The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. History Mathematical topics typically emerge and evolve through interactions among many researchers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Cantor's work initially polarized the mathematicians of his day. Basic concepts and notation Some ontology
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Order theoryFor a topical guide to this subject, see Outline of order theory. Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. Order theory captures the intuition of orders that arises from such examples in a general setting. Basic definitions This section introduces ordered sets by building upon the concepts of set theory, arithmetic, and binary relations. Partially ordered sets Orders are special binary relations. a ≤ a (reflexivity) if a ≤ b and b ≤ a then a = b (antisymmetry) if a ≤ b and b ≤ c then a ≤ c (transitivity). a ≤ b or b ≤ a (totality). Duality
Antisymmetric relationif R(a,b) and R(b,a), then a = b, or, equivalently, if R(a,b) with a ≠ b, then R(b,a) must not hold. As a simple example, the divisibility order on the natural numbers is an antisymmetric relation. In mathematical notation, this is: Antisymmetry is different from asymmetry, which requires both antisymmetry and irreflexivity. Examples The relation "x is even, y is odd" between a pair (x, y) of integers is antisymmetric: Every asymmetric relation is also an antisymmetric relation. See also Symmetry in mathematics ReferencesReflexive relationRelated terms A relation that is irreflexive, or anti-reflexive, is a binary relation on a set where no element is related to itself. An example is the "greater than" relation (x>y) on the real numbers. A relation is called quasi-reflexive if every element that is related to some element is related to itself. The reflexive reduction of a binary relation ~ on a set S is the smallest relation ~′ such that ~′ shares the same reflexive closure as ~. Examples Examples of reflexive relations include: Examples of irreflexive relations include: "is not equal to""is coprime to" (for the integers>1, since 1 is coprime to itself)"is a proper subset of""is greater than""is less than" Number of reflexive relations The number of reflexive relations on an n-element set is 2n2−n. See also Notes References Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. External links
Pearltrees Hits Android, as it Prepares to Become a File ManagerPearltrees is a content curation startup that we’ve been tracking for some time now, and today it’s launching an app on Android. However, there’s a twist, as the launch points towards an expansion of exactly what this service is all about. As with the Web and iOS versions of Pearltrees, the Android app allows you to create, share and explore mindmap-style ‘trees’ of content. So, I could create a tree of articles, images and notes related to a particular theme and then if you searched Pearltrees for that theme, you’d find my tree and related ones by other people. It’s a highly visual, logical way of organizing and sharing ideas and information, and the Android app benefits from the OS’ built-in sharing capabilities. A ‘post-PC’ file manager? It’s fair to say that Pearltrees hasn’t found mainstream fame quite yet as an alternative to more traditional social bookmarking services. It’s easy to see how the service could be repositioned as a ‘mobile file manager for those who think visually’.
CognitionCognition is a faculty for the processing of information, applying knowledge, and changing preferences. Cognition, or cognitive processes, can be natural or artificial, conscious or unconscious. These processes are analyzed from different perspectives within different contexts, notably in the fields of linguistics, anesthesia, neuroscience, psychiatry, psychology, philosophy, anthropology, systemics, and computer science.[page needed] Within psychology or philosophy, the concept of cognition is closely related to abstract concepts such as mind, intelligence. It encompasses the mental functions, mental processes (thoughts), and states of intelligent entities (humans, collaborative groups, human organizations, highly autonomous machines, and artificial intelligences). Etymology Origins Wilhelm Wundt (1832-1920) heavily emphasized the notion of what he called introspection; examining the inner feelings of an individual. Psychology Social process Serial position
Archivez vous-même un site dans The Wayback MachineArchivez vous-même un site dans The Wayback Machine The Wayback Machine, la section du site Internet Archive qui a pour mission de conserver ad-vitam eternam une copie des tous les sites web du monde, se voit doter d'une nouvelle fonctionnalité. Sur la page d'accueil, il est possible d'entrer n'importe quelle URL et la page visée sera alors sauvegardée dans les archives de la fondation. Notez que cela fonctionne aussi avec les PDF. De quoi garder immortaliser pour toute la vie un article de blog débile, un document PDF leaké ou un tweet de politique qui pourraient être supprimé ;-) Source Vous avez aimé cet article ?