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Predicate (mathematics)
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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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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Element (mathematics)
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.Contents1 Sets 2 Notation and terminology 3 Cardinality
Cardinality
of sets 4 Examples 5 References 6 Further reading 7 External linksSets[edit] Writing A = 1 , 2 , 3 , 4 displaystyle A= 1,2,3,4 means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example 1 , 2 displaystyle 1,2 , are subsets of A. Sets can themselves be elements. For example, consider the set B = 1 , 2 , 3 , 4 displaystyle B= 1,2, 3,4 . The elements of B are not 1, 2, 3, and 4
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Binary Relation
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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International Standard Book Number
"ISBN" redirects here. For other uses, see ISBN (other).International Standard Book
Book
NumberA 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar codeAcronym ISBNIntroduced 1970; 48 years ago (1970)Managing organisation International ISBN AgencyNo. of digits 13 (formerly 10)Check digit Weighted sumExample 978-3-16-148410-0Website www.isbn-international.orgThe International Standard Book
Book
Number (ISBN) is a unique[a][b] numeric commercial book identifier. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.[1] An ISBN is assigned to each edition and variation (except reprintings) of a book. For example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, and 10 digits long if assigned before 2007
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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 album by Vesta Williams "Special" (Garbage song), 1998 "Special
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Formal Language
In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it. The alphabet of a formal language is the set of symbols, letters, or tokens from which the strings of the language may be formed.[1] The strings formed from this alphabet are called words, and the words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, also called its formation rule. The field of formal language theory studies primarily the purely syntactical aspects of such languages—that is, their internal structural patterns. Formal language
Formal language
theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages
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Formation Rule
In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. These rules only address the location and manipulation of the strings of the language. It does not describe anything else about a language, such as its semantics (i.e. what the strings mean). (See also formal grammar).Contents1 Formal language 2 Formal systems 3 Propositional and predicate logic 4 See alsoFormal language[edit] Main article: Formal language A formal language is an organized set of symbols the essential feature being that it can be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any reference to any meanings of any of its expressions; it can exist before any interpretation is assigned to it—that is, before it has any meaning
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Formal Proof
A formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of natural deduction is a generalization of the concept of proof.[1] The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof
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Formal Semantics (logic)
In logic, the semantics of logic is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic notion of entailment. Overview[edit] The truth conditions of various sentences we may encounter in arguments will depend upon their meaning, and so logicians cannot completely avoid the need to provide some treatment of the meaning of these sentences. The semantics of logic refers to the approaches that logicians have introduced to understand and determine that part of meaning in which they are interested; the logician traditionally is not interested in the sentence as uttered but in the proposition, an idealised sentence suitable for logical manipulation.[citation needed] Until the advent of modern logic, Aristotle's Organon, especially De Interpretatione, provided the basis for understanding the significance of logic
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Well-formed Formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.[1] A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic.Contents1 Introduction 2 Propositional calculus 3 Predicate logic 4 Atomic and open formulas 5 Closed formulas 6 Properties applicable to formulas 7 Usage of the terminology 8 See also 9 Notes 10 References 11 External linksIntroduction[edit] A key use of formulas is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated
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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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Class (set Theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. Outside set theory, the word "class" is sometimes used synonymously with "set"
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Opaque Predicate
In computer programming, an opaque predicate is a predicate—an expression that evaluates to either "true" or "false"—for which the outcome is known by the programmer a priori, but which, for a variety of reasons, still needs to be evaluated at run time. Opaque predicates have been used as watermarks, as it will be identifiable in a program's executable. They can also be used to prevent an overzealous optimizer from optimizing away a portion of a program. Another use is in obfuscating the control or dataflow of a program to make reverse engineering harder. External links[edit]"A Method for Watermarking Java Programs via Opaque Predicates"This computer-programming-related article is a stub
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Classical Logic
Classical logic (or standard logic[1][2]) is an intensively studied and widely used class of formal logics
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Axiom
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'[1][2] The term has subtle differences in definition when used in the context of different fields of study
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