Line 1: |
Line 1: |
− | 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]].
| + | <font size="3">☞</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. |
| | | |
− | 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.
| + | 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 [[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. | + | 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 <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== | | ==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. | + | * 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=== |
| | | |
− | * [[Zvi Kohavi|Kohavi, Zvi]] (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978. | + | * [[Inquiry Live]] |
| + | * [[Logic Live]] |
| | | |
− | * [[Robert R. Korfhage|Korfhage, Robert R.]] (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.
| + | ===Peer nodes=== |
| | | |
− | * [[Mathematical Society of Japan]], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM. | + | * [http://intersci.ss.uci.edu/wiki/index.php/Boolean-valued_function Boolean-Valued Function @ InterSciWiki] |
| + | * [http://mywikibiz.com/Boolean-valued_function Boolean-Valued Function @ MyWikiBiz] |
| + | * [http://ref.subwiki.org/wiki/Boolean-valued_function Boolean-Valued Function @ Subject Wikis] |
| + | * [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] |
| | | |
− | * [[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.
| + | ===Logical operators=== |
| | | |
− | ==See also== | + | {{col-begin}} |
− | {| | + | {{col-break}} |
− | | valign=top |
| + | * [[Exclusive disjunction]] |
− | * [[Boolean algebra]] | + | * [[Logical conjunction]] |
| + | * [[Logical disjunction]] |
| + | * [[Logical equality]] |
| + | {{col-break}} |
| + | * [[Logical implication]] |
| + | * [[Logical NAND]] |
| + | * [[Logical NNOR]] |
| + | * [[Logical negation|Negation]] |
| + | {{col-end}} |
| + | |
| + | ===Related topics=== |
| + | |
| + | {{col-begin}} |
| + | {{col-break}} |
| + | * [[Ampheck]] |
| * [[Boolean domain]] | | * [[Boolean domain]] |
− | * [[Boolean logic]] | + | * [[Boolean function]] |
− | | valign=top |
| + | * [[Boolean-valued function]] |
| + | * [[Differential logic]] |
| + | {{col-break}} |
| + | * [[Logical graph]] |
| + | * [[Minimal negation operator]] |
| + | * [[Multigrade operator]] |
| + | * [[Parametric operator]] |
| + | * [[Peirce's law]] |
| + | {{col-break}} |
| * [[Propositional calculus]] | | * [[Propositional calculus]] |
| + | * [[Sole sufficient operator]] |
| * [[Truth table]] | | * [[Truth table]] |
| + | * [[Universe of discourse]] |
| * [[Zeroth order logic]] | | * [[Zeroth order logic]] |
− | |}
| + | {{col-end}} |
| | | |
− | ===Equivalent concepts=== | + | ===Relational concepts=== |
| | | |
− | * [[Characteristic function]] | + | {{col-begin}} |
− | * [[Indicator function]] | + | {{col-break}} |
− | * [[Predicate]], in some senses. | + | * [[Continuous predicate]] |
− | * [[Proposition]], in some senses. | + | * [[Hypostatic abstraction]] |
| + | * [[Logic of relatives]] |
| + | * [[Logical matrix]] |
| + | {{col-break}} |
| + | * [[Relation (mathematics)|Relation]] |
| + | * [[Relation composition]] |
| + | * [[Relation construction]] |
| + | * [[Relation reduction]] |
| + | {{col-break}} |
| + | * [[Relation theory]] |
| + | * [[Relative term]] |
| + | * [[Sign relation]] |
| + | * [[Triadic relation]] |
| + | {{col-end}} |
| | | |
− | ===Related concepts=== | + | ===Information, Inquiry=== |
| | | |
− | * [[Boolean function]] | + | {{col-begin}} |
| + | {{col-break}} |
| + | * [[Inquiry]] |
| + | * [[Dynamics of inquiry]] |
| + | {{col-break}} |
| + | * [[Semeiotic]] |
| + | * [[Logic of information]] |
| + | {{col-break}} |
| + | * [[Descriptive science]] |
| + | * [[Normative science]] |
| + | {{col-break}} |
| + | * [[Pragmatic maxim]] |
| + | * [[Truth theory]] |
| + | {{col-end}} |
| + | |
| + | ===Related articles=== |
| + | |
| + | {{col-begin}} |
| + | {{col-break}} |
| + | * [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://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems] |
| + | {{col-break}} |
| + | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction] |
| + | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus] |
| + | * [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems] |
| + | {{col-break}} |
| + | * [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems] |
| + | * [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems] |
| + | * [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry] |
| + | {{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. |
| | | |
− | {{aficionados}}<sharethis />
| + | * [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]] |
| + | [[Category:Peer Educational Resource]] |
| [[Category:Combinatorics]] | | [[Category:Combinatorics]] |
| [[Category:Computer Science]] | | [[Category:Computer Science]] |