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A '''boolean-valued function''', in some usages a '''predicate''' or a '''proposition''', is a function of the type <math>f : X \to \mathbb{B}</math>, where <math>X</math> is an arbitrary set, where <math>\mathbb{B}</math> is a generic 2-element set, typically <math>\mathbb{B} = \left \{ 0, 1 \right \}</math>, and where the latter is frequently interpreted for logical applications as <math>\mathbb{B} = \left \{ false, true \right \}</math>. In ], a '''boolean-valued function''' is a ] <i>f</i> from an arbitrary ] <i>X</i> to a Boolean domain <b>B</b>. A '''boolean domain''' is a 2-element set, typically <b>B</b> = {0, 1}, whose elements are interpreted as ]s, typically 0 = false and 1 = true.


⚫ In the ]s, mathematics, ], ], and their applied disciplines, a boolean-valued function may also be referred to as a '']'', '']'', '']'', or '']''. 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.
A '''boolean domain''' '''B''' is a generic 2-element ], say, '''B''' = {0, 1}, whose elements are interpreted as ]s, typically 0 = false and 1 = true.


⚫ In ] theories of ], a '''truth predicate''' is a predicate on the ]s of a ], 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 ]s, ], ], ], and their applied disciplines, a boolean-valued function may also be referred to as a ], ], ], or ]. 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 ] theories of ], a '''truth predicate''' is a predicate on the ]s of a ], 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.

⚫ ==References==


⚫ == References ==
* ] (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003. * ] (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

* ] (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978. * ] (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.

* ] (1974), ''Discrete Computational Structures'', Academic Press, New York, NY. * ] (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.

* ], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume). * ], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).


==See also== == See also ==

* ]

===Equivalent concepts===

* ]
* ]
* ], in some senses.
* ], in some senses.

===Related concepts===

* ] * ]


Revision as of 01:15, 31 July 2006

In mathematics, a boolean-valued function is a function f from an arbitrary set X to a Boolean domain B. A boolean domain is a 2-element set, typically B = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true.

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.

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 (volume).

See also

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