Difference between revisions of "Boolean-valued function"

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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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A '''boolean domain''' '''B''' is a generic 2-element set, say, '''B'''&nbsp;=&nbsp;{0,&nbsp;1}, whose elements are interpreted as [[logical value]]s, for example, 0&nbsp;=&nbsp;false and 1&nbsp;=&nbsp;true.
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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 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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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 [[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.
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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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* [[Zvi Kohavi|Kohavi, Zvi]] (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970.  2nd edition, McGraw–Hill, 1978.
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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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* [[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.
  
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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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* 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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* [[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.
  
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==See also==
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==Syllabus==
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{|
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| valign=top |
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===Focal nodes===
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* [[Boolean algebra]]
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* [[Inquiry Live]]
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* [[Logic Live]]
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 +
===Peer nodes===
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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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===Logical operators===
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{{col-begin}}
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{{col-break}}
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* [[Exclusive disjunction]]
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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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===Relational concepts===
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{{col-begin}}
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{{col-break}}
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* [[Continuous predicate]]
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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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===Information, Inquiry===
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{{col-begin}}
 +
{{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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===Equivalent concepts===
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===Related articles===
  
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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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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
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* [[Proposition]], in some senses.
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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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===Related concepts===
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==Document history==
  
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* [[Boolean function]]
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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.
  
−
{{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]
 +
* [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function], [http://mywikibiz.com/ MyWikiBiz]
 +
* [http://planetmath.org/BooleanValuedFunction Boolean-Valued Function], [http://planetmath.org/ PlanetMath]
 +
* [http://wikinfo.org/w/index.php/Boolean-valued_function Boolean-Valued Function], [http://wikinfo.org/w/ Wikinfo]
 +
* [http://en.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://en.wikiversity.org/ Wikiversity]
 +
* [http://beta.wikiversity.org/wiki/Boolean-valued_function Boolean-Valued Function], [http://beta.wikiversity.org/ Wikiversity Beta]
 +
* [http://en.wikipedia.org/w/index.php?title=Boolean-valued_function&oldid=67166584 Boolean-Valued Function], [http://en.wikipedia.org/ Wikipedia]
  
 +
[[Category:Inquiry]]
 +
[[Category:Open Educational Resource]]
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[[Category:Peer Educational Resource]]
 
[[Category:Combinatorics]]
 
[[Category:Combinatorics]]
 
[[Category:Computer Science]]
 
[[Category:Computer Science]]

Latest revision as of 21:14, 5 November 2015

☞ This page belongs to resource collections on Logic and Inquiry.

A boolean-valued function is a function of the type \(f : X \to \mathbb{B},\) where \(X\!\) is an arbitrary set and where \(\mathbb{B}\) 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 \(f : \mathbb{N}^+ \to \mathbb{B}\), where \(\mathbb{N}^+ = \{ 1, 2, 3, \ldots \},\). In other words, \(f\!\) is an infinite sequence of 0's and 1's.

A binary sequence of length \(k\!\) is a boolean-valued function \(f : [k] \to \mathbb{B}\), where \([k] = \{ 1, 2, \ldots k \}.\)

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

Template:Col-breakTemplate:Col-breakTemplate:Col-end

Related topics

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Relational concepts

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Information, Inquiry

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Related articles

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

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.

Retrieved from "https://mywikibiz.com/index.php?title=Boolean-valued_function&oldid=466831"