Ah, but it isn't that simple. Consider f, not as a function from X to Y, but just as a function. In set-theoretic terms, as a set of ordered pairs, such that nothing appears as the first element of any two distinct pairs. Now the domain is the set of elements appearing as the first element of any pair, and this is exactly the domain in any other sense. But the only reasonable definition for the range is the set of all elements appearing in the second element of any ordered pair; and this is the image in the other sense. Another way to look at it: if Z is the image of f: X -> Y, then f can be regarded as a function X -> Z (or X -> W for any W containing Y as a subset), and this is really the same function. So the image (often called the range) is a property of the function, while the "codomain" is a property of the function as viewed in a particular context, namely as a function X -> Y. Franklin T. Adams-Watters -----Original Message----- From: dasimov@earthlink.net When I first learned the rigorous definition of a function f: X -> Y, the terminology was unequivocal: X is the domain of f, and Y is the range of f. The latter is unconditionally the case, irrespective of whether f is onto. Nowadays, so many high school course and ill-considered calculus texts have redefined "range" to mean f(X) -- what I was taught is called the image of f -- that in teaching I've leaned away from using the word range to mean either thing. Now I call the classical range by the term "codomain" (and I still call image "image"). It's a shame when the cognoscenti feel obliged to follow the ignorami. ___________________________________________________ Try the New Netscape Mail Today! Virtually Spam-Free | More Storage | Import Your Contact List http://mail.netscape.com