HOME  TheInfoList.com 
Extension (predicate Logic) The EXTENSION of a predicate – a truthvalued function – is the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation . CONTENTS * 1 Examples * 2 Relationship with characteristic function * 3 See also * 4 References EXAMPLESFor example, the statement "d2 is the weekday following d1" can be seen as a truth function associating to each tuple (d2, d1) the value true or false. The extension of this truth function is, by convention, the set of all such tuples associated with the value true, i.e. {(Monday, Sunday), (Tuesday, Monday), (Wednesday, Tuesday), (Thursday, Wednesday), (Friday, Thursday), (Saturday, Friday), (Sunday, Saturday)} By examining this extension we can conclude that "Tuesday is the weekday following Saturday" (for example) is false. Using setbuilder notation , the extension of the nary predicate {displaystyle Phi } can be written as { ( x 1 , . . [...More...]  "Extension (predicate Logic)" on: Wikipedia Yahoo 

Extensional Set In mathematics , a SETOID (X, ~) is a set (or type ) X equipped with an equivalence relation ~. A Setoid may also be called ESET, BISHOP SET, or EXTENSIONAL SET. Setoids are studied especially in proof theory and in typetheoretic 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) [...More...]  "Extensional Set" on: Wikipedia Yahoo 

Predicate (mathematical Logic) In mathematical logic , a PREDICATE is commonly understood to be a Booleanvalued function P: X→ {true, false}, called the predicate on X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a relation , then a predicate is simply the characteristic function or the indicator function of a relation. However, not all theories have relations, or are founded on set theory , and so one must be careful with the proper definition and semantic interpretation of a predicate. CONTENTS * 1 Simplified overview * 2 Formal definition * 3 See also * 4 References * 5 External links SIMPLIFIED OVERVIEWInformally, a PREDICATE is a statement that may be true or false depending on the values of its variables [...More...]  "Predicate (mathematical Logic)" on: Wikipedia Yahoo 

Special SPECIAL or SPECIALS may refer to: CONTENTS * 1 Music * 2 Film and television * 3 Other uses * 4 See also MUSIC * Special (album) , a 1992 album by Vesta Williams * "Special" (Garbage song) , 1998 * "Special" (Mew song) , 2005 * "Special" (Stephen Lynch song) , 2000 * The Specials The Specials , a British band * "Special", a song by Violent Femmes on The Blind Leading the Naked * "Special", a song on [...More...]  "Special" on: Wikipedia Yahoo 

NLab The NLAB is a wiki for researchlevel notes, expositions and collaborative work, including original research, in mathematics , physics , and philosophy , with a focus on methods from category theory and homotopy theory . The nLab espouses the "npoint of view" (a deliberate pun on 's "neutral point of view") that category theory and particularly higher ncategory theory provide a useful unifying viewpoint for mathematics, physics and philosophy. OVERVIEWThe nLab was originally conceived to provide a repository for ideas (and even new research) generated in the comments on posts at the n Category Category Café , a group blog run (at the time) by John Baez , David Corfield and Urs Schreiber . Eventually the nLab developed into an independent project, which has since grown to include whole research projects and encyclopedic material [...More...]  "NLab" on: Wikipedia Yahoo 

Logic LOGIC (from the Ancient Greek : λογική, translit. logikḗ ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is generally held to consist of the systematic study of the form of valid inference . A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion . (In ordinary discourse, inferences may be signified by words like therefore, hence, ergo and so on.) There is no universal agreement as to the exact scope and subject matter of logic (see § Rival conceptions , below), but it has traditionally included the classification of arguments, the systematic exposition of the 'logical form' common to all valid arguments, the study of inference , including fallacies , and the study of semantics , including paradoxes [...More...]  "Logic" on: Wikipedia Yahoo 

Mathematical Logic MATHEMATICAL LOGIC is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics , the foundations of mathematics , and theoretical computer science . The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory , model theory , recursion theory , and proof theory . These areas share basic results on logic, particularly firstorder logic , and definability . In computer science (particularly in the ACM Classification ) mathematical logic encompasses additional topics not detailed in this article; see Logic Logic in computer science for those. Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics [...More...]  "Mathematical Logic" on: Wikipedia Yahoo 

Nary In logic , mathematics , and computer science , the ARITY /ˈærɪti/ ( listen ) of a function or operation is the number of arguments or operands that the function takes. The arity of a relation (or predicate ) is the dimension of the domain in the corresponding Cartesian product Cartesian product . (A function of arity n thus has arity n+1 considered as a relation.) The term springs from words like unary, binary, ternary, etc. Unary functions or predicates may be also called "monadic"; similarly, binary functions may be called "dyadic". In mathematics arity may also be named rank, but this word can have many other meanings in mathematics. In logic and philosophy, arity is also called adicity and degree. In linguistics , arity is usually named valency . In computer programming , there is often a syntactical distinction between operators and functions ; syntactical operators usually have arity 0, 1, or 2 (the ternary operator ?: is also common) [...More...]  "Nary" on: Wikipedia Yahoo 

Function (mathematics) In mathematics , a FUNCTION is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function. Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics . There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input [...More...]  "Function (mathematics)" on: Wikipedia Yahoo 

Truthvalue In logic and mathematics , a TRUTH VALUE, sometimes called a LOGICAL VALUE, is a value indicating the relation of a proposition to truth . CONTENTS * 1 Classical logic * 2 Intuitionistic and constructive logic * 3 Multivalued logic * 4 Algebraic semantics * 5 In other theories * 6 See also * 7 References * 8 External links CLASSICAL LOGIC ⊤ true ·∧· conjunction ¬ ↕ ↕ ⊥ false ·∨· disjunction Negation interchanges true with false and conjunction with disjunction In classical logic , with its intended semantics, the truth values are true (1 or T), and untrue or false (0 or ⊥); that is, classical logic is a twovalued logic . This set of two values is also called the Boolean domain . Corresponding semantics of logical connectives are truth functions , whose values are expressed in the form of truth tables [...More...]  "Truthvalue" on: Wikipedia Yahoo 

Set (mathematics) In mathematics , a SET is a welldefined collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education , elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree. The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano Bernard Bolzano in his work The Paradoxes of the Infinite [...More...]  "Set (mathematics)" on: Wikipedia Yahoo 

Tuple In mathematics a TUPLE is a finite ordered list (sequence) of elements . An NTUPLE is a sequence (or ordered list) of n elements, where n is a nonnegative integer. There is only one 0tuple, an empty sequence. An ntuple is defined inductively using the construction of an ordered pair . Mathematicians usually write tuples by listing the elements within parentheses " ( ) {displaystyle ({text{ }})} " and separated by commas; for example, ( 2 , 7 , 4 , 1 , 7 ) {displaystyle (2,7,4,1,7)} denotes a 5tuple. Sometimes other symbols are used to surround the elements, such as square brackets "" or angle brackets "< >". Braces "{ }" are only used in defining arrays in some programming languages such as Java , but not in mathematical expressions, as they are the standard notation for sets . The term tuple can often occur when discussing other mathematical objects, such as vectors . In computer science , tuples come in many forms [...More...]  "Tuple" on: Wikipedia Yahoo 

Relation (mathematics) In mathematics , a BINARY RELATION on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms CORRESPONDENCE, DYADIC RELATION and 2PLACE RELATION are synonyms for binary relation. An example is the "divides " relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). In this relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13 [...More...]  "Relation (mathematics)" on: Wikipedia Yahoo 

Setbuilder Notation In set theory and its applications to logic , mathematics , and computer science , SETBUILDER NOTATION is a mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy. Defining sets by properties is also known as SET COMPREHENSION, SET ABSTRACTION or as defining a set's INTENSION . Setbuilder notation is sometimes simply referred to as set notation, although this phrase may be better reserved for the broader class of means of denoting sets . CONTENTS * 1 Sets defined by enumeration * 2 Sets defined by a predicate * 2.1 Specifying the domain * 2.2 Examples * 3 More complex expressions on the left side of the notation * 4 Equivalent predicates yield equal sets * 5 Set existence axiom * 6 Z notation Z notation * 7 Parallels in programming languages * 8 See also * 9 Notes SETS DEFINED BY ENUMERATIONA set is an unordered collection of elements [...More...]  "Setbuilder Notation" on: Wikipedia Yahoo 

Extensionality In logic , EXTENSIONALITY, or EXTENSIONAL EQUALITY, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality , which is concerned with whether the internal definitions of objects are the same. CONTENTS * 1 Example * 2 In mathematics * 3 See also * 4 References EXAMPLEConsider the two functions f and g mapping from and to natural numbers , defined as follows: * To find f(n), first add 5 to n, then multiply by 2. * To find g(n), first multiply n by 2, then add 10.These functions are extensionally equal; given the same input, both functions always produce the same value. But the definitions of the functions are not equal, and in that intensional sense the functions are not the same. Similarly, in natural language there are many predicates (relations) that are intensionally different but are extensionally identical [...More...]  "Extensionality" on: Wikipedia Yahoo 

Extensional Logic INTENSIONAL LOGIC is an approach to predicate logic that extends firstorder logic , which has quantifiers that range over the individuals of a universe (extensions ), by additional quantifiers that range over terms that may have such individuals as their value (intensions ). The distinction between intensional and extensional entities is parallel to the distinction between sense and reference . CONTENTS * 1 Its place inside logic * 2 Modal logic * 3 Type theoretical intensional logic * 4 See also * 5 Notes * 6 References * 7 External links ITS PLACE INSIDE LOGIC Logic Logic is the study of proof and deduction as manifested in language (abstracting from any underlying psychological or biological processes) [...More...]  "Extensional Logic" on: Wikipedia Yahoo 