discrete mathematics homework help

discrete mathematics homework help

Exploring the Fundamentals of Discrete Mathematics

1. Introduction to Discrete Mathematics

Instead, discrete mathematics lies at the crossroads of a number of different sciences that share in common the study of discrete objects and the invidious methods of counting discrete objects. This essay aims to teach you this basic ‘theory of counting’, which is often an end in itself and can also be applied in many other areas. This essay draws on the material covered in School Certificate and Higher School Certificate mathematics. When we published the first edition of this essay, discrete mathematics was just beginning to show its relevance to computing. Although the essay was aimed at a relatively broad audience, it nevertheless developed the more important mathematical ideas in a somewhat technical way. We know that it has been used to stimulate bright, talented students to take a deeper look at mathematics.

Discrete Mathematics is a flexible, challenging subject that is best learned through active participation. You may appreciate the beauty of mathematics, but we believe that discrete mathematics has to offer you more than that. Fundamentally, discrete mathematics is the mathematics of not being able to go halfway down a staircase. It may be unsettling, but it’s fundamental. This essay, being concerned about the theory, is abstract (or theoretical) in nature. We are not especially concerned about algorithmic questions or about programming per se. At the level of this essay, discrete mathematics is also equally far from the ostentatious world of the string theorists or the subatomic physicists.

2. Set Theory and Logic

B = {2, 3, 4, 5, 6} as the set of even numbers. A = {l, e, t, t, e, r} as a neglected set. B = {a, a, a} is a distinct set.

Set: A set is defined as a collection of distinct and well-defined objects or set of coordinates. These objects are called the elements or members of a set. For example, let us assume x and y is the set, then {a, e, i, o, u} is the set with elements. Further, {e} is a set with only a single element, e ∈ {a, e, i, o, u}. ∅ is a null and finite set or empty set. It is an empty set if it has no elements. Thus, the set contains the value {}. For example, suppose A is any set of positive integers and B = the set of negative integers. More precisely, the given is A = {a, e, i, o, u}, B = {a}, and C = {c} is a set of vowels, two elements a and three. Further, a complete collection of maps only in set.

Set Theory forms the basis of discrete math and above the most important part of set theory is how to represent the representation of sets.

Each partition of the study of discrete mathematics and how it is used or applicable in the study of computers makes reference to sets and set operations. This section, therefore, provides the study of those basic operations. In particular, this section discusses the following: What those sets or ‘partitions’ are, and also the representation of sets; and the operations of and on sets.

3. Combinatorics and Probability

Probability is a simple and fascinating art of estimating the number or proportion of something. The need for this art extends to all branches where statistics have ever been applied. Probability is the mathematical theory that uses this kind of information and relations. It is the mathematical treatment of uncertainty and is one of the most important and useful areas of discrete mathematics. Ensuring accuracy in such preliminary calculations is therefore mandatory for resolving a case-based scenario. In simpler terms, probability theory deals with the number of occurrences of an event and the number of possible events involving any uncertain outcomes. The former is called a favorable event and the latter a sample space.

Combinatorics is the study and enumeration of possible arrangements involving a finite number of objects taken some at a time. It is also concerned with the ways by which some of them are to be selected or arranged. Combinatorics, or the theory of counting, is one of the major tools for studying discrete mathematics. It has wide theoretical importance and is also a very useful tool for computer scientists, particularly when one needs to optimize a combinatorial task faced with numerous constraints. The basic ideas in combinatorics are counting and arrangement. In real life, there are numerous problems requiring counting and arrangement of certain elements, such as snooker balls in a bundle of 15 colored balls and one white ball. The number of ways, n (where n is a whole number), really depends on the cases.

4. Graph Theory

Graph theory is at the foundation of a spectrum of computational problems that—when addressed—enable the formulation of solutions for applications. A graph is a data structure that has a number of applications. These problem classes originate from diverse domains such as routing in computer networks to knowledge sharing or retrieval on a large scale. Degree of a node in a graph is the number of edges connected to it. A graph that has different connections of edges for all the vertices is known as a simple graph. If g has no self-loop and no multiple edges. In a directed graph, an edge from vertex vi to vertex vj is an ordered pair, i-j. A directed graph is simple when the maximum of any two given vertices exists at most one edge. Detecting short cycles, whether a graph has a Hamiltonian cycle.

In discrete mathematics, an interesting area of study is known as graph theory. A unique way of representing relationships and connections between objects, graph theory fundamentally studies the mathematical structures of certain kinds of abstract graphs – which are a type of mathematical model quantifying interconnecting linkages or the processing of relationship data. Graphs are mathematical structures used to show and analyze the different relationships between discrete objects. Vertices are the points that represent the objects, while edges signify the connections or relationships between the pair of points. There are different types of graphs such as simple graphs, directed graphs, undirected graphs, connected graphs, weighted graphs, and full graphs. Applications of graph theory are quite extensive and oversee a wide variety of disciplines, mostly including computer science, and covering a wide array of areas such as social networking, biology, geography, communication, linguistics, airline traffic, business, and computer architecture, among others.

5. Applications in Computer Science

Presently, already mentioned set theory, combinatorics, graph theory, and logic, amongst other topics in discrete mathematics, are regularly examined so that researchers may explore and develop algorithms for computer software, coding theory, data compression, and network optimization, to name just a few ways discrete mathematics influences researchers in software and programming. In the realm of robotics, discrete mathematics is employed to produce the coding necessary for moving robots, manufacturing automobiles, and developed mathematical models of economic behavior, decision analysis, and statistical risk models in gamblers and investors. Realizing some of its practical value, companies like Microsoft ask prospective employees discrete mathematics-related questions to determine their skill level in this key area of computer science. Many popular websites have “computational puzzles” to solve, involving discrete mathematics, in order to be invited to a contest at Google, often ending in employment.

A considerable advantage of employing discrete mathematics to solve problems lies in its logical – sometimes rather than reduce or eliminate the range of solutions altogether in making a decision, one can simply evaluate all the possibilities and make the optimal choice. Enter computer science. According to the Oxford English Dictionary, computer science is defined as “the branch of science which deals with the nature of information and how it can be manipulated; the design, testing, and application of computer systems.” What better way to manipulate information on a computer? At its deepest level, discrete mathematics is a fundamental tool in computer science, the latter of which actually has an extensive impact on our everyday lives, shaping the way in which we communicate and interact.