I'm not saying that viewing a function as X -> Y is meaningless, just that there is another way to view it. And, yes, questions of whether a function is onto make sense only in this sort of context. Certainly, when talking about all functions from X to Y, one does not mean only the ones that are onto. Regarding defining a specific function, I am thinking in terms of ZF set theory. There, a function is just a set of ordered pairs (of sets) with the requisite uniqueness property. Such a function does not bring with it any concept of its codomain, just the image - which is normally called the range. The only general range would be the set of all sets - which does not exist. Franklin T. Adams-Watters -----Original Message----- From: dasimov@earthlink.net I don't know about this. (For me, "just a function" always includes a domain & range, in the traditional terminology -- but I know what you're saying.) If every function had its codomain by definition shrunken down to its image, then every function would be onto, and questions about whether certain functions are, or could be, onto would become meaningless. But above all: In order to define a specific function, one needs to begin with a set to which the second element of its ordered pairs belong; it is not until after the function is defined that one can even ask what its image is. At that point, you have the option of defining a new function whose ordered pairs have their second elements in the image of the first function. But you also have the option of retaining the original such set (codomain) if there is some reason to do so. ___________________________________________________ Try the New Netscape Mail Today! Virtually Spam-Free | More Storage | Import Your Contact List http://mail.netscape.com