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Extension (predicate Logic)
The extension of a predicate – a truth-valued function – is the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation.Contents1 Examples 2 Relationship with characteristic function 3 See also 4 ReferencesExamples[edit] For 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
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Predicate (mathematical Logic)
In mathematical logic, a predicate is commonly understood to be a Boolean-valued 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
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NLab
The nLab is a wiki for research-level 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 "n-point of view"[1] (a deliberate pun on's "neutral point of view") that category theory and particularly higher n-category theory provide a useful unifying viewpoint for mathematics, physics and philosophy. Overview[edit] The nLab was originally conceived to provide a repository for ideas (and even new research) generated in the comments on posts at the n- Category Café, a group blog run (at the time) by John Baez, David Corfield and Urs Schreiber
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Function (mathematics)
In mathematics, a function[1] 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"[2] 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
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Set (mathematics)
In mathematics, a set is a 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 . The concept of a set is one of the most fundamental 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
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Tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. There is only one 0-tuple, an empty sequence, or empty tuple, as it is referred to. An n-tuple 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 5-tuple. 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 and Visual Basic, but not in mathematical expressions, as they are the standard notation for sets
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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
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 2-place 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)
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Set-builder Notation
The set of all even integers, expressed in set-builder notation.In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy.[1] Defining sets by properties is also known as set comprehension, set abstraction or as defining a set's intension. Set-builder 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.Contents1 Sets defined by enumeration 2 Sets defined by a predicate2.1 Specifying the domain 2.2 Examples3 More complex expressions on the left side of the notation 4 Equivalent predicates yield equal sets 5 Set existence axiom 6 Z notation 7 Parallels in programming languages 8 See also 9 NotesSets defined by enumeration[edit] A set is an unordered collection of elements
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N-ary
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. (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,[1][2] but this word can have many other meanings in mathematics. In logic and philosophy, arity is also called adicity and degree.[3][4] In linguistics, arity is usually named valency.[5] 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)
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
(album), a 1992
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Truth-value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.[1]Contents1 Classical logic 2 Intuitionistic and constructive logic 3 Multi-valued logic 4 Algebraic semantics 5 In other theories 6 See also 7 References 8 External linksClassical logic[edit] ⊤ true  ·∧· conjunction¬↕↕ ⊥ false·∨· disjunction Negation interchanges true with false and conjunction with disjunctionIn 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 two-valued logic. This set of two values is also called the Boolean domain
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Extensional Set
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.[1] 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)
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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.[1] 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 first-order 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
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Logic
Logic
Logic
(from the Ancient Greek: λογική, translit. logikḗ[1]), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth,[2] and is now 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
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Extensional Logic
Intensional logic is an approach to predicate logic that extends first-order 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)
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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.Contents1 Example 2 In mathematics 3 See also 4 ReferencesExample[edit] Consider 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
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