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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 the parts are unrelated, only that no common relation has been established yet for the generalization. The concept of generalization has broad application in many connected disciplines, and might sometimes have a more specific meaning in a specialized context (e.g. generalization in psychology, generalization in learning). In general, given two related concepts ''A'' and ''B,'' ''A'' is a "generalization" of ''B'' (equiv., ''B'' is a

special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case ...

of ''A'') if and only if both of the following hold: * Every instance of concept ''B'' is also an instance of concept ''A.'' * There are instances of concept ''A'' which are not instances of concept ''B''. For example, the concept ''animal'' is a generalization of the concept ''bird'', since every bird is an animal, but not all animals are birds (dogs, for instance). For more, see Specialisation (biology).

Hypernym and hyponym

The connection of ''generalization'' to ''specialization'' (or ''particularization'') is reflected in the contrasting words hypernym and hyponym. A hypernym as a

generic Generic or generics may refer to: In business * Generic term, a common name used for a range or class of similar things not protected by trademark * Generic brand, a brand for a product that does not have an associated brand or trademark, other ...

stands for a class or group of equally ranked items, such as the term ''tree'' which stands for equally ranked items such as ''peach'' and ''oak'', and the term ''ship'' which stands for equally ranked items such as ''cruiser'' and ''steamer''. In contrast, a hyponym is one of the items included in the generic, such as ''peach'' and ''oak'' which are included in ''tree'', and ''cruiser'' and ''steamer'' which are included in ''ship''. A hypernym is superordinate to a hyponym, and a hyponym is subordinate to a hypernym.

Examples

Biological generalization

An animal is a generalization of a mammal, a bird, a fish, an amphibian and a reptile.

Cartographic generalization of geo-spatial data

Generalization has a long history in cartography as an art of creating maps for different scale and purpose. Cartographic generalization is the process of selecting and representing information of a map in a way that adapts to the scale of the display medium of the map. In this way, every map has, to some extent, been generalized to match the criteria of display. This includes small cartographic scale maps, which cannot convey every detail of the real world. As a result, cartographers must decide and then adjust the content within their maps, to create a suitable and useful map that conveys the geospatial information within their representation of the world. Generalization is meant to be context-specific. That is to say, correctly generalized maps are those that emphasize the most important map elements, while still representing the world in the most faithful and recognizable way. The level of detail and importance in what is remaining on the map must outweigh the insignificance of items that were generalized—so as to preserve the distinguishing characteristics of what makes the map useful and important.

Mathematical generalizations

* A polygon is a generalization of a 3-sided triangle, a 4-sided quadrilateral, and so on to ''n'' sides. * A hypercube is a generalization of a 2-dimensional square, a 3-dimensional cube, and so on to ''n'' dimensions. * A quadric, such as a hypersphere, ellipsoid, paraboloid, or hyperboloid, is a generalization of a conic section to higher dimensions. * A Taylor series is a generalization of a

MacLaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...

. * The binomial formula is a generalization of the formula for

(1+x)^n

References