Boolean-valued function

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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 &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 semiotic 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 $$[k] = \{ 1, 2, \ldots k \}.$$

Focal nodes

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 * Logic Live

Peer nodes

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 * Boolean-Valued Function @ OER Commons


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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

 * Jon Awbrey, &ldquo;Semiotic Information&rdquo;


 * Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;


 * Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;


 * Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;


 * Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;


 * Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;


 * Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;


 * Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;

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, MyWikiBiz
 * Boolean-Valued Function, MathWeb Wiki
 * Boolean-Valued Function, PlanetMath
 * Boolean-Valued Function, PlanetPhysics


 * Boolean-Valued Function, Wikiversity Beta
 * Boolean-Valued Function, Wikinfo
 * Boolean-Valued Function, Textop Wiki
 * Boolean-Valued Function, Wikipedia