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<font size="3">☞</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]].

In the [[formal science]]s — [[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 [[semiotic]] 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.

==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.

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>

==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.

* [[Zvi Kohavi|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.

* [[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.

==Syllabus==

===Focal nodes===

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* [[Inquiry Live]]
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* [[Logic Live]]
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===Peer nodes===

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* [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]
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* [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://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function @ Wikiversity Beta]
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===Logical operators===

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* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
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* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
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===Related topics===

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* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
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* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
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* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[Zeroth order logic]]
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===Relational concepts===

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* [[Continuous predicate]]
* [[Hypostatic abstraction]]
* [[Logic of relatives]]
* [[Logical matrix]]
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* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
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* [[Relation theory]]
* [[Relative term]]
* [[Sign relation]]
* [[Triadic relation]]
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===Information, Inquiry===

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* [[Inquiry]]
* [[Dynamics of inquiry]]
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* [[Semeiotic]]
* [[Logic of information]]
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* [[Descriptive science]]
* [[Normative science]]
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* [[Pragmatic maxim]]
* [[Truth theory]]
{{col-end}}

===Related articles===

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, “Semiotic Information”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, “Introduction To Inquiry Driven Systems”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, “Prospects For Inquiry Driven Systems”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, “Inquiry Driven Systems : Inquiry Into Inquiry”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, “Propositional Equation Reasoning Systems”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, “Differential Logic : Introduction”]

* [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, “Differential Propositional Calculus”]

* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, “Differential Logic and Dynamic Systems”]

==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.

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* [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]
* [http://planetmath.org/encyclopedia/BooleanValuedFunction.html Boolean-Valued Function], [http://planetmath.org/ PlanetMath]
* [http://planetphysics.org/encyclopedia/BooleanValuedFunction.html Boolean-Valued Function], [http://planetphysics.org/ PlanetPhysics]
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* [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]
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[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Discrete Mathematics]]
[[Category:Logic]]
[[Category:Mathematics]]

Boolean-valued function - Revision history

Jon Awbrey: update

Jon Awbrey: + article