Notational difference, functions and mappings, talking about sets and classes

Question

A Function is a set of pairs such that no two pairs have the same first member.

My question summarized: What if I want to consider proper classes of pairs?

The closest question to mine I could find was Is there any difference between mapping and function? I have something different in mind however.

I use to use the term "function" when I'm talking about a relation that maps one set into another, and I use to use the term "mapping when" I have a collection of functions where the collection might not be a set but a proper class.

For something to be called a function (or a mapping) do I need to have domain and image sets? If I want to describe a collection of functions that map sets into other sets, can I still call the whole thing a function? Regardless of that whole thing being a proper class?

Also I need a term that is on the one hand correct, on the other hand easily understandable. Which I guess is one of the reasons I use to use the term mapping. My target audience are computer science, linguistics and philosophy people.

Answer 1

Ups, wikipedia (http://en.wikipedia.org/wiki/Map_%28mathematics%29) seems to help:

Map (mathematics)

In logic

In formal logic, the term is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory.

Which is pretty clear. And thus I think a sufficient answer for my query, and also matches my intuition. Still there could be a more clear answer. Does anybody have a proper reference for this? The referenced wikipedia article could be far more extensive.

So a mapping is some abstract definition of a function, that can be applied to concrete structures or models. A function is an implementation of some mapping. Every function is a mapping but not the other way round (since some classes are not sets).

Intuitively I'd say that also morphisms are some sort of mappings and not the other way round, as functions are some sort of morphisms. Naively written for classes: $\textit{functions}\subsetneq\textit{morphisms}\subsetneq\textit{mappings}$

Answer 2

It's completely standard to say function, even when talking about proper classes. If you want to distinguish this fact, then you should say class function.

I can't say that there's any difference in notation either, except when (for example) sets are denoted by a lowercase letter, and proper classes by capital letters, then a class function should probably adhere to this convention as well.