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Boolean-valued function (view source)

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A '''boolean-valued function''', in some usages a '''predicate''' or a '''proposition''', is a [[function (mathematics)|function]] of the type ''f'' : ''X'' → '''B''', where ''X'' is an arbitrary [[set]] and where '''B''' is a [[boolean domain]].
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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
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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.
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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]].
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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 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.
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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.
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==Examples==
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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.
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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==
 
==References==
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* [[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.
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* Kohavi, Zvi (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970.  2nd edition, McGraw–Hill, 1978.
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* Korfhage, Robert R. (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.
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* Mathematical Society of Japan, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993.  Cited as EDM.
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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.
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==Syllabus==
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===Focal nodes===
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* [[Zvi Kohavi|Kohavi, Zvi]] (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970.  2nd edition, McGraw–Hill, 1978.
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* [[Inquiry Live]]
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* [[Logic Live]]
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* [[Robert R. Korfhage|Korfhage, Robert R.]] (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.
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===Peer nodes===
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* [[Mathematical Society of Japan]], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM.
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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]
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* [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]
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* [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function @ Wikiversity Beta]
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* [[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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===Logical operators===
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==See also==
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{{col-begin}}
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{|
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{{col-break}}
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| valign=top |
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* [[Exclusive disjunction]]
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* [[Boolean algebra]]
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* [[Logical conjunction]]
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* [[Logical disjunction]]
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* [[Logical equality]]
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{{col-break}}
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* [[Logical implication]]
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* [[Logical NAND]]
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* [[Logical NNOR]]
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* [[Logical negation|Negation]]
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{{col-end}}
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===Related topics===
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{{col-begin}}
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{{col-break}}
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* [[Ampheck]]
 
* [[Boolean domain]]
 
* [[Boolean domain]]
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* [[Boolean logic]]
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* [[Boolean function]]
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| valign=top |
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* [[Boolean-valued function]]
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* [[Differential logic]]
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{{col-break}}
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* [[Logical graph]]
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* [[Minimal negation operator]]
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* [[Multigrade operator]]
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* [[Parametric operator]]
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* [[Peirce's law]]
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{{col-break}}
 
* [[Propositional calculus]]
 
* [[Propositional calculus]]
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* [[Sole sufficient operator]]
 
* [[Truth table]]
 
* [[Truth table]]
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* [[Universe of discourse]]
 
* [[Zeroth order logic]]
 
* [[Zeroth order logic]]
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|}
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{{col-end}}
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===Equivalent concepts===
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===Relational concepts===
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* [[Characteristic function]]
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{{col-begin}}
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* [[Indicator function]]
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{{col-break}}
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* [[Predicate]], in some senses.
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* [[Continuous predicate]]
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* [[Proposition]], in some senses.
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* [[Hypostatic abstraction]]
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* [[Logic of relatives]]
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* [[Logical matrix]]
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{{col-break}}
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* [[Relation (mathematics)|Relation]]
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* [[Relation composition]]
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* [[Relation construction]]
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* [[Relation reduction]]
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{{col-break}}
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* [[Relation theory]]
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* [[Relative term]]
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* [[Sign relation]]
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* [[Triadic relation]]
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{{col-end}}
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===Related concepts===
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===Information, Inquiry===
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* [[Boolean function]]
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{{col-begin}}
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{{col-break}}
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* [[Inquiry]]
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* [[Dynamics of inquiry]]
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{{col-break}}
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* [[Semeiotic]]
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* [[Logic of information]]
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{{col-break}}
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* [[Descriptive science]]
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* [[Normative science]]
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{{col-break}}
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* [[Pragmatic maxim]]
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* [[Truth theory]]
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{{col-end}}
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===Related articles===
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{{col-begin}}
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{{col-break}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
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* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
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* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
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{{col-break}}
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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]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
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{{col-break}}
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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]
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* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
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{{col-end}}
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==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.
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{{aficionados}}<sharethis />
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* [http://intersci.ss.uci.edu/wiki/index.php/Boolean-valued_function Boolean-Valued Function], [http://intersci.ss.uci.edu/ InterSciWiki]
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* [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function], [http://mywikibiz.com/ MyWikiBiz]
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* [http://planetmath.org/BooleanValuedFunction 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]
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* [http://en.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://en.wikiversity.org/ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://beta.wikiversity.org/ Wikiversity Beta]
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* [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]]
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[[Category:Open Educational Resource]]
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[[Category:Peer Educational Resource]]
 
[[Category:Combinatorics]]
 
[[Category:Combinatorics]]
 
[[Category:Computer Science]]
 
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