Boolean function

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. Boolean functions are the subject of Boolean algebra and switching theory.[5]

A binary decision diagram and truth table of a ternary Boolean function

A Boolean function takes the form , where is known as the Boolean domain and is a non-negative integer called the arity of the function. In the case where , the function is a constant element of . A Boolean function with multiple outputs, with is a vectorial or vector-valued Boolean function (an S-box in cryptography).[6]

There are different Boolean functions with arguments; equal to the number of different truth tables with entries.

Every -ary Boolean function can be expressed as a propositional formula in variables , and two propositional formulas are logically equivalent if and only if they express the same Boolean function.

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