I'm pretty sure this definition is incorrect. The range of a relation is synonymous with the image of a relation. In this case, the range is a subset of T and not necessarily T itself. However, the codomain is defined as being the set T. --Spoon737 19:06, 16 August 2008 (UTC)
Shrug. Depends on who you read. There's two ways of defining it:
1. Codomain is the entirety of T, Image is the elements of T which are related to by elements of S, Range is the same as Codomain (which is how I've defined it).
2. Codomain is the entirety of T, Image is the elements of T which are related to by elements of S, Range is the same as Image (which is how you've defined it).
A lot of authors shy away from specifying the terms they're going to use because of the potential comprehensional minefield. IMO you come down on one side or the other. I came down on the side of 1. above and that's what I've been using. Feel free to add another entry for "codomain" and amend the one for "range" to indicate that it has two different meanings. Then we can go and change the links to "range" to point them to "codomain" instead, and attempt not to use "range".
I'm taking a break for a while, so I may not be back on line for a few hours, maybe not till tomorrow.--Prime.mover 19:19, 16 August 2008 (UTC)