NetNews Usenet Archive 1992 #30/NN_1992_30.iso/spool/sci/math/16742

Internet Message Format | 1992-12-11 | 2.6 KB

Path: sparky!uunet!wupost!usc!news.aero.org!Aero.org!campbell
From: [e-mail] (Andrew Campbell)
Newsgroups: sci.math
Subject: Re: Function Terminology
Date: 12 Dec 1992 00:26:10 GMT
Organization: The Aerospace Corporation
Lines: 40
Distribution: world
Message-ID: <[e-mail]>
References: <[e-mail]> <[e-mail]> <[e-mail]>
NNTP-Posting-Host: amanuensis.aero.org

In article <[e-mail]>, [e-mail] (Herman Rubin) writes:
|> In article <[e-mail]> [e-mail] (Vaughan R. Pratt) writes:
|> >In article <[e-mail]> [e-mail] (David G Radcliffe) writes:
|> >>Suppose I have a function f: A --> B, and C is a subset of B which
|> >>contains the image set of f.  I define a function g: A --> C by
|> >>setting g(a) = f(a) for all a in A.  Usually, f and g can be considered
|> >>as the same function, but sometimes the distinction is important.
|>
|> >>Is there a standard term or notation for this?
|>

and Rubin replies

|> >While I've not heard of one, one could borrow notation used to make the
|> >analogous distinction for the integer 2 viewed as the real 2. and write
|> >f:A->B as f. and f:A->C as just f without the point.
|>
|> The function is exactly the same.  A function, in whatever foundational
|> system is used, is something which takes arguments in a domain and
|> operates on them.  The image set depends only on f and A.  Which
|> superset of the range is used does not affect the function.
|> --


My view is different. A function has a domain and a co-domain. If the
codomain is changed a different function is obtained (as Pratt wants).
The term `range' is usually used (as Rubin does) for the image of the
domain. In the given setup, I have seen the term `co-restricted' (to C)
used, in analogy to `restricted'. I believe I recall various notations
for this, such as C\f (analogous to f/D or f|D for the restriction of f
to D, a subset of the domain). However, the solution generally applied
in situations where not too many co-restrictions are needed is simply
to name the new function "after f" and specify its co-domain. For example:
"let f':A->C be given by f'(a) = f(a) for all a in A.", or, using the
`wiggly arrow' notation, "... consider f':A --> C, a ~> f(a) ... "
--
 zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz
|L. Andrew Campbell     internet: [e-mail]   |
|M1-102 PO Box 92957    organization: The Aerospace Corp.       |
|Los Angeles CA 90009   telephone: [phone]               |