Boolean-valued function

Function
x ↦ f (x)
History of the function concept
Examples of domains and codomains
  • X {\displaystyle X} → B {\displaystyle \mathbb {B} } , B {\displaystyle \mathbb {B} } → X {\displaystyle X} , B n {\displaystyle \mathbb {B} ^{n}} → X {\displaystyle X}
  • X {\displaystyle X} → Z {\displaystyle \mathbb {Z} } , Z {\displaystyle \mathbb {Z} } → X {\displaystyle X}
  • X {\displaystyle X} → R {\displaystyle \mathbb {R} } , R {\displaystyle \mathbb {R} } → X {\displaystyle X} , R n {\displaystyle \mathbb {R} ^{n}} → X {\displaystyle X}
  • X {\displaystyle X} → C {\displaystyle \mathbb {C} } , C {\displaystyle \mathbb {C} } → X {\displaystyle X} , C n {\displaystyle \mathbb {C} ^{n}} → X {\displaystyle X}
 Classes/properties 
  Constructions
  Generalizations  
  • 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.

See also

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.