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Cardinalities
In mathematics, the cardinality of a finite set is the number of its elements, and is therefore a measure of size of the set. Since the discovery by Georg Cantor, in the late 19th century, of different sizes of infinite sets, the term ''cardinality'' was coined for generalizing to infinite sets the concept of the number of elements. More precisely, two sets have the same cardinality if there exists a one-to-one correspondence between them. In the case of finite sets, the common operation of ''counting'' consists of establishing a one-to-one correspondence between a given set and the set of the first natural numbers, for some natural number . In this case, is the cardinality of the set. A set is ''infinite'' if the counting operation never finishes, that is, if its cardinality is not a natural number. The basic example of an infinite set is the set of all natural numbers. The great discovery of Cantor is that infinite sets of apparently different size may have the same cardina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-to-one Correspondence
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mapped f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter \aleph (aleph) marked with subscript indicating their rank among the infinite cardinals. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of number of elements. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apples And Oranges V2
An apple is a round, edible fruit produced by an apple tree (''Malus'' spp.). Fruit trees of the orchard or domestic apple (''Malus domestica''), the most widely grown in the genus, are cultivated worldwide. The tree originated in Central Asia, where its wild ancestor, ''Malus sieversii'', is still found. Apples have been grown for thousands of years in Eurasia before they were introduced to North America by European colonists. Apples have cultural significance in many mythologies (including Norse and Greek) and religions (such as Christianity in Europe). Apples grown from seeds tend to be very different from those of their parents, and the resultant fruit frequently lacks desired characteristics. For commercial purposes, including botanical evaluation, apple cultivars are propagated by clonal grafting onto rootstocks. Apple trees grown without rootstocks tend to be larger and much slower to fruit after planting. Rootstocks are used to control the speed of growth and the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Etymonline
Etymonline, or ''Online Etymology Dictionary'', sometimes abbreviated as OED (not to be confused with the ''Oxford English Dictionary'', which the site often cites), is a free online dictionary that describes the etymology, origins of English language, English words, written and compiled by Douglas R. Harper. Description Douglas R. Harper is an American Civil War historian and copy editor for LNP Media Group. He compiled the etymology dictionary to record the history and evolution of more than 50,000 words, including slang and technical terms. The core of its etymology information stems from ''The Barnhart Dictionary of Etymology'' by Robert Barnhart, Ernest Klein's ''Comprehensive Etymology Dictionary of the English Language'', ''The Middle English Compendium'', ''The Oxford English Dictionary'', and the 1889–1902 ''Century Dictionary''. Harper also researches on digital archives. On the ''Etymonline'' homepage, Harper says that he considers himself "essentially and for the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Sins
The seven deadly sins (also known as the capital vices or cardinal sins) function as a grouping of major vices within the teachings of Christianity. In the standard list, the seven deadly sins according to the Catholic Church are pride, greed, wrath, envy, lust, gluttony, and sloth. In Catholicism, the classification of deadly sins into a group of seven originated with Tertullian and continued with Evagrius Ponticus. The concepts were partly based on Greco-Roman and Biblical antecedents. Later, the concept of seven deadly sins evolved further, as shown by historical context based on the Latin language of the Roman Catholic Church, though with significant influence from the Greek language and associated religious traditions. Knowledge of this concept is evident in various treatises; in paintings and sculpture (for example, architectural decorations on churches in some Catholic parishes); and in some older textbooks. Further knowledge has been derived from patterns of confessi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Directions
The four cardinal directions or cardinal points are the four main compass directions: north (N), south (S), east (E), and west (W). The corresponding azimuths ( clockwise horizontal angle from north) are 0°, 90°, 180°, and 270°. The four ordinal directions or intercardinal directions are northeast (NE), southeast (SE), southwest (SW), and northwest (NW). The corresponding azimuths are 45°, 135°, 225°, and 315°. The intermediate direction of every pair of neighboring cardinal and intercardinal directions is called a secondary intercardinal direction. These eight shortest points in the compass rose shown to the right are: # West-northwest (WNW) # North-northwest (NNW) # North-northeast (NNE) # East-northeast (ENE) # East-southeast (ESE) # South-southeast (SSE) # South-southwest (SSW) # West-southwest (WSW) Points between the cardinal directions form the points of the compass. Arbitrary horizontal directions may be indicated by their azimuth angle value. Determinatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Numbers (linguistics)
In linguistics, and more precisely in traditional grammar, a cardinal numeral (or cardinal number word) is a part of speech used to count. Examples in English are the words ''one'', ''two'', ''three'', and the compounds ''three hundred ndforty-two'' and ''nine hundred ndsixty''. Cardinal numerals are classified as definite, and are related to ordinal numbers, such as the English ''first'', ''second'', ''third'', etc. See also * Arity * Cardinal number for the related usage in mathematics * English numerals (in particular the ''Cardinal numbers'' section) * Distributive number * Multiplier * Nominal numeral * Numeral for examples of number systems * Ordinal number * Valency * Roman numerals * Latin numerals * Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, is a numeral system, system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal number (linguistics), or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford English Dictionary
The ''Oxford English Dictionary'' (''OED'') is the principal historical dictionary of the English language, published by Oxford University Press (OUP), a University of Oxford publishing house. The dictionary, which published its first edition in 1884, traces the historical development of the English language, providing a comprehensive resource to scholars and academic researchers, and provides ongoing descriptions of English language usage in its variations around the world. In 1857, work first began on the dictionary, though the first edition was not published until 1884. It began to be published in unbound Serial (literature), fascicles as work continued on the project, under the name of ''A New English Dictionary on Historical Principles; Founded Mainly on the Materials Collected by The Philological Society''. In 1895, the title ''The Oxford English Dictionary'' was first used unofficially on the covers of the series, and in 1928 the full dictionary was republished in 10 b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nominal Number
Nominal numbers are numerals used as labels to Unique identifier, identify items uniquely. Importantly, the actual values of the numbers which these numerals represent are less relevant, as they do not indicate quantity, rank, or any other measurement. Labelling referees Smith and Kumar as referees "1" and "2" is a use of nominal numbers. Any set of numbers (a subset of the natural numbers) will be consistent labels as long as a ''distinct'' number is uniquely used for each distinct term which needs to be labelled. Nonetheless, sequences of integers may naturally be used as the simplest way to begin labelling; for example, 1, 2, 3, and so on. Definition The term "nominal number" may be quite recent and of limited use. It appears to have originated in school textbooks derived from the statistical term "nominal data", defined as data indicating "...merely statements of qualitative category of membership." This usage comes from the sense of wikt:nominal, nominal as "name". Mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Numbers (linguistics)
In linguistics, ordinal numerals or ordinal number words are words representing position or rank in a sequential order; the order may be of size, importance, chronology, and so on (e.g., "third", "tertiary"). They differ from cardinal numerals, which represent quantity (e.g., "three") and other types of numerals. In traditional grammar, all Numeral (linguistics), numerals, including ordinal numerals, are grouped into a separate part of speech (, hence, "noun numeral" in older English grammar books). However, in modern interpretations of English grammar, ordinal numerals are usually conflated with adjectives. Ordinal numbers may be written in English with numerals and letter suffixes: 1st, 2nd or 2d, 3rd or 3d, 4th, 11th, 21st, 101st, 477th, etc., with the suffix acting as an ordinal indicator. Written dates often omit the suffix, although it is nevertheless pronounced. For example: 5 November 1605 (pronounced "the fifth of November ... "); November 5, 1605, ("November (the) F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Medieval Latin
Medieval Latin was the form of Literary Latin used in Roman Catholic Church, Roman Catholic Western Europe during the Middle Ages. It was also the administrative language in the former Western Roman Empire, Roman Provinces of Mauretania, Numidia (Roman province), Numidia and Africa (Roman province), Africa Proconsularis under the Vandals, the Exarchate of Africa, Byzantines and the Kingdom of Altava, Romano-Berber Kingdoms, until it declined after the Arab conquest of North Africa, Arab Conquest. Medieval Latin in Southern and Central Visigothic Kingdom, Visigothic Hispania, conquered by the Arabs immediately after North Africa, experienced a similar fate, only recovering its importance after the Reconquista by the Northern Christian Kingdoms. In this region it served as the primary written language, though local languages were also written to varying degrees. Latin functioned as the main medium of scholarly exchange, as the liturgical language of the Roman Catholic Church, Churc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
