What Is the Empty Set in Set Theory

When can nothing be something? It seems like a silly question, and quite paradoxical.   In the mathematical field of set theory, it is routine for nothing to be something other than nothing. How can this be? When we form a set with no elements, we no longer have nothing. We have a set with nothing in it. There is a special name for the set which contains no elements. This is called the empty or null set. A Subtle Difference The definition of the empty set is quite subtle and requires a little bit of thought. It is important to remember that we think of a set as a collection of elements. The set itself is different from the elements that it contains. For example, we will look at {5}, which is a set containing the element 5. The set {5} is not a number. It is a set with the number 5 as an element, whereas 5 is a number. In a similar way, the empty set is not nothing. Instead, it is the set with no elements. It helps to think of sets as containers, and the elements are those things that we put in them. An empty container is still a container and is analogous to the empty set. The Uniqueness of the Empty Set The empty set is unique, which is why it is entirely appropriate to talk about the empty set, rather than an empty set. This makes the empty set distinct from other sets. There are infinitely many sets with one element in them. The sets {a}, {1}, {b} and {123} each have one element, and so they are equivalent to one another.   Since the elements themselves are different from one another, the sets are not equal. There is nothing special about the examples above each having one element. With one exception, for any counting number or infinity, there are infinitely many sets of that size. The exception is for the number zero. There is only one set, the empty set, with no elements in it. The mathematical proof of this fact is not difficult. We first assume that the empty set is not unique, that there are two sets with no elements in them, and then use a few properties from set theory to show that this assumption implies a contradiction. Notation and Terminology for the Empty Set The empty set is denoted by the symbol ∅, which comes from a similar symbol in the Danish alphabet. Some books refer to the empty set by its alternate name of null set. Properties of the Empty Set Since there is only one empty set, it is worthwhile to see what happens when the set operations of intersection, union, and complement are used with the empty set and a general set that we will denote by X. It is also interesting to consider subset of the empty set and when is the empty set a subset. These facts are collected below: The intersection of any set with the empty set is the empty set. This is because there are no elements in the empty set, and so the two sets have no elements in common. In symbols, we write X ∠© ∅ ∅.The union of any set with the empty set is the set we started with. This is because there are no elements in the empty set, and so we are not adding any elements to the other set when we form the union. In symbols, we write X U ∅ X.The complement of the empty set is the universal set for the setting that we are working in. This is because the set of all elements that are not in the empty set is just the set of all elements.The empty set is a subset of any set. This is because we form subsets of a set X by selecting (or not selecting) elements from X. One option for a subset is to use no elements at all from X. This gives us the empty set.

The Effects Of Musical Instruments On Music - 890 Words

All noise making objects produce certain tones that help distinguish them from another item. For instance, when a child hears a cartoon playing in another room then they are residing in, the brain can tell if the voices they hear are from SpongeBob, or Looney Tunes. The same process happens with musical instruments, although the actual term to describe it is timbre. Timbre describes all characteristics of music that have nothing to do with pitch, loudness, or length. An example of timbre within music would be when a piccolo and a clarinet are playing a tone with dynamic levels identically, a trained ear can separate the two. Timbre is often referred to as tone color or tone quality. There are an unnumbered amount of expressions that are used to describe timbre such as, dim, nasally, mellow, brittle, and flat. Without changes in timbre an audience would quickly become uninterested in a piece. Therefore, composers often use timbre to affect mood of the audience. By changing dynamics, but keeping the timbre the same the musician can provoke different emotions onto the audience. In performing a piece with bright timbres, and loud dynamics the musician causes a disposition of conflict, but through varying the dynamic level to soft the audience moods suddenly shift to compassion. A composer creates musical landscapes in this way. Moreover, without the use of voices and instruments timbre would have not use in music. To further understand the use of voices and instruments inShow MoreRelatedCorrelation Between Music and IQ in Children Essay631 Words   |  3 Pages1. Introduction This project investigates the effect which the playing of a musical instrument – namely the drums – has on one’s intelligence and academic ability. A search of the literature reveals that studies of this kind have been conducted by other researchers in many parts of the world and has led to the conclusion that music is an extremely healthy activity to be involved in. 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Response To Beijing Games Book Free Essays

The field of sports has definitely gained an unquestionable significance and obvious implications among people and within societies. Such relevant characteristics concerning the sports discipline have resulted in making it as an important topic to be presented in a literary work. This is because facts and stories about sports are essential to be discussed in order for its underlying principles and messages to be learned and ultimately realized by the public. We will write a custom essay sample on Response To Beijing Games Book or any similar topic only for you Order Now Sports field in today’s contemporary setting has achieved an apparent prominence. The said fame is attributed to the success of the quintessential sports event – the Olympics. The advent and emergence of the Olympics have paved the way for recognition of various kinds of sports, their respective players or athletes and most importantly, the acknowledgement of the host countries where the said sports occasion is held every four years. The purpose, essence and valuable effects of Olympics to its host nation were what Susan Brownell (2008) effectively and successfully exemplified in her book â€Å"Beijing Games: What the Olympics Mean to China. † The Brownell material provided the public with clear insight about the meaning and effects of the 2008 summer Olympics to China. In the book, the author attained her goals and the narrative’s worth when they efficiently imparted to the readers the importance of hosting the Olympic Games to the country of China. In doing so, the â€Å"Beijing Games† book revealed that the meaning of Olympics to China lies on the reality that Olympics as Western civilization component can, in fact, possibly occur and affect the Far East culture. The attempt of Brownell and her book to emphasize the triumphant connection that transpired between Olympics and China led to the recognition of the beneficial impact of Olympic Games to China as well as the nation’s worldwide influence. The book therefore served as a very inspiring and awakening literary work where the effect and how Olympics change its host country as well as how China impacted and altered the Olympic were eventually realized. Written by Brownell months before the 2008 summer Olympic Games in Beijing, the book â€Å"Beijing Games† is evidently topical to China in showing the nation as one of the powerful and influential countries in Asia or the Far East part of the world. In general, the work acted as defender of the history and sports background of China as well as its political structure. This was carried out by the book with its specific criticism against the Western culture’s manner of interpreting and performing the Olympics (Brownell, 2008). â€Å"Beijing Games’ was not purely sports-oriented as several historical, political and social facts and events relating to China were discussed such as the 1989 Tiananmen Square tragic incident. Hence, in organizing the book, the author made seven chapters that talked about a number of issues. These include the disparities between the respective historical encounters with Olympic sports or the physical endeavors of China and the rest of Western countries (Brownell, 2008). The book also presented how Chinese regarded their affiliation with the International Olympic Committee or IOC, the coverage made by Western media concerning China as a nation and its sports. Foremost these topics, the Brownell book is to be significantly noted by its intriguing yet powerful presentation on whether Beijing is destined to change the Olympic’s history and nature or vice versa (Brownell, 2008). In order to understand and appreciate better the Beijing Games book, it is empirically essential to analyze how the material appeared and considered by the readers. Additionally, it is valuable to react to the book by discussing both its main and weak points. In responding to the book, the general concepts which pertain to China and the Olympics are first to be evaluated. On the whole, the book is to be commended for its efforts at disclosing the ability of the Far East civilization to hold a global sports event such as the Olympics thereby making people realize that the quintessential sports activity should not be made exclusive to Western countries. Since China embarked on what seemed like a remarkable sports undertaking, the book asserted that it is but just practical to praise and commend the country for its firm principle and practice of bringing Olympics to the other side of the world. Due to its emphasis on the capability of China, the book is to be equally acclaimed because the author has successfully disputed a misleading and incorrect notion about the sports history and background in China. The book then did well in making the world recognize that Beijing aspiration to host the 2008 summer Olympics is attributed to an enormous desire to overcome a lasting wrong impression about China’s inability towards sports or little or none at all connection with the said field. In asserting China’s power and influence in sports, the book’s notable criticism against the Western control of what is projected to be a global sports activity has worked to the advantage of China and the field of sports in general. This is because through the book, the international public was provided with the fact that the country has, in fact, rich and original sports background or nature. Based from the efforts of the author or the book, various political and social concerns that prevented sports from flourishing in China were presented, all aimed at ending the many forms of discriminations and misrepresentation on the function of Olympics sports in China and the apparent neglect done by Western cultures. Personally, I considered the book to be generally acceptable and commendable. While its organizational presentation appeared to be loosening, its principles and objectives made â€Å"Beijing Games† as one literary material worth reading and referred to other sectors of the public aside from sports. This is because the book is supposed to impart important messages or lessons to anyone. That is, the field of sports is a universal discipline which needs to be equally adhered to and uphold by worldwide audience. The realization of the meaning of Olympics to China showed how the book ultimately achieved its very essence and goals. Reference Brownell, S (2008). Beijing Games: What the Olympics Mean to China. New York: Rowman Littlefield How to cite Response To Beijing Games Book, Papers