[Haskell-cafe] set terminology

Dan Weston westondan at imageworks.com
Fri Feb 15 17:25:02 EST 2008

Since no one else has replied, I will take a stab. This is the 
terminology I have seen/heard:

A mapping in a category is typed. It can map only from a "source" object 
to a "target" object. There may be zero, one, or multiple such mappings 
(functions) from a given source to a given target (but at least one if 
source and target are the same, namely the identity map).

For a particular source and target, where the source and target happen 
not just to be opaque objects but have internal structure (with subset 
operations), mappings are called functions, the source is called the 
domain, and the target is called the codomain.

Elements X in the domain are mapped to some element Y in the codomain. 
The set of all such Y is the range, and the set of all such X is the 
corange. (Wikipedia [1] suggests that there is ambiguity with the word 
"domain", but I have never heard that elsewhere).

Any given subset S of the corange (called a preimage) maps to the 
corresponding image of S, which is a subset of the range. Preimage and 
image apply to singleton sets as well, so (by trivial isomorphism) these 
words apply to mapped elements themselves. In this case, the usual arrow 
symbol gets a little vertical cap on the left end.

In any case, I would not get too hung up on the terminology. It is much 
more important to understand what is meant in any given setting.

[1] http://en.wikipedia.org/wiki/Function_%28mathematics%29


PR Stanley wrote:
> Hi
> domain, source --- are the two different things? I'm sure I read 
> somewhere that the source \subseteq domain in mappings. The same was 
> said about range and target -- target \subseteq range.
> Any ideas?
> Thanks, Paul
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