Set theory

Set theory Set Theory and Georg Cantor Georg Ferdinand Ludwig Phillipp Cantor, or Georg Cantor, was one of the groundbreaking mathematicians to approach the concept of infinity. He worked intensively with set theory, working with the cardinality of sets, one-to-one correspondence, transcendental numbers, and different types of infinity. Over the course of the study, we shall take a journey through Cantors life, works, and arguments. First, Richard Dedikind proposed the proposition of infinity. He, instead of constructing it, began to recognize it, avoiding an argument made by Gauss: I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a FaÃ' «on de parler, the real meaning being a limit which certain ratios have approached indefinitely near, while others are permitted to increase without restriction. Georg Ferdinand Ludwig Phillipp Cantor was born in 1845 in Saint Petersburg, Russia. He was a talented violinist having inherited skills from his father and mother. His father worked in the Saint Petersburg stock exchange. Cantor lived in Russia until he turned eleven. He got sick that year and the family moved to Germany to experience warmer winters. Cantor graduated from Darmstadt in 1860; in 1862, he was enrolled in the Federal Polytechnic Institution in Zurich. When his father died, he received an inheritance that enabled him to attend the university ofg Berlin in 1862. He received his PhD in 1867 for his math paper on number theory. Cantor first began teaching at a girls school. He then moved to the University of Halle where he would be promoted to Extraordinary Professor in 1872 and full professor in 1879. He achieved this status at the young age of 32. Unsatisfied, he wanted to pursue a better job. But his colleague, Leopold Kronecker fundamentally disagreed with Cantors studies. He believed it was incorrect to propose a set with certain qualities without giving certain examples. Georg Cantor suffered from his first bout with depression in 1884. Because of this he took a break from math and began to teach philosophy. He did begin to work with math again, but it was not of the same caliber as before. He tried to reconcile with Kronecker who enthusiastically accepted, but their views on mathematics and philosophy still opposed each other. Many people suggest that because of this conflict Cantor was depressed, but others think it was a cause of his bipolarity. Cantor retired from mathematics in 1913 and suffered from poverty because of WW1. He died on January 1918 in the asylum where he spent his final years. As a mathematician, Cantor contributed many things to the mathematical field. H developed Set theory. He developed countability, denumerability, and 1-to-1 correspondences between sets. He was the first mathematician to theorize different sizes of infinity. Back then infinity was more of a philosophical topic rather than a mathematical topic. Plus, he received a lot of criticism from Leopold Kronecker. So how is a set defined? Cantor defined a set as, â€Å"a collection into a whole, of definite, well-distinguished objects (called the elements of the set) of our perception or of our thought†. For example, every even number from 1 to 100 can be considered a set. Every prime number from 1 to 1000 can be considered a set. Even the amount of vegetables in the world can be considered a set. A set is just a group. In a set, order is not important, for the sets {1,2,3,4,5} and the sets {4,5,2,3,1} are considered equal. To write that set L is equal to set H, you could write L=H. For that to be true, all the elements in set L have to be in set H, and the elements would all have to be equal. If set L contained {1,2,3}, the set H must contain {1,2,3}. However, if L has only some of the elements of H, we call L a subset of H. To show that something is an element of L, we use the symbol â€Å"Ï µÃ¢â‚¬ . If mÏ µL, it represents â€Å"m is an element of set L†. To represent unions between sets, we use. L M means the union of sets L and M. We use the symbol when describing an intersection between sets. We use this notation when trying to find an element between two sets. To get a better representation of the use, let O be the set of odd integers from 1-10 and let P be the set of prime integers from 1-10. When we see O P, the elements of that intersection would be {3, 5, 7}. If we make a union between the sets, the elements of the union would be {1, 2, 3, 5, 7, 9}. You can think of union and intersections in the form of a Venn diagram. An intersection would be only the area where the circles intersect. A union would be the entire thing: the middle and the sides. Other important facts about set theory are cardinality and ordinal numbers. The cardinal number of a set represents the amount of elements in a set. An elements ordinal number shows where the number is in a sequence. Sometimes in well-ordered sets you can have each element with its ordinal number. Cantor developed the term enumerability. When a set is enumerable, it means that is cardinal number is the same size as the natural numbers or is the same size as a subset of the natural numbers. In a countable set, there exists an injective function. An injective function is when you can associate distinct values with distinct arguments. This is also referred to as a 1-to-1 function. In addition to injective functions, there is a surjective function where for the function f(x)= y, there exists more than one x value to one y value. Bijection is when for f(x) = y between sets, there exists one and only one value of y. a bijective function is different from an injective function because in an injective function, you can map all them elements from set A to set B with some elements in B left over when with a bijective function all the elements in set A must map over to set B with only one corresponding element. So where does this all tie into Cantors work? Well, to start off he was the first one to actually work with set theory. Through his work, he was able to prove that the set of odd integers is equal to the set of integers overall. For this proof, let us assume that the amount of even integers is equal to the amount of odd integers. Now, people will think, â€Å"But arent the odd integers a subset of the integers?† True, but subsets can have the same cardinality as the whole set. The way Cantor proved this was through proving the odd integers equal to the number of integers with a bijective function: f (x) = y = 2x+1, where x is an element of the entire set of integers. This way, -3 would go to –5, -2 would go to -3, -1 would go to -1, and 0 would go to -1. Through this, Cantor made a groundbreaking discovery. It would lead on to understanding different kinds of infinity. Cantor came up with two great theorems. The first one, Cantors Theorem showed that the power set of a set is larger than the set itself. A power set contains all the subsets of a set. Consider a set whose elements are {1, 2}. The power set of this set would be {{}, {1}, {2}, {1, 2}}. The cardinality of this power set is 4. 4 is greater than two. As we described before, we showed that two sets have the same cardinality if they have the same number of elements and there exists a 1 to 1 correspondence. He proved his theorem by finding a subset, B, that was not in A. Consider a set, A, and its power set P(A). The subset B would be represented by: F(x) is a general bijective function that maps the elements of set As power set to the elements of set A. This shows that for any element x of A, x is an element of B if and only if x does not equal f(x). But then that would mean x is an element of B where x isnt an element of f(x) and then x is not an element of B? Impossible! One of the most famous proofs of set theory was the diagonal proof by Cantor. He applied it to show that the real numbers were more numerous than the naturals, therefore proving the existence of uncountable sets. To prove it, we will use contradiction. Consider a list of the real numbers that could be put into a 1-to-1 correspondence with the naturals. 1 .5657678†¦ 2 .3364625†¦ 3 .2425364†¦ 4 .3544657†¦ 5 .3535465†¦ 6 .1324354†¦ 7 .2000000†¦ Because of their 1 to one correspondence, should we try to construct another element in the list of real numbers, it would already be accounted for. But what the diagonal argument did was it took the first digit of the fist element, the second digit of the second element and so on and so on, all the way to the nth digit and added one to each individual digit mod ten. What would happen is we would add one to the first digit 5 mod ten and get six. Then we would add 1 to the second digit 3 mod ten and get 4. The pattern of numbers follows a diagonal formation, such as the numbers highlighted below. 1 .5657678†¦ 2 .3364625†¦ 3 .2425364†¦ 4 .3544657†¦ 5 .3535465†¦ 6 .1324354†¦ 7 .2000000†¦ The digits we would get are 6, , 3, 5, 5, 6, and 1. From these digits, we make a decimal with each digit in the spot respective to the element they were taken from. For example, 6 would be the first digit because it was taken from the 1st element. 4 would be the next one for it was taken from the second element, and so on and so on. Following that pattern, we would construct the number .6435561†¦. This beauty of this proof is we have just constructed a number that isnt part of the list! Why? For example, if we looked at the mth digit of this new number and the mth digit of the mth element of the list, we would see that they differ by that one number, thereby having created a new number. What we have done here is just made a way to make an infinite list strictly larger than the naturals therefore proving the existence of uncountable sets. What makes this proof so much more amazing is that there are so many ways to represent it. I used decimals to represent it. However, other peop le might use two variables and just switch them when changing by one. Other people might only use 0 and 1. Cantors work became an important part of other mathematicians work. It became an important part in Russells Paradox, Godels Incompleteness theorem, and Turings Entscheidungsproblem (German for â€Å"decision problem†) Through Cantors groundbreaking work, mathematicians were finally able to approach the concept of infinity. No longer was the topic reserved for the philosophers. Infinity could be used as a mathematical field.

K-Mart :: essays research papers

K-Mart Kmart is the #3 discount retailer in the United States behind Wal-Mart and Target. Kmart sells name brand and private label merchandise, mostly to low and mid - income families. It has more than 1,800 stores and currently employs more than 220,000 associates in all 50 states, Guam, Puerto Rico and the U.S. Virgin Islands and owns an e-tailer BlueLight.com. As of Jan.30, 2002,Kmart had 124 Kmart Supercenters that combine a full grocery, deli, bakery, video rental and 24 hour/seven-days-a-week availability along with the general merchandise selection of a Kmart discount store. A core strength for the company continues to be the expansion of Kmart Exclusive brands such as Martha Stewart Everyday, Sesame Street, Jaclyn Smith, Kathy Ireland, and Route 66. These brands-nationally available only at Kmart- have progressively added to their assortments. Despite that, the company filed for Chapter 11 Bankruptcy protection on January 22, 2002, after a year in which its financial performance declined from unimpressive to bleak. The filing came a day after Kmart's major food distributor, Fleming Cos., said it had cut off most shipments to Kmart because the discounter failed to make its regular weekly payment for deliveries. Fleming said Kmart, its largest customer, owed $78 million. This bankruptcy filing let Kmart rid itself of unprofitable stores and shrink the payroll. The company has closed 284 unprofitable stores and laid off about 22,000 workers to pare costs. Kmart stores have fallen as consumers have slowed their spending and as rivals like Wal-Mart Stores and Target have tried to siphon off Kmart’s customers. In this paper, I will try to suggest some solutions to improve Kmart as a company as a whole and help increase its sales. First of all, Kmart needs to improve its marketing technigues. The problem that it’s facing right now is decreasing sales. What I think it should do is lower the prices. It will definitely attract more customers. For example, the reason that one of Kmart’s biggest competitors, Wal-Mart is doing so well is because Wal-Mart is known for its very low prices. Also, I think that Kmart should increase eye-capturing advertising. Not only should they increase the advertising, but they should try to appeal to different kind of customers. Kmart should make sure that its advertisements attract not only middle-aged population-mostly housewives, but it ought to attract more hipper/younger consumers. In addition, it needs to provide better quality of items. One main reason why Target is a

3d Tv Business Plan

Business plan of 3D TV Executive summery LOGO[pic] 1. Problem: 3D TV market in China has great potential, but Chinese brands and foreign brands in terms of product quality and functionality,they are still a certain gap. What is the opportunity? Estimated 2013 flat-panel TV consumer demand of 42 million units, of which 20 million units of demand for 3D TV, smart TV demand 800 million units. Consumer demand will return to the essential characteristics of the TV – picture quality and sound effects; 3D channel launch volume growth accelerated 3D Smart TV scale.Has not yet appeared on the market directly by the 2D signal into 3D signal TV 2. Solution: Create a brand for innovative 3D TV market in China. shui is the newest high-tech product, movie technology brought right to your living room, 3D effects to bring you the visual impact you can enjoy anytime, anywhere. If you want to see the common chanel , just change the signal to finish the transfert by 3D-2D. 3. Market facing fulfi lling demand advantage From consumer groups, the average age of 3D TV consumption is becoming younger.To the age of marriage for young people will be more and more, they are the main purchaser of goods, all around us, you can see that they desire to buy the TV before marriage, but is missing in the second-tier cities such as Chongqing, Jinan, Hangzhou 3D television propaganda, in the second-tier cities, such as shopping malls, relatively speaking inside the 3D TV can not meet consumer multi-faceted, multi-selection requirements. I precisely for their needs and the setting up of the project, at the same time, the majority of manufacturers have also noticed that.Changhong, Konka, Hisense and other vendors . 4. Product with potential commercial value The main body of the project is to build a high, medium and low-grade TV. Required to achieve the purpose: to meet the vast majority of consumer demand for 3D TV. TV has great profit margins, very attractive terms from the manufacturers, m erchants. Enterprises have power, market demand naturally easily from share part of the profits, part of the relatively solid profit. 5. Our company?We will build two manufacturing plants in Zhejiang, to build our R & D base and the company is located in Hangzhou, the plant scale probably need about 50,000 square meters, fifteen thousand more than a thousand employees, the company has a chairman and general manager,R & D Director, financial Director, sales Director, human Resources Director, logistics Officer, plant department manager. 6. Group Prisident : zhang weizhong – training / EMBA + Entrepreneur . Hongyu – experience / sales managers in SUNING. Coodinator developpment product GUOMEI in ShangHai. The business profile 1. Description of my businessH2O Corporation is a Chinese multinational electronics company headquartered in ruian Zhejiang, China,which is 100 km northeast of the wenzhou. H2O is a China ‘s first self-developed 3D TV technoloy company, we not only sell our products to the market, to output our product philosopy : Innovation and change our lives! In addition, we will also build our own sales outlet based in the Pearl River Delta region, to establish a national entity with network the trinity-marketing standards mode. 2. Market and customer The concept of 3D TV launch time is not very long, but in the 1980s began to study abroad.Japan's Panasonic and Sony invested heavily in higher technological content of products in this study. Currently on the market there are a lot of the concept of 3D TV, the effect is less obvious than ordinary liquid crystal effect. The main reason is really 3D resources, many movies and TV shows 3D sense is not very strong; another TV product technology is not as advertised or we imagine that there is a strong sense of three-dimensional. The actual market share about 10% to 20%, and consumer groups are mainly concentrated in the cities and the high-spending families.Future production with the larg e number of 3D films and television programs, 3D TV will be the rapid development of the market share will average 80% -120% of the rapid growth of 3D TV on the market in 2015 is expected to account for 45-60% of the market share. Means that 12-18% of Chinese households will have 3D TV by 2015 †¦ From a technical point of view, has a self-luminous, wider color gamut, color is more real and natural, more levels of detail, the inherent advantages of the 1080 line dynamic clarity, soft eye Plasma TV can better show the 3D image effect.The domestic color TV market, market sales of the top five are domestic brands, totaled 66. 02 percent share. Which Konka TV retail volume market share of 14. 27%. LCD TV sales, made eight brands in volume accounted for 70. 06 percent of the market share. Among them, Hisense ranked first in market share of 13. 18%. It is understood that, since the beginning of this year, foreign brands have launched two price war to try to impact on the domestic colo r TV market, but did not gain market advantage.The impact of foreign investment, but also indirectly promote domestic brands from Competition to Cooperation. Current market for our new company, is both an opportunity and also a challenge, and our market position is the position in which the competitors in the market for consumer products, the degree of attention strongly shape the enterprise product unique, gives the impression of a distinct personality or image, so that the product on the market to determine the appropriate location. We are not on a product itself, what to do and what to do, but in the minds of potential consumers.In other words, we need to determine an appropriate position in the minds of potential consumers to the product, and the positioning of our products in high quality, new technologies, and affordable. Our market positioning can be divided into a predetermined bit of re-positioning of existing products and potential products. The re-positioning of existing products may lead to a change of product name, price and packaging, but the purpose of these appearance changes in order to ensure that the products worth buying the image of the left in the minds of potential consumers.Pre-positioning of potential products, requires marketers must start from scratch to develop all of the 4Ps Features meets the selected target market. Conducting market positioning, on the one hand to understand the competitor's product with what characteristics, on the other hand to study the degree of attention of the consumers of the various properties of the product. Our consumers will first be targeted at: Fever type: is the pursuit of the latest technology, the latest product of a group of people, but a limited number of their new creation of the new invention is extremely interested in and willing to try the immaturity of the technology and products, even yourself involved in personality or to the manufacturers recommendations. †¢ Pioneer type: more visio n and pursuit of a group of people, a sensitive technologies, products, the willing nearing mature technologies and products to improve work efficiency and quality of life, walking in front of the most people.Our competitors Chinese brand: TCL,KONKA,HISENSE, CHANGHONG Foreign brand: SONY,SANSUNG,LG,TOSHIBA,SHARP [pic] [pic] Product Our Technology 3D TV-based competitiveness in the market, our company is also actively cooperate with the market demand, to the main 3D TV brand belongs to China, and our company's core technology is the use of advanced production technology, production of a kind in the 3Dbetween the signal and 2D signal conversion machine, it's very small size to install TV back of the motherboard.When people need to convert, just use the remote control to control it, this machine will then pass the signal to the monitor, the TV will start the conversion according to the instructions, the whole process takes only a minute. PRODUCT [pic][pic][pic] [pic][pic][pic] Competit ive Comparison Our products are unique in the existing market. While there are other TV companies,none use these technology to produce 3D TV as ours. Our ideas come from listening to what other consumers have to say. Financial projection [pic] Benchmark[pic] Profit yearly [pic] 1 Marketing et salesProduct positioning â€Å"One and the only one† At present, there is only one 3D television channel in china. Not only it does not broadcast all-weather, but also the audiences must pay more cost to receive it. In addition, these programs of 3D television channel in china always have broadcasted in the general channel and had no more attraction. In consequence, the 3D televisions in Chinese family just are the common television in usual except watching 3D DVD or playing 3D TV games sometimes, can not show the 3D visual effects. There is not any brand’s 3D television can change the situation right now.But the innovation of our 3D television breaks up the situation. Our 3D tele vision is the first and the unique 3D television can change the signal 2D from general television channel to 3D by itself in the world. The appearance of our 3D television is a revolutionary change for the 3D television’s market. Our 3D television is the one and the only one product in the market. Make a price From the report of the Chinese 3D television’s market in April 2012(resource: http://big5. askci. com/news/201205/31/311653860124. shtml ), we will analyze and make sure our main product and its price.From the 3D television’s screen size, the large screen size 3D television is a mainstream product which is all over 20% by the consumer attention of screen size for 42 inch, 46 inch and 55 inch. The 46 inch screen size is the most popular. Proportion of top three concerns is more than 70%, showing the popularity of large screen size 3D television. [pic] In April 2012, China 3 d LCD TV market mainstream size product attention proportional distribution (Source: ZDC) From the price of a product, 5001-8000 RMB price segment 3D television has been in most of the concerns of consumers, concerned about the ratio reached 34. %, 3001-5000 RMB has also attracted the attention of consumer more than twenty percent. [pic] In April 2012, China 3 d LCD TV market price of different period of product attention proportional distribution (source: ZDC) Compare the current average price of the main dimensions of the 3D TV market in China, in addition to 46 and 55 inches and larger products, other dimensions of 3D TV price in 10000 RMB within. 42 and 47 inches from the Chinese local brands and LG, while 40 and 46 inches from Samsung, Sony, Sharp and other manufacturers, so their price is relatively high. [pic]In august 2011, size of 3D in Chinese TV market’s product average price(source: ZDC) From different brand, the foreign’s brands, such as: Samsung, LG, Sony, Sharp†¦, occupy the most of high-end market and the super large size 3D telev ision’s market (plus 52 Inch); the Chinese’s brands, such as: TCL, Hisense, Konka, Changhong, Skyworth†¦, occupy the most of low-end market. Normally, the price of foreign’s brands ‘3D televisions are made mostly triple than the Chinese’s brands‘3D televisions in the same size. We will take an example of the price of the most popular screen size 46 Inch in different brands: |Samsung |LG |Sony | |Land rent |? 3,000,000 |? 3,000,000 |? 3,000,000 | |Office rent |? 500,000 |? 500,000 |? 500,000 | |construction |? 2,000,000 |? 100,000 |? 100,000 | |Raw Materials |? 10,000,000 |? 12,500,000 |? 15,000,000 | |Equipment |? 0,000,000 |? 100,000 |? 500,000 | |salary |? 35,000,000 |? 38,000,000 |? 30,000,000 | |Advertisement |? 10,000,000 |? 8,000,000 |? 5,000,000 | |Research |? 10,000,000 |? 10,000,000 |? 10,000,000 | |Current funds |? 10,000,000 |? 10,000,000 |? 10,000,000 | |Insurance |? 1,000,000 |? ,000,000 |? 1,000,000 | |Other |? 2,000,000 |? 2,000,000 |? 2,000,000 | |Total |? 113,500,000 |? 85,200,000 |? 77,100,000 | In the second year and third year, the investment demand will gradually decrease. With the improvement of the profit, this plan calls for the bank loans to be repaid in the third year, these personal investments will be shared out bonus at the end of years according to the 20% of the investment. Risk prevention Possible risk and solution Risk: same industry competitors? Like every new company, new brand enter the fiercely competitive market at the very start, we must meet the strong competition counterparts to stop us to gain market share, such as: the others brands’ products significantly depreciate sales promotion, negative campaigning from the others brands’ and they will stand together to threat the dealers stop supplying their products if the dealers sell our products.Solution: So we must make a right price and reinforce the market communication to get market recognition and enough consum ers as soon as possible, thereby to enter the market and seek more development opportunities. Risk: material purchase? Suppliers casually raise prices of raw and processed materials Solution: As far as possible to expand purchase channels, compare the suppliers based on the quality, price and credibility. Choose the best supplier and establish a stable supplier chain.Risk: environmental pollution? The environmental pollution principal means the scrap TV pollution. It has a lot of heavy metals harmful to humans. If casually discarded harmful substances as rainwater into the underground or like a garbage landfill incineration, these harmful substances will stay here for a long time in the natural and the destruction of the ecological environment, and may at any time through a number of channels to enter the inside of the human body, so as to endanger human health.Solution: We will install a specialized decomposition of the old TV production line imported from Japan . ——à ¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€œ Looking at the changes in the market, to fully grasp the pulse of the market, coldly towards market competition, the courage to challenge, ebullient, make their own taste and style, to create first-class brand —-slogan _____ zhang weizhong chairman

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