Difference between revisions of "Boolean-valued function"

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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
 
<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
  
A '''boolean-valued function''' is a [[function (mathematics)|function]] of the type <math>f : X \to \mathbb{B},</math> where <math>X\!</math> is an arbitrary [[set]] and where <math>\mathbb{B}</math> is a [[boolean domain]].
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A '''boolean-valued function''' is a function of the type <math>f : X \to \mathbb{B},</math> where <math>X\!</math> is an arbitrary set and where <math>\mathbb{B}</math> is a [[boolean domain]].
  
In the [[formal science]]s &mdash; [[mathematics]], [[mathematical logic]], [[statistics]] &mdash; and their applied disciplines, a boolean-valued function may also be referred to as a [[characteristic function]], [[indicator function]], [[predicate]], or [[proposition]].  In all of these uses it is understood that the various terms refer to a mathematical object and not the corresponding [[semiotic]] sign or syntactic expression.
+
In the formal sciences &mdash; mathematics, mathematical logic, statistics &mdash; and their applied disciplines, a boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition.  In all of these uses it is understood that the various terms refer to a mathematical object and not the corresponding sign or syntactic expression.
  
In [[semantics|formal semantic]] theories of [[truth]], a '''truth predicate''' is a predicate on the [[sentence]]s of a [[formal language]], interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true.  A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value.
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In formal semantic theories of truth, a '''truth predicate''' is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true.  A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value.
  
 
==Examples==
 
==Examples==
  
A '''binary sequence''' is a boolean-valued function <math>f : \mathbb{N}^+ \to \mathbb{B}</math>, where <math>\mathbb{N}^+ = \{ 1, 2, 3, \ldots \},</math>.  In other words, <math>f\!</math> is an infinite [[sequence]] of 0's and 1's.
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A '''binary sequence''' is a boolean-valued function <math>f : \mathbb{N}^+ \to \mathbb{B}</math>, where <math>\mathbb{N}^+ = \{ 1, 2, 3, \ldots \},</math>.  In other words, <math>f\!</math> is an infinite sequence of 0's and 1's.
  
 
A '''binary sequence''' of '''length''' <math>k\!</math> is a boolean-valued function <math>f : [k] \to \mathbb{B}</math>, where <math>[k] = \{ 1, 2, \ldots k \}.</math>
 
A '''binary sequence''' of '''length''' <math>k\!</math> is a boolean-valued function <math>f : [k] \to \mathbb{B}</math>, where <math>[k] = \{ 1, 2, \ldots k \}.</math>
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==References==
 
==References==
  
* [[Frank Markham Brown|Brown, Frank Markham]] (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA.  2nd edition, Dover Publications, Mineola, NY, 2003.
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* Brown, Frank Markham (2003), ''Boolean Reasoning : The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA.  2nd edition, Dover Publications, Mineola, NY, 2003.
  
* [[Zvi Kohavi|Kohavi, Zvi]] (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970.  2nd edition, McGraw–Hill, 1978.
+
* Kohavi, Zvi (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970.  2nd edition, McGraw–Hill, 1978.
  
* [[Robert R. Korfhage|Korfhage, Robert R.]] (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.
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* Korfhage, Robert R. (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.
  
* [[Mathematical Society of Japan]], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993.  Cited as EDM.
+
* Mathematical Society of Japan, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993.  Cited as EDM.
  
* [[Marvin L. Minsky|Minsky, Marvin L.]], and [[Seymour A. Papert|Papert, Seymour, A.]] (1988), ''[[Perceptrons]], An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969.  Revised, 1972.  Expanded edition, 1988.
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* Minsky, Marvin L., and Papert, Seymour, A. (1988), ''Perceptrons, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969.  Revised, 1972.  Expanded edition, 1988.
  
 
==Syllabus==
 
==Syllabus==
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===Focal nodes===
 
===Focal nodes===
  
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* [[Inquiry Live]]
 
* [[Inquiry Live]]
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* [[Logic Live]]
 
* [[Logic Live]]
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===Peer nodes===
 
===Peer nodes===
  
{{col-begin}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Boolean-valued_function Boolean-Valued Function @ InterSciWiki]
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* [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function @ MyWikiBiz]
 
* [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function @ MyWikiBiz]
* [http://mathweb.org/wiki/Boolean-valued_function Boolean-Valued Function @ MathWeb Wiki]
 
* [http://netknowledge.org/wiki/Boolean-valued_function Boolean-Valued Function @ NetKnowledge]
 
* [http://wiki.oercommons.org/mediawiki/index.php/Boolean-valued_function Boolean-Valued Function @ OER Commons]
 
{{col-break}}
 
* [http://p2pfoundation.net/Boolean-Valued_Function Boolean-Valued Function @ P2P Foundation]
 
* [http://semanticweb.org/wiki/Boolean-valued_function Boolean-Valued Function @ SemanticWeb]
 
 
* [http://ref.subwiki.org/wiki/Boolean-valued_function Boolean-Valued Function @ Subject Wikis]
 
* [http://ref.subwiki.org/wiki/Boolean-valued_function Boolean-Valued Function @ Subject Wikis]
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* [http://en.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function @ Wikiversity]
 
* [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function @ Wikiversity Beta]
 
* [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function @ Wikiversity Beta]
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===Logical operators===
 
===Logical operators===
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===Related articles===
 
===Related articles===
  
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;]
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{{col-begin}}
 
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{{col-break}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
 
+
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
 
+
{{col-break}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
 
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
 
+
{{col-break}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
 
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* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
 
+
{{col-end}}
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;]
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==Document history==
 
==Document history==
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Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
 
Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
  
{{col-begin}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Boolean-valued_function Boolean-Valued Function], [http://intersci.ss.uci.edu/ InterSciWiki]
{{col-break}}
+
 
* [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function], [http://mywikibiz.com/ MyWikiBiz]
 
* [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function], [http://mywikibiz.com/ MyWikiBiz]
* [http://mathweb.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://mathweb.org/ MathWeb Wiki]
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* [http://planetmath.org/BooleanValuedFunction Boolean-Valued Function], [http://planetmath.org/ PlanetMath]
* [http://planetmath.org/encyclopedia/BooleanValuedFunction.html Boolean-Valued Function], [http://planetmath.org/ PlanetMath]
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* [http://wikinfo.org/w/index.php/Boolean-valued_function Boolean-Valued Function], [http://wikinfo.org/w/ Wikinfo]
* [http://planetphysics.org/encyclopedia/BooleanValuedFunction.html Boolean-Valued Function], [http://planetphysics.org/ PlanetPhysics]
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* [http://en.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://en.wikiversity.org/ Wikiversity]
{{col-break}}
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* [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://beta.wikiversity.org/ Wikiversity Beta]
 
* [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://beta.wikiversity.org/ Wikiversity Beta]
* [http://wikinfo.org/index.php/Boolean-valued_function Boolean-Valued Function], [http://wikinfo.org/ Wikinfo]
 
* [http://textop.org/wiki/index.php?title=Boolean-valued_function Boolean-Valued Function], [http://textop.org/wiki/ Textop Wiki]
 
 
* [http://en.wikipedia.org/w/index.php?title=Boolean-valued_function&oldid=67166584 Boolean-Valued Function], [http://en.wikipedia.org/ Wikipedia]
 
* [http://en.wikipedia.org/w/index.php?title=Boolean-valued_function&oldid=67166584 Boolean-Valued Function], [http://en.wikipedia.org/ Wikipedia]
{{col-end}}
 
  
 
[[Category:Inquiry]]
 
[[Category:Inquiry]]

Latest revision as of 21:16, 5 November 2015

This page belongs to resource collections on Logic and Inquiry.

A boolean-valued function is a function of the type where is an arbitrary set and where is a boolean domain.

In the formal sciences — mathematics, mathematical logic, statistics — and their applied disciplines, a boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses it is understood that the various terms refer to a mathematical object and not the corresponding sign or syntactic expression.

In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value.

Examples

A binary sequence is a boolean-valued function , where . In other words, is an infinite sequence of 0's and 1's.

A binary sequence of length is a boolean-valued function , where

References

  • Brown, Frank Markham (2003), Boolean Reasoning : The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
  • Kohavi, Zvi (1978), Switching and Finite Automata Theory, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.
  • Korfhage, Robert R. (1974), Discrete Computational Structures, Academic Press, New York, NY.
  • Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM.
  • Minsky, Marvin L., and Papert, Seymour, A. (1988), Perceptrons, An Introduction to Computational Geometry, MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988.

Syllabus

Focal nodes

Peer nodes

Logical operators

Related topics

Relational concepts

Information, Inquiry

Related articles

Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.