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Set (mathematics)
In mathematics, a set is a 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 . The concept of a set is one of the most fundamental 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
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Square Number
In mathematics, a square number or perfect square is an integer that is the square of an integer;[1] 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 non-negative. Another way of saying that a (non-negative) 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 square-free. For a non-negative 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
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Vertical Bar
؋ ​₳ ​ ฿ ​₿ ​ ₵ ​¢ ​₡ ​₢ ​ $ ​₫ ​₯ ​֏ ​ ₠ ​€ ​ ƒ ​₣ ​ ₲ ​ ₴ ​ ₭ ​ ₺ ​₾ ​ ₼ ​ℳ ​₥ ​ ₦ ​ ₧ ​₱ ​₰ ​£ ​ 元 圆 圓 ​﷼ ​៛ ​₽ ​₹ ₨ ​ ₪ ​ ৳ ​₸ ​₮ ​ ₩ ​ ¥ 円Uncommon typographyasterism ⁂fleuron, hedera ❧index, fist ☞interrobang ‽irony punctuation ⸮lozenge ◊tie ⁀RelatedDiacritics Logic
Logic
symbolsWhitespace charactersIn other scriptsChinese Hebrew Japanese Korean Category Portal Bookv t eThe vertical bar (  ) is a computer character and glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke
Sheffer stroke
(in logic), verti-bar, vbar, stick, vertical line, vertical slash, bar, obelisk, or pipe, and several variants on these names
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Partition Of A Set
In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation
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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.[1][2] 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 three-dimensional 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 four-dimensional space but not the one that was found necessary to describe electromagnetism
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Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space
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0 (number)
0 (zero; /ˈzɪəroʊ/) is both a number[1] 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) (/nɔːt/), nil, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—oh or o (/oʊ/)
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Flag Of France
The flag of France
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 Tricolour has become one of the most influential flags in history, with its three-colour scheme being copied by many other nations, both in Europe and the rest of the world.[1] The royal government used many flags, the best known being a blue shield and gold fleur-de-lis (the Royal Arms of France) on a white background, or state flag. Early in the French Revolution, the Paris militia, which played a prominent role in the storming of the Bastille, wore a cockade of blue and red,[citation needed] the city's traditional colours
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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 [straight or curved] 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
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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 pre-existing meaning
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Set Notation
Sets are fundamental objects in mathematics. Intuitively, a set is merely a collection of elements or members. There are various conventions for textually denoting sets. In any particular situation, an author typically chooses from among these conventions depending on which properties of the set are most relevant to the immediate context or on which perspective is most useful.Contents1 Denoting a set as an object 2 Focusing on the membership of a set 3 Metaphor in denoting sets 4 Other conventions 5 See also 6 ReferencesDenoting a set as an object[edit] Where it is desirable to refer to a set as an indivisible entity, one typically denotes it by a single capital letter. In referring to an arbitrary, generic set, a typical notational choice is S. When several sets are being discussed simultaneously, they are often denoted by the first few capitals: A, B, C, and so forth
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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,[1] a group of mainly French 20th-century mathematicians who under this pseudonym wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935
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Curly Bracket
؋ ​₳ ​ ฿ ​₿ ​ ₵ ​¢ ​₡ ​₢ ​ $ ​₫ ​₯ ​֏ ​ ₠ ​€ ​ ƒ ​₣ ​ ₲ ​ ₴ ​ ₭ ​ ₺ ​₾ ​ ₼ ​ℳ ​₥ ​ ₦ ​ ₧ ​₱ ​₰ ​£ ​ 元 圆 圓 ​﷼ ​៛ ​₽ ​₹ ₨ ​ ₪ ​ ৳ ​₸ ​₮ ​ ₩ ​ ¥ 円Uncommon typographyasterism ⁂fleuron, hedera ❧index, fist ☞interrobang ‽irony punctuation ⸮lozenge ◊tie ⁀RelatedDiacritics Logic symbolsWhitespace charactersIn other scriptsChinese Hebrew Japanese Korean Category Portal Bookv t eThis article contains special characters. Without proper rendering support, you may see question marks, boxes, or other symbols.A bracket is a tall punctuation mark typically used in matched pairs within text, to set apart or interject other text
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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 "one-place" 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, Rex, and so on
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Integer
An integer (from the Latin
Latin
integer meaning "whole")[note 1] 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,[1][2] and their additive inverses (the negative integers, i.e., −1, −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold Z displaystyle mathbb Z ( Unicode
Unicode
U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers").[3][4] 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
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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")
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