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Partially Ordered Set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is Reflexive relation, reflexive, antisymmetric relation, antisymmetric, and Transitive relation, transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hasse Diagram Of Powerset Of 3
Hasse is both a surname and a given name. Notable people with the name include: Surname: * Clara H. Hasse (1880–1926), American botanist * Helmut Hasse (1898–1979), German mathematician * Henry Hasse (1913–1977), US writer of science fiction * Johann Adolph Hasse (1699–1783), German composer * Maria Hasse (1921–2014), German mathematician * Peter Hasse (c. 1585–1640), German organist and composer Given name or nickname: * Hans Alfredson (born 1931), Swedish actor, film director, writer and comedian * Hans Backe (born 1952), Swedish football manager * Hasse Borg (born 1953), Swedish footballer * Hasse Börjes (born 1948), Swedish speed skater * Hasse Ekman (1915-2004), Swedish film director and actor * Hans Wind (1919–1995), Finnish flying ace See also * Hasse bound, on the number of points on an elliptic curve * Hasse diagram, a diagram used in set theory {{given name, type=both ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Setoid
In mathematics, a setoid (''X'', ~) is a set (or type) ''X'' equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set. Setoids are studied especially in proof theory and in type-theoretic foundations of mathematics. Often in mathematics, when one defines an equivalence relation on a set, one immediately forms the quotient set (turning equivalence into equality). In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of intensional equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set). Proof theory In proof theory, particularly the proof theory of constructive mathematics based on the Curry–Howard correspondence, one often identifies a mathematical proposition with its set of proofs (if any). A given proposition may have many proofs, of course; according to the principle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mutually Exclusive
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both. In the coin-tossing example, both outcomes are, in theory, collectively exhaustive, which means that at least one of the outcomes must happen, so these two possibilities together exhaust all the possibilities. However, not all mutually exclusive events are collectively exhaustive. For example, the outcomes 1 and 4 of a single roll of a six-sided die are mutually exclusive (both cannot happen at the same time) but not collectively exhaustive (there are other possible outcomes; 2,3,5,6). Logic In logic, two propositions \phi and \psi are mutually exclusive if it is not logically possible for them to be true at the same time; that is, \lnot (\phi \land \psi) is a tautology. To say that more than two propositions are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Comparability
In mathematics, two elements ''x'' and ''y'' of a set ''P'' are said to be comparable with respect to a binary relation ≤ if at least one of ''x'' ≤ ''y'' or ''y'' ≤ ''x'' is true. They are called incomparable if they are not comparable. Rigorous definition A binary relation on a set P is by definition any subset R of P \times P. Given x, y \in P, x R y is written if and only if (x, y) \in R, in which case x is said to be to y by R. An element x \in P is said to be , or (), to an element y \in P if x R y or y R x. Often, a symbol indicating comparison, such as \,,\, \geq, and many others) is used instead of R, in which case x < y is written in place of which is why the term "comparable" is used. Comparability with respect to induces a canonical binary relation on ; specifically, the induced by is defined to be the set of all pairs such that i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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If, And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Complementary Relation
In set theory, the complement of a set , often denoted by A^c (or ), is the set of elements not in . When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^c= U \setminus A = \. The absolute complement of is usually denoted by A^c. Other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Total Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Converse Relation
In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if X and Y are sets and L \subseteq X \times Y is a relation from X to Y, then L^ is the relation defined so that yL^x if and only if xLy. In set-builder notation, :L^ = \. Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation is also called the transpose relation. It has also been called the opposite or dual of the original relation, the inverse of the original relation,Gerard O'Regan (2016): ''Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications'' or the reciprocal L^ of the relation L. Other notations for the converse relation include L^, L^, \breve, L^, or L^. The notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Reflexive Closure
In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R, i.e. the set R \cup \. For example, if X is a set of distinct numbers and x R y means "x is less than y", then the reflexive closure of R is the relation "x is less than or equal Definition The reflexive closure S of a relation R on a set X is given by S = R \cup \ In plain English, the reflexive closure of R is the union of R with the identity relation In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ... on X. Example As an example, if X = \ R = \ then the relation R is already reflexive by itself, so it does not differ from its reflexive closure. However, if any of the reflexive pairs in R was absent, it would be inserted for the reflexive closure. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |