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Set (mathematics) In mathematics , a SET is a welldefined collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education , elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree. The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano Bernard Bolzano in his work The Paradoxes of the Infinite [...More...]  "Set (mathematics)" on: Wikipedia Yahoo 

Dimension (mathematics) In physics and mathematics , the DIMENSION of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube , a cylinder or a sphere is threedimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics , space and time are different categories and refer to absolute space and time . That conception of the world is a fourdimensional space but not the one that was found necessary to describe electromagnetism [...More...]  "Dimension (mathematics)" on: Wikipedia Yahoo 

0 (number) 0 (ZERO; /ˈzɪəroʊ/ ) is both a number and the numerical digit used to represent that number in numerals . The number 0 fulfills a central role in mathematics as the additive identity of the integers , real numbers , and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems . Names for the number 0 in English include ZERO, NOUGHT (UK), NAUGHT (US) (/ˈnɔːt/ ), NIL, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—OH or O (/ˈoʊ/ ). Informal or slang terms for zero include ZILCH and ZIP. Ought and aught (/ˈɔːt/ ), as well as cipher, have also been used historically [...More...]  "0 (number)" on: Wikipedia Yahoo 

Plane (mathematics) In mathematics , a PLANE is a flat, twodimensional surface that extends infinitely far. A plane is the twodimensional analogue of a point (zero dimensions), a line (one dimension) and threedimensional space . Planes can arise as subspaces of some higherdimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry Euclidean geometry . When working exclusively in twodimensional Euclidean space Euclidean space , the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, geometry , trigonometry , graph theory , and graphing are performed in a twodimensional space, or, in other words, in the plane [...More...]  "Plane (mathematics)" on: Wikipedia Yahoo 

If And Only If In logic and related fields such as mathematics and philosophy , IF AND ONLY IF (shortened IFF) is a biconditional logical connective between statements. In that it is biconditional , the connective can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false). It is controversial whether the connective thus defined is properly rendered by the English "if and only if", with its preexisting meaning. There is nothing to stop one from stipulating that we may read this connective as "only if and if", although this may lead to confusion [...More...]  "If And Only If" on: Wikipedia Yahoo 

Flag Of France The FLAG OF FRANCE (French : Drapeau français) is a tricolour flag featuring three vertical bands coloured blue (hoist side ), white , and red . It is known to English speakers as the FRENCH TRICOLOUR or simply the TRICOLOUR (French : TRICOLORE). The royal government used many flags, the best known being a blue shield and gold fleurdelis (the Royal Arms of France ) on a white background, or state flag. Early in the French Revolution French Revolution , the Paris militia, which played a prominent role in the storming of the Bastille , wore a cockade of blue and red, the city's traditional colours. According to French general Gilbert du Motier, Marquis de Lafayette , white was the "ancient French colour" and was added to the militia cockade to create a tricolour, or national, cockade. This cockade became part of the uniform of the National Guard , which succeeded the militia and was commanded by Lafayette [...More...]  "Flag Of France" on: Wikipedia Yahoo 

Straight Line The notion of LINE or STRAIGHT LINE was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature ) with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined in this manner: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which will leave from its imaginary moving some vestige in length, exempt of any width [...More...]  "Straight Line" on: Wikipedia Yahoo 

Ellipsis ؋ ₳ ฿ ₿ ₵ ¢ ₡ ₢ $ ₫ ₯ ֏ ₠ € ƒ ₣ ₲ ₴ ₭ ₺ ₾ ₼ ℳ ₥ ₦ ₧ ₱ ₰ £ 元 圆 圓 ﷼ ៛ ₽ ₹ ₨ ₪ ৳ ₸ ₮ ₩ ¥ 円 UNCOMMON TYPOGRAPHY asterism ⁂ hedera ❧ index, fist ☞ interrobang ‽ irony punctuation ⸮ lozenge ◊ tie ⁀ RELATED* * Diacritics * Logic symbols * Whitespace characters IN OTHER SCRIPTS * Chinese * Hebrew * Japanese * Korean * Category [...More...]  "Ellipsis" on: Wikipedia Yahoo 

Partition Of A Set In mathematics , a PARTITION OF A SET is a grouping of the set's elements into nonempty subsets , in such a way that every element is included in one and only one of the subsets. CONTENTS * 1 Definition * 2 Examples * 3 Partitions and equivalence relations * 4 Refinement of partitions * 5 Noncrossing partitions * 6 Counting partitions * 7 See also * 8 Notes * 9 References DEFINITIONA partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets (i.e., X is a disjoint union of the subsets). Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: * The family P does not contain the empty set (that is P {displaystyle emptyset notin P} ). * The union of the sets in P is equal to X (that is A P A = X {displaystyle textstyle bigcup _{Ain P}A=X} ) [...More...]  "Partition Of A Set" on: Wikipedia Yahoo 

Vertical Bar ؋ ₳ ฿ ₿ ₵ ¢ ₡ ₢ $ ₫ ₯ ֏ ₠ € ƒ ₣ ₲ ₴ ₭ ₺ ₾ ₼ ℳ ₥ ₦ ₧ ₱ ₰ £ 元 圆 圓 ﷼ ៛ ₽ ₹ ₨ ₪ ৳ ₸ ₮ ₩ ¥ 円 UNCOMMON TYPOGRAPHY asterism ⁂ hedera ❧ index, fist ☞ interrobang ‽ irony punctuation ⸮ lozenge ◊ tie ⁀ RELATED* * Diacritics * Logic Logic symbols * Whitespace characters IN OTHER SCRIPTS * Chinese * Hebrew * Japanese * Korean * Category Category * Portal Portal * Book Book * v * t * e The VERTICAL BAR ( ) is a computer character and glyph with various uses in mathematics, computing, and typography [...More...]  "Vertical Bar" on: Wikipedia Yahoo 

Square Number In mathematics , a SQUARE NUMBER or PERFECT SQUARE is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as 3 × 3. The usual notation for the square of a number n is not the product n × n, but the equivalent exponentiation n2, usually pronounced as "n squared". The name square number comes from the name of the shape; see below . Square numbers are nonnegative . Another way of saying that a (nonnegative) integer is a square number, is that its square root is again an integer. For example, √9 = 3, so 9 is a square number. A positive integer that has no perfect square divisors except 1 is called squarefree . For a nonnegative integer n, the nth square number is n2, with 02 = 0 being the zeroth one. The concept of square can be extended to some other number systems [...More...]  "Square Number" on: Wikipedia Yahoo 

Extension (semantics) In any of several studies that treat the use of signs —for example, in linguistics , logic , mathematics , semantics , and semiotics —the EXTENSION of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension , which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question. In philosophical semantics or the philosophy of language , the 'extension' of a concept or expression is the set of things it extends to, or applies to, if it is the sort of concept or expression that a single object by itself can satisfy. Concepts and expressions of this sort are monadic or "oneplace" concepts and expressions. So the extension of the word "dog" is the set of all (past, present and future) dogs in the world: the set includes Fido, Rover, Lassie Lassie , Rex, and so on [...More...]  "Extension (semantics)" on: Wikipedia Yahoo 

Integer An INTEGER (from the Latin Latin integer meaning "whole") is a number that can be written without a fractional component . For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1⁄2, and √2 are not. The set of integers consists of zero (0 ), the positive natural numbers (1 , 2 , 3 , …), also called whole numbers or counting numbers, and their additive inverses (the NEGATIVE INTEGERS, i.e., −1 −1 , −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold Z {displaystyle mathbb {Z} } (Unicode U+2124 ℤ) standing for the German word Zahlen ( , "numbers"). Z is a subset of the set of all rational numbers Q, in turn a subset of the real numbers R. Like the natural numbers, Z is countably infinite . The integers form the smallest group and the smallest ring containing the natural numbers [...More...]  "Integer" on: Wikipedia Yahoo 

Natural Number In mathematics , the NATURAL NUMBERS are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers " and words used for ordering are "ordinal numbers ". Some definitions, including the standard ISO 800002 , begin the natural numbers with 0 , corresponding to the NONNEGATIVE INTEGERS 0, 1, 2, 3, …, whereas others start with 1, corresponding to the POSITIVE INTEGERS 1 , 2 , 3 , …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the WHOLE NUMBERS, but in other writings, that term is used instead for the integers (including negative integers) [...More...]  "Natural Number" on: Wikipedia Yahoo 

Surjection In mathematics , a function f from a set X to a set Y is SURJECTIVE (or ONTO), or a SURJECTION, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y. It is not required that x is unique ; the function f may map one or more elements of X to the same element of Y. A surjective function from domain X to codomain Y. The function is surjective because every point in the codomain is the value of f(x) for at least one point x in the domain. The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki , a group of mainly French 20thcentury mathematicians who under this pseudonym wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French prefix sur means over or above and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain [...More...]  "Surjection" on: Wikipedia Yahoo 

Colon (punctuation) ؋ ₳ ฿ ₿ ₵ ¢ ₡ ₢ $ ₫ ₯ ֏ ₠ € ƒ ₣ ₲ ₴ ₭ ₺ ₾ ₼ ℳ ₥ ₦ ₧ ₱ ₰ £ 元 圆 圓 ﷼ ៛ ₽ ₹ ₨ ₪ ৳ ₸ ₮ ₩ ¥ 円 UNCOMMON TYPOGRAPHY asterism ⁂ hedera ❧ index, fist ☞ interrobang ‽ irony punctuation ⸮ lozenge ◊ tie ⁀ RELATED* * Diacritics * Logic symbols * Whitespace characters IN OTHER SCRIPTS * Chinese * Hebrew * Japanese * Korean * Category Category * Portal Portal * Book Book * v * t * e The COLON ( : ) is a punctuation mark consisting of two equally sized dots centered on the same vertical line. A colon precedes an explanation or an enumeration , or list [...More...]  "Colon (punctuation)" on: Wikipedia Yahoo 