Boolean-valued function

☞ 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===
* Inquiry Live
* Logic Live
===Peer nodes===
* Boolean-Valued Function @ InterSciWiki
* Boolean-Valued Function @ MyWikiBiz
* Boolean-Valued Function @ Subject Wikis
* Boolean-Valued Function @ Wikiversity
* Boolean-Valued Function @ Wikiversity Beta
===Logical operators===
* Exclusive disjunction
* Logical conjunction
* Logical disjunction
* Logical equality
* Logical implication
* Logical NAND
* Logical NNOR
* Negation
===Related topics===
* 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
===Relational concepts===
* Continuous predicate
* Hypostatic abstraction
* Logic of relatives
* Logical matrix
* Relation
* Relation composition
* Relation construction
* Relation reduction
* Relation theory
* Relative term
* Sign relation
* Triadic relation
===Information, Inquiry===
* Inquiry
* Dynamics of inquiry
* Semeiotic
* Logic of information
* Descriptive science
* Normative science
* Pragmatic maxim
* Truth theory
===Related articles===
* 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
==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.
* Boolean-Valued Function, InterSciWiki
* Boolean-Valued Function, MyWikiBiz
* Boolean-Valued Function, PlanetMath
* Boolean-Valued Function, Wikinfo
* Boolean-Valued Function, Wikiversity
* Boolean-Valued Function, Wikiversity Beta
* Boolean-Valued Function, Wikipedia
Category:Inquiry
Category:Open Educational Resource
Category:Peer Educational Resource
Category:Combinatorics
Category:Computer Science
Category:Discrete Mathematics
Category:Logic
Category:Mathematics