<P> Functions are commonly defined as a type of relation . A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y . A function from X to Y can be described as a relation from X to Y that is left - total and right - unique . However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions . Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets . The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation . </P> <P> Many operations in set theory, such as the power set, have the class of all sets as their domain, and therefore, although they are informally described as functions, they do not fit the set - theoretical definition outlined above, because a class is not necessarily a set . However some definitions of relations and functions define them as classes of pairs rather than sets of pairs and therefore do include the power set as a function . </P> <P> In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single - valued relations . For example, the function f such that f (x) = 1 / x does not define a value for x = 0, since division by zero is not defined . Hence f is only a partial function from the real line to the real line . The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph . </P> <P> In other parts of mathematics, non-single - valued relations are similarly conflated with functions: these are called multivalued functions, with the corresponding term single - valued function for ordinary functions . </P>

Whats a function and whats not a function