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Definitions
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes).Lyons, John. "Semantics, vol. I." Cambridge: Cambridge (1977). p.158 and on. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what the mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. Basic terminology In modern usage, a definition is something, typically expressed in words, tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word
A word is a basic element of language that carries semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguistics, linguists on its definition and numerous attempts to find specific criteria of the concept remain controversial. Different standards have been proposed, depending on the theoretical background and descriptive context; these do not converge on a single definition. Some specific definitions of the term "word" are employed to convey its different meanings at different levels of description, for example based on phonology, phonological, grammar, grammatical or orthography, orthographic basis. Others suggest that the concept is simply a convention used in everyday situations. The concept of "word" is distinguished from that of a morpheme, which is the smallest unit of language that has a meaning, even if it cannot stand on its own. Words a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), 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 polygon, simple (not self-intersecting), or complex polygon, complex (self-intersecting, or crossed). Simple quadrilaterals are either convex polygon, convex or concave polygon, concave. The Internal and external angle, interior angles of a simple (and Plane (geometry), planar) quadrilateral ''ABCD'' add up to 360 Degree (angle), degrees, that is :\angle A+\angle B+\angle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necessary And Sufficient Conditions
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of . (Equivalently, it is impossible to have without , or the falsity of ensures the falsity of .) Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one (possibly one of several conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Wittgenstein taught at the University of Cambridge. Despite his position, only one book of his philosophy was published during his entire life: the 75-page ''Logisch-Philosophische Abhandlung'' (''Logical-Philosophical Treatise'', 1921), which appeared, together with an English translation, in 1922 under the Latin title ''Tractatus Logico-Philosophicus''. His only other published works were an article, "Some Remarks on Logical Form" (1929); a book review; and a children's dictionary. #Works, His voluminous manuscripts were edited and published posthumously. The first and best-known of this posthumous series is the 1953 book ''Philosophical Investigations''. A 1999 survey among American university and college teachers ranked the ''Investigations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. When quantified, A \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that A \subseteq B, # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''. The validity of this technique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classics
Classics, also classical studies or Ancient Greek and Roman studies, is the study of classical antiquity. In the Western world, ''classics'' traditionally refers to the study of Ancient Greek literature, Ancient Greek and Roman literature and their original languages, Ancient Greek and Latin. Classics may also include as secondary subjects Greco-Roman Ancient philosophy, philosophy, Ancient history, history, archaeology, anthropology, classical architecture, architecture, Ancient art, art, Classical mythology, mythology, and society. In Western culture, Western civilization, the study of the Ancient Greek and Roman classics was considered the foundation of the humanities, and they traditionally have been the cornerstone of an elite higher education. Etymology The word ''classics'' is derived from the Latin adjective ''wikt:classicus, classicus'', meaning "belonging to the highest class of Citizenship, citizens." The word was originally used to describe the members of the Patri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enumerative Definition
An enumerative definition of a concept or term is a special type of extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets and only practical for relatively small sets. Example An example of an enumerative definition for the set extant monotreme species (for which the intensional definition is "species of currently-living mammals that lay eggs") would be: : platypuses : echidnae: :: short-beaked echidna :: long-beaked echidnae: ::: Sir David's long-beaked echidna ::: eastern long-beaked echidna ::: western long-beaked echidna See also * Definition * Extension * Extensional definition * Set notation * Enumeration An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the element (mathematics), elements of a Set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Minister
A prime minister or chief of cabinet is the head of the cabinet and the leader of the ministers in the executive branch of government, often in a parliamentary or semi-presidential system. A prime minister is not the head of state, but rather the head of government, serving as the chief of the executive under either a monarch or a president in a republican form of government. In parliamentary systems of government (be they constitutional monarchies or parliamentary republics), the Prime Minister (or occasionally a similar post with a different title, such as the Chancellor of Germany) is the most powerful politician and the functional leader of the state, by virtue of commanding the confidence of the legislature. The head of state is typically a ceremonial officer, though they may exercise reserve powers to check the Prime Minister in unusual situations. Under some presidential systems, such as South Korea and Peru, the prime minister is the leader or the most s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aristotle
Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, and the arts. As the founder of the Peripatetic school of philosophy in the Lyceum (classical), Lyceum in Athens, he began the wider Aristotelianism, Aristotelian tradition that followed, which set the groundwork for the development of modern science. Little is known about Aristotle's life. He was born in the city of Stagira (ancient city), Stagira in northern Greece during the Classical Greece, Classical period. His father, Nicomachus (father of Aristotle), Nicomachus, died when Aristotle was a child, and he was brought up by a guardian. At around eighteen years old, he joined Plato's Platonic Academy, Academy in Athens and remained there until the age of thirty seven (). Shortly after Plato died, Aristotle left Athens and, at the request ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the Diamonds (suit), diamonds suit in playing cards which resembles the projection of an Octahedron#Orthogonal projections, octahedral diamond, or a lozenge (shape), lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after calisson, the French sweet—also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple polygon, simple (non-self-intersecting), and is a special case of a parallelogram and a Kite (geometry), kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from , meaning something that spins, which derives from the verb , roman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posterior Analytics
The ''Posterior Analytics'' (; ) is a text from Aristotle's '' Organon'' that deals with demonstration, definition, and scientific knowledge. The demonstration is distinguished as ''a syllogism productive of scientific knowledge'', while the definition marked as ''the statement of a thing's nature, ... a statement of the meaning of the name, or of an equivalent nominal formula''. Content In the '' Prior Analytics'', syllogistic logic is considered in its formal aspect; in the ''Posterior'' it is considered in respect of its matter. The "form" of a syllogism lies in the necessary connection between the premises and the conclusion. Even where there is no fault in the form, there may be in the matter, i.e. the propositions of which it is composed, which may be true or false, probable or improbable. When the premises are certain, true, and primary, and the conclusion formally follows from them, this is demonstration, and produces scientific knowledge of a thing. Such syllogisms ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
