Talk:Injection to Image is Bijection

question answered

In response to the "questionable" template invocation:

"Restriction of Injection is Injection doesn't deal with codomain but is invoked like it does"

I have amended Restriction of Injection is Injection so that it does - please feel free to check, then this questionable can be cleared. --prime mover (talk) 10:33, 24 January 2013 (UTC)

You say just "restriction", parenthetically. Should this be "codomain restriction" or some such? --Dfeuer (talk) 10:57, 24 January 2013 (UTC)

No it shouldn't. Nothing's parenthetical. --prime mover (talk) 11:04, 24 January 2013 (UTC)

There is value in the distinction between "codomain restriction" and "domain restriction" because it resolves the inherent ambiguity resulting from established mathematical lingo for these. So I encourage this development - the issue has resided in the back of my mind for some time now. Admittedly this is liable for hypocrisy since I'm not inclined to actually go and do that myself. How I wish for an FTP connection from my brain to the wiki. That'd speed up the content generation for PW a lot, and take the burden of trivialities off my shoulders. --Lord_Farin (talk) 11:10, 24 January 2013 (UTC)

Alas, less talking, more results. Off we go. --Lord_Farin (talk) 11:12, 24 January 2013 (UTC)

I can't see it myself, as the way restriction is defined, both the subset of the domain and codomain are speficied, with the added lagniappe that if you specify just the domain restriction, the codomain is automatically specified to be restricted to the image of that restricted domain. Having a specific definition of "codomain restriction" seems to me like overcomplication. --prime mover (talk) 11:46, 24 January 2013 (UTC)

The $\restriction$ notation is unambiguous. In what sense is the codomain automatically specified? I've never heard of such a thing. I am accustomed to the term "restriction" being used for the domain unless otherwise specified, so I would think "The restriction of $\sin$ to $[-1,1]$ would mean $\restriction_{[-1,1] \times \R}\sin$. --Dfeuer (talk) 11:51, 24 January 2013 (UTC)

The definition as given on restriction specifically gives the definition for both domain and codomain. By specifying the full cartesian product of the specific subsets of both domain and codomain, one does not need to create a separate definition for "codomain restriction". The codomain is "automatically specified" by the second factor in the $\restriction_{A \times B}$. Furthermore, by the definition as given, $f \restriction_A$ unqualified specifically means $f \restriction_{A \times f \left({A}\right)}$ so to need to refer to a "codomain restriction" is superfluous.

If your understanding of a restriction differs from the above, then bear this in mind, and (if necessary) add an "also defined as" to the page in question. --prime mover (talk) 12:43, 24 January 2013 (UTC)