A boolean-valued function is a function of the type where is an arbitrary set and where is a boolean domain.
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 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.
A binary sequence is a boolean-valued function , where . In other words, is an infinite sequence of 0's and 1's.
A binary sequence of length is a boolean-valued function , where
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