Function
Function
In mathematics and logic, an n-adic function is a law of correspondence between an ordered set of n things (called arguments of the function, or values of the independent variables) and another thing (the value of the function, or value of the dependent variable), of such a sort that, given any ordered set of n arguments which belongs to a certain domain (the range of the function), the value of the function is uniquely determined. The value ot the function is spoken of as obtained by applying the function to the arguments. The domain of all possible values of the function is called the range of the dependent variable. If F denotes a function and X1, X2, . . . , Xn denote the first argument, second argument, etc., respectively, the notation F(X1, X2, . . . , Xn) is used to denote the corresponding value of the function; or the notation may be [F](X1, X2 . . . , Xn), to provide against ambiguities which might otherwisc arise if F were a long expression rather than a single letter.
In particular, a monadic function is a law of correspondence between an argument (or value of the independent variable) and a value of the function (or value of the dependent variable), of such a sort that, given any argument belonging to a certain domain (the range of the function, or range of the independent variable), the value of the function is uniquely determined. If F denotes a monadic function and X denotes an argument, the notation F(X) is used for the corresponding value of the function.
Instead of a monadic function, dyadic function, etc., one may also speak of a function of one variable, a function of two variables, etc. The terms singulary or unary (= monadic), binary (= dyadic), etc., are also in use. The phrase, “function from A to B” is used in the case of a monadic function to indicate that A and B (or some portion of B) are the ranges of the independent and dependent variables respectively — in the case of a polyadic function to indicate that B (or some portion of B) is the range of the dependent variable while the range of the function consists of ordered sets of n things out of A.
It is sometimes necessary to distinguish between functions in intension and functions in extension, the distinction being that two n-adic functions in extension are considered identical if they have the same range and the same value for every possible ordered set of n arguments, whereas some more severe criterion of identity is imposed in the case of functions in intension. In most mathematical contexts the term function (also the roughly synonymous terms operation, transformation) is used in the sense of function in extension.
(In the case of propositional functions, the distinction between intension and extension is usually made somewhat differently, two propositional functions in extension being identical if they have materially equivalent values for even set of arguments.)
Sometimes it is convenient to drop the condition that the value of a function is unique and to require rather that an ordered set of arguments shall determine a set of values of the function. In this case one speaks of a many-valued function.
Often the word function is found used loosely for what would more correctly be called an ambiguous or undetermined value of a function, an expression containing one or more free variables being said, for example, to denote a function. Sometimes also the word function is used in a syntactical sense — e.g., to mean an expression containing free variables.
See the article Propositional function. — Alonzo Church