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Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Example Of A Set
Example may refer to: * ''exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, example.edu, second-level domain names reserved for use in documentation as examples * HMS Example (P165), HMS ''Example'' (P165), an Archer-class patrol and training vessel of the Royal Navy Arts * ''The Example'', a 1634 play by James Shirley * The Example (comics), ''The Example'' (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson * Example (musician), the British dance musician Elliot John Gleave (born 1982) * Example (album), ''Example'' (album), a 1995 album by American rock band For Squirrels See also * * Exemplar (other), a prototype or model which others can use to understand a topic better * Exemplum, medieval collections of short stories to be told in sermons * Eixample, an inner suburb of Barcelona wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extensionality
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same. Example Consider the two functions ''f'' and ''g'' mapping from and to natural numbers, defined as follows: * To find ''f''(''n''), first add 5 to ''n'', then multiply by 2. * To find ''g''(''n''), first multiply ''n'' by 2, then add 10. These functions are extensionally equal; given the same input, both functions always produce the same value. But the definitions of the functions are not equal, and in that intensional sense the functions are not the same. Similarly, in natural language there are many predicates (relations) that are intensionally different but are extensionally identical. For example, suppose that a town has one person named Joe, who is also the oldest person in the town. Then, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Italic Type
In typography, italic type is a cursive font based on a stylised form of calligraphic handwriting. Owing to the influence from calligraphy, italics normally slant slightly to the right. Italics are a way to emphasise key points in a printed text, to identify many types of creative works, to cite foreign words or phrases, or, when quoting a speaker, a way to show which words they stressed. One manual of English usage described italics as "the print equivalent of Underline, underlining"; in other words, underscore in a manuscript directs a typesetter to use italic. The name comes from the fact that calligraphy-inspired typefaces were first designed in Italy, to replace documents traditionally written in a handwriting style called chancery hand. Aldus Manutius and Ludovico Arrighi (both between the 15th and 16th centuries) were the main type designers involved in this process at the time. Along with blackletter and Roman type, it served as one of the major typefaces in the history ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Capital Letters
Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing systems that distinguish between the upper and lowercase have two parallel sets of letters, with each letter in one set usually having an equivalent in the other set. The two case variants are alternative representations of the same letter: they have the same name and pronunciation and are treated identically when sorting in alphabetical order. Letter case is generally applied in a mixed-case fashion, with both upper and lowercase letters appearing in a given piece of text for legibility. The choice of case is often prescribed by the grammar of a language or by the conventions of a particular discipline. In orthography, the uppercase is primarily reserved for special purposes, such as the first letter of a sentence or of a proper noun (ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel's Incompleteness Theorems
Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ... that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistency, consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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