Boolean-valued function

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A boolean-valued function is a function of the type  f : X -> B , {\displaystyle f:X\to \mathbb {B} ,} where  X {\displaystyle X\\!} is an arbitrary set and where  B {\displaystyle \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

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A binary sequence is a boolean-valued function  f : N + -> B {\displaystyle f:\mathbb {N} ^{+}\to \mathbb {B} } , where  N + = { 1 , 2 , 3 , … } , {\displaystyle \mathbb {N} ^{+}=\\{1,2,3,\ldots \\},} . In other words,  f {\displaystyle f\\!} is an infinite sequence of 0's and 1's.

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

## References

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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.

* 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

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### Focal nodes

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* Inquiry Live
* Logic Live

### Peer nodes

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* Boolean-Valued Function @ InterSciWiki
* Boolean-Valued Function @ Subject Wikis
* Boolean-Valued Function @ Wikiversity
* Boolean-Valued Function @ Wikiversity Beta

### Logical operators

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* Exclusive disjunction
* Logical conjunction
* Logical disjunction
* Logical equality

* Logical implication
* Logical NAND
* Logical NNOR
* Negation

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### Related topics

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* Ampheck
* Boolean domain
* Boolean function
* Boolean-valued function
* Differential logic

* Logical graph
* Minimal negation operator
* Multigrade operator
* Parametric operator
* Peirce's law

* 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

* Relation
* Relation composition
* Relation construction
* Relation reduction

* Relation theory
* Relative term
* Sign relation
* Triadic relation

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### Information, Inquiry

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* Inquiry
* Dynamics of inquiry

* Semeiotic
* Logic of information

* Descriptive science
* Normative science

* Pragmatic maxim
* Truth theory

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### Related articles

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* Cactus Language
* Futures Of Logical Graphs
* Propositional Equation Reasoning Systems

* Differential Logic : Introduction
* Differential Propositional Calculus
* Differential Logic and Dynamic Systems

* Prospects for Inquiry Driven Systems
* Introduction to Inquiry Driven Systems
* Inquiry Driven Systems : Inquiry Into Inquiry

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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.

* Boolean-Valued Function, InterSciWiki
* Boolean-Valued Function, PlanetMath
* Boolean-Valued Function, Wikinfo
* Boolean-Valued Function, Wikiversity
* Boolean-Valued Function, Wikiversity Beta
* Boolean-Valued Function, Wikipedia