Boolean-valued function: Difference between revisions

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			{{Short description\|A function that outputs either true or false for given inputs.}}
	A '''boolean-valued function''', in some usages a ] or a ], is a ] of the type f : X → '''B''', where X is an arbitrary ] and where '''B''' is a ].
			{{Cleanup rewrite\|date=March 2011}}
			{{Functions}}

	A '''~~boolean~~ ~~domain~~''' '''B''' is a generic 2-element set, ~~say,~~ '''B''' = {0, 1}, whose elements are interpreted as ]s, for example, 0 = false and 1 = true.		A '''Boolean-valued function''' (sometimes called a ] or a ]) is a ] of the type f : X → '''B''', where X is an arbitrary ] and where '''B''' is a ], i.e. a generic two-element set, (for example '''B''' = {0, 1}), whose elements are interpreted as ]s, for example, 0 = ] and 1 = ], i.e., a single ] of ].

	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 the ]s, ], ], ], and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic 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 ] 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 ] 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 ].

⚫	==References==

⚫	* ] (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.

⚫	* ] (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.

⚫	* ], and ] (1988), ''], An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988.

	==See also==		==See also==
			{{colbegin\|colwidth=18em}}
	{{col-begin}}
			* ]
	{{col-break}}
			* ]
	* ]		* ]
	* ]		* ]
	* ]		* ]
	{{col-break}}
	* ]		* ]
	* ]		* ]
	* ]
	{{col-break}}
	* ]		* ]
	{{col-end}}

	===Equivalent concepts===

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

	===Related concepts===

	* ]
	* ]		* ]
			{{colend}}

		⚫	==References==
⚫	]
		⚫	* ] (2003), '''', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
		⚫	* ] (1978), '''', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978. 3rd edition, McGraw–Hill, 2010.
		⚫	* ] (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.
		⚫	* ], and ] (1988), ''], An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988.

		⚫	]
	]
	]
	]

Latest revision as of 02:34, 4 January 2025

A function that outputs either true or false for given inputs.

This article may need to be rewritten to comply with Misplaced Pages's quality standards. You can help. The talk page may contain suggestions. (March 2011)

Function
x ↦ f (x)
History of the function concept
Types by domain and codomain
`X` → 𝔹 𝔹 → `X` 𝔹 → `X` `X` → ℤ ℤ → `X` `X` → ℝ ℝ → `X` ℝ → `X` `X` → ℂ ℂ → `X` ℂ → `X`
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
List of specific functions
v t e

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.

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.

Category:

Boolean algebra

Latest revision as of 02:34, 4 January 2025

See also

References