiar with the concept of relations, operations of relations, properties of relations and equivalence class of relations. In this unit, we also consider the various manipulation , such as union , intersection, compliment and inverse that can be performed on relations. Also we proved the fundamental ideas of partition' and covering.

Warshall's algorithm, named after Stephen Warshall, who describe this algorithm in 1960, is an efficient method for computing the transitive closure of a relation. In general, algorithm can find the transitive closure of a relation on a set with n elements using 2n3(n = -1) bit operations.

A recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms. The term difference equation sometimes refers - to a specific type of recurrence _ relation. Note however that "difference equation" is frequently used to refer to any recurrence relation. Some simply defined recurrence relations can have very complex ......

A generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general
linear recurrence problem.

A relation is mainly a correspondence between the members of two sets, associating members of the first set with those of the second. It is possible that a given relation associates with any member of the first set several different members of the second set. It is possible that some
elements of the first set are not associated with any from the second.

In a database huge amount of data is stored in the form of records. Each record contains a field called a key to that record. The key has a value that identifies a record in the computer storage. The address of a record in the storage is obtained by performing some reproducible
arithmetic or logical operation on the internal bit representation of its key. Any transformation which maps the interna ......

A graph G = (V, E) consists of a set of vertices V = {VI, V2, .. . } and a set of edges E = {e1,e2, ... }. Each edge ek is denoted by a pair of vertices (Vi, Vj). These vertices Vi, Vj are end vertices of edge ek.

Isomorphism is similar to the concept of 'congruent' or 'equivalent' in geometry. Two graphs G and G' are called isomorphic if there is a one to one correspondence between their vertices and their edges preserve incidence relationship. In other words if an edge e is incident on vertices VI and V2 in G, then the corresponding edge e' in G' must be incident on the vertices VI' and V2' that correspon ......

Tree is a special graph. The concept of tree is very important in graph theory and also in many applications of graph theory. Also tree is an important data structure in computer science.
Tree is a connected graph without any circuits.

Semigroups are the simplest algebraic structures which satisfy the properties of closure and associativity. They are very important in the theory of sequential machines, formal languages and in certain applications relating to computer arithmetic such as multiplication.
A monoid in addition to being a semigroup, also satisfies the identity property. Monoids are used in a number of applications, b ......

From the definition of a subgroup it is clear that not every subset of a group is a subgroup. To find those subsets which can qualify to become subgroups is an interesting problem. An important relationship exists between the subgroups and the group itself. This relationship is
explained by a theorem known as Lagrange's theorem. This theorem has important application in the development of efficie ......

The concept of isomorphism shows that two algebraic systems which are isomorphic to one another are structurally indistinguishable and that the results of operations in one system call be obtained from those of the other by simply relabeling the names of the elements and symbols for
operations. This concept has useful applications in the sense that the results of one system permit an identical in ......

Error-detection and correction techniques have become increasingly important in the design of computer systems. Most systems contain telephone and communication lines which cause transmitted messages to be corrupted by the presence of noise. Peripheral equipment
associated with such systems is by far the most unreliable component of these systems and both error detection and error correction are ......

'Probable' or 'chance' is a word we often come across in our day-to-day life. We say that there is a high chance of raining today; that is we very much expect to have downpour today.
This expectation comes from our knowledge' about the conditions of the weather. In general the expectation is based on the present. knowledge and belief about the system.

We have already observed that, in a chance experiment, it is often not the actual outcome that concems us but some quantity that depends upon the outcome. In a random experiment, we may be interested quite often in the numerical measure of the different outcomes.
Through the notion of random variable, we can develop methods for the study of experiments whose outcomes may be described numerically. ......

The distribution function of a discrete random variable grows only by jumps whereas the distribution function of a continuous random variable has no jumps but grows continuously.
Thus, a continuous random variable X is characterized by a distribution function F,,(x) that is a continuous function of x for all x, -00 < x < 00.

In many practical problems, however, it is important to consider two or more random variables defined on the same sample space or probability space. So, we discuss discrete random vector and jointly distributed random variables. Also, we are interested in studying the relationship existing among the jointly distributed random variables. To have some understanding of exactly what this relation meas ......

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