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In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}). [1] [2] Alternative names are switching function, used especially in older computer science literature, [3] [4] and truth function (or logical function), used in logic.
Similarly, a Boolean function of degree at most depends on at most coordinates, making it a junta (a function depending on a constant number of coordinates), where is an absolute constant equal to at least 1.5, and at most 4.41, as shown by Wellens. [5]
Boolean algebra also deals with functions which have their values in the set {0,1}. A sequence of bits is a commonly used example of such a function.
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.
Boolean function; Boolean-valued function; Boolean-valued model; Boolean satisfiability problem; Boolean differential calculus; Indicator function (also called the characteristic function, but that term is used in probability theory for a different concept) Espresso heuristic logic minimizer; Logical matrix; Logical value; Stone duality; Stone ...
In computer science, a Boolean expression is an expression used in programming languages that produces a Boolean value when evaluated. A Boolean value is either true or false.A Boolean expression may be composed of a combination of the Boolean constants True/False or Yes/No, Boolean-typed variables, Boolean-valued operators, and Boolean-valued functions.
There are 2 n+1 symmetric n-ary Boolean functions. Instead of the truth table, traditionally used to represent Boolean functions, one may use a more compact representation for an n-variable symmetric Boolean function: the (n + 1)-vector, whose i-th entry (i = 0, ..., n) is the value of the function on an input vector with i ones.
In contrast, converting between Booleans and integers (or any other types) still required explicit tests or function calls, as in ALGOL 60. This approach (Boolean is an enumerated type) was adopted by most later languages which had enumerated types, such as Modula, Ada, and Haskell.