Clippings

Specification A function is a mathematical entity with the following properties:

• f has domain and codomain, each of which must be a set.
• For every element x of the domain, f has a value at x, which is an element of the codomain and is denoted f(x).
• The domain, the codomain, and the value f(x) for each x in the domain are all determined completely by the function.
• Conversely, the data consisting of the domain, the codomain, and the value f(x) for each element x of the domain completely determine the function f.

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Functions in theory and practice

The concept of function can be explicity defined in terms of its domain, codomain and graph. Precisely, a function f : S → T could be defined as an ordered triple (S, Γ, T) with the property that Γ is a subset of the cartesian product S × T with the functional property (Γ is the graph of f). Then for x ∈ S, f(x) is the unique element y ∈ T for which (x,y) ∈ Γ. Such a definition clearly satisfies [the specification above].

The description of functions in [the specification above] is closer to the way a matematician thinks of a function than the definition [in the preceding paragraph]. For a matematician, a function has a domain and a codomain, and if x is in the domain, then there is a well defined value f(x) in the codomain. It is wrong to think that a function is actually an ordered triple as described in the preceding paragraph in the same sense that it is wrong for a programmer writing in a high level language to think of the numbers he deals with as being expressed in binary notation. The possible definition of functon in the preceding paragraph is an implementation of the specification for function, and just as with program specifications the expectation is that one normally works with the specification, not the implementation, in mind.