Boolean-valued function

Function

x ↦ f (x)
Examples by domain and codomain
| X| → | B,| B | →| X, |  Bn| →| B

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X| → | Z,| Z | →| X
X| → | R,| R | →| X, | Rn| →| X
X| → | C,| C | →| X, | Cn| →| X
Classes/properties
Constant * Identity * Linear * Polynomial * Rational * Algebraic * Analytic * Smooth * Continuous * Measurable * Injective * Surjective * Bijective
Constructions
Restriction * Composition * λ * Inverse
Generalizations
Partial * Multivalued * Implicit

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A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information.

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 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.

## See also

* Bit
* Boolean data type
* Boolean algebra (logic)
* Boolean domain
* Boolean logic
* Propositional calculus
* Truth table
* Logic minimization
* Indicator function
* Predicate
* Proposition
* Finitary boolean function
* Boolean function

## 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. 3rd edition, McGraw–Hill, 2010.
* 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.

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