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Generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that th ...
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Cartographic Generalization
Cartographic generalization, or map generalization, includes all changes in a map that are made when one derives a smaller-scale map from a larger-scale map or map data. It is a core part of cartographic design. Whether done manually by a cartographer or by a computer or set of algorithms, generalization seeks to abstract spatial information at a high level of detail to information that can be rendered on a map at a lower level of detail. The cartographer has license to adjust the content within their maps to create a suitable and useful map that conveys spatial information, while striking the right balance between the map's purpose and the precise detail of the subject being mapped. Well generalized maps are those that emphasize the most important map elements while still representing the world in the most faithful and recognizable way. History During the first half of the 20th century, cartographers began to think seriously about how the features they drew depended on scale. ...
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Generalization (learning)
Generalization is the concept that humans and other animals use past learning in present situations of learning if the conditions in the situations are regarded as similar. The learner uses generalized patterns, principles, and other similarities between past experiences and novel experiences to more efficiently navigate the world.Banich, M. T., Dukes, P., & Caccamise, D. (2010). Generalization of knowledge: Multidisciplinary perspectives. Psychology Press. For example, if a person has learned in the past that every time they eat an apple, their throat becomes itchy and swollen, they might assume they are allergic to all fruit. When this person is offered a banana to eat, they reject it upon assuming they are also allergic to it through generalizing that all fruits cause the same reaction. Although this generalization about being allergic to all fruit based on experiences with one fruit could be correct in some cases, it may not be correct in all. Both positive and negative effects h ...
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Generalization Process Using Trees
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that th ...
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Cartography
Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an imagined reality) can be modeled in ways that communicate spatial information effectively. The fundamental objectives of traditional cartography are to: * Set the map's agenda and select traits of the object to be mapped. This is the concern of map editing. Traits may be physical, such as roads or land masses, or may be abstract, such as toponyms or political boundaries. * Represent the terrain of the mapped object on flat media. This is the concern of map projections. * Eliminate characteristics of the mapped object that are not relevant to the map's purpose. This is the concern of generalization. * Reduce the complexity of the characteristics that will be mapped. This is also the concern of generalization. * Orchestrate the elements of t ...
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Falsifiability
Falsifiability is a standard of evaluation of scientific theories and hypotheses that was introduced by the philosopher of science Karl Popper in his book '' The Logic of Scientific Discovery'' (1934). He proposed it as the cornerstone of a solution to both the problem of induction and the problem of demarcation. A theory or hypothesis is falsifiable (or refutable) if it can be ''logically'' contradicted by an empirical test that can potentially be executed with existing technologies. Popper insisted that, as a logical criterion, it is distinct from the related concept "capacity to be proven wrong" discussed in Lakatos' falsificationism. Even being a logical criterion, its purpose is to make the theory predictive and testable, thus useful in practice. Popper opposed falsifiability to the intuitively similar concept of verifiability. Verifying the claim "All swans are white" would theoretically require observing all swans, which in actuality, is not possible. In contrast, o ...
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Generic Antecedents
Generic antecedents are representatives of classes, referred to in ordinary language by another word (most often a pronoun), in a situation in which gender is typically unknown or irrelevant. These mostly arise in generalizations and are particularly common in abstract, theoretical or strategic discourse. Examples (with the antecedent in boldface and the referring pronoun in italics) include "readers of Wikipedia appreciate ''their'' encyclopedia," "the customer ''who'' spends in this market." The question of appropriate style for using pronouns to refer to such generic antecedents in the English language became politicized in the 1970s, and remains a matter of substantial dispute. Treatment in various languages Many languages share the following issue with English: the generic antecedent is a representative individual of a class, whose gender is unknown or irrelevant, but pronouns are gender-specific. In languages such as English that distinguish natural gender in pronouns ...
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Deductive
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and "Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is ''sound'' if it is ''valid'' and all its premises are true. Some theorists define deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning. Psychology is interested in deductive reasoning as a psychological process, i.e. how people ''actually'' draw ...
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Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usua ...
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Hypersphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which is ...
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Quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in ''D'' + 1 variables; for example, in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a ''degenerate quadric'' or a ''reducible quadric''. In coordinates , the general quadric is thus defined by the algebraic equationSilvio LevQuadricsin "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'', CRC Press, from The Geometry Center at University of Minnesota : \sum_^ x_i Q_ x_j + \sum_^ P_i x_i + R = 0 which may be compactly written in vector and matrix notation as: : x Q x^\mathrm + P x^\mathrm ...
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Quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices A, B, C and D is sometimes denoted as \square ABCD. Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ''ABCD'' add up to 360 degrees of arc, that is :\angle A+\angle B+\angle C+\angle D=360^. This is a special case of the ''n''-gon interior angle sum formula: ''S'' = (''n'' − 2) × 180°. All non-self-crossing quadrilaterals tile the plan ...
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Polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any num ...
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