of a number ''n'' of '''Booleanboolean variables''' ''b''<sub>i</sub> from the two-element [[Booleanboolean algebra]] '''B''' = {0, 1}, and such that ''F'' also takes values in {0, 1}'''B'''. A function on aan generalarbitrary [[___domainset (mathematics)|___domain of a function]]''X'' taking values in {0,'''B''' 1} would beis called a '''''[[boolean|Boolean-valued function]]''''', so that Booleanboolean functions are a special case. Such a function with ___domain {1, 2, 3, ... …} is commonly called a '''binary sequence''', i.e.that is, an [[infinite sequence]] of 0's and 1's;. by By restricting to {1, 2, 3, ... …, ''nk'' } a Booleanboolean function is in a natural way coded by a sequence of length ''nk''.
There are <math>2^{2^n}</math> such functions; these play a basic role in questions of [[complexity theory]] as well as the design of circuits and chips for [[digital computer]]s. The properties of boolean functions play a critical role in [[cryptography]], particularly in the design of [[symmetric key algorithm]]s (see [[S-box]]).
A '''Booleanboolean operation''' on Booleanboolean-valued functions combines values point-wise (for example, by [[XOR]], or other [[Booleanboolean operator]]).
