Kees
Posts:
136
Registered:
8/24/05
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Re: Why f:domain->codomain instead of f:domain->range?
Posted:
Jan 31, 2006 7:10 AM
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> In <446va0Fme8sU1@news.dfncis.de> Marc Olschok
> <invalid@nowhere.com> writes:
>
> >kj <socyl@987jk.com.invalid> wrote:
> >>
> >>
> >>
> >>
> >> I never understood why the notation
> >>
> >> f:X->Y
> >>
> >> instead of
> >>
> >> f:X->f(X)
> >>
> >> In other words, I don't understand the utility of
> the notion of a
> >> codomain. Why not make the definition of
> "function" be so that
> >> every function is surjective? I'm sure there are
> very good reasons
> >> for this, but I don't see them.
> >>
> >> The only explanation I can think of is that there
> are often times
> >> when it is much easier to describe a function's
> codomain than than
> >> its range. Is this it? Or are there more
> fundamental reasons
> >> behind this practice?
>
> >That is already quite a good reason.
>
> I disagree. Let me put my question differently. In
> the theory of
> sets, a function is typically defined as a set F of
> ordered pairs
> (aka a "relation") such that the first element of
> each pair occurs
> only in one pair in F (i.e. if the pairs (a, b) and
> (a, c) are in
> F, then b = c). Under this definition, what we
> nowadays call f:R->R
> and g:R->[-1, 1], with f(x) = g(x) = sin(x), are
> identical sets.
> But the definition of functions using domains and
> codomains forces
> us to regard f and g as different somehow. What's
> the point?
>
> >> I suspect the answer has to do with category
> theory (or its
> >> antecedents), but this is a wild, ignorant guess.
>
> >Your guess is probably quite good. Specifying the
> codomain explicitely
> >certainly makes the bookkeeping easier once you
> start being interested
> >in composing maps, because the information is
> static:
>
> >(1) two maps f and g are composable iff the codomain
> of f equals the
> >codomain of g.
>
> >(2) if f and g are composable then their composite
> inherits its domain
> >from f and its codomain from g.
>
> >Now, while you could rephrase (1) in terms of range
> and domain, the range
> >of a possible composite is not determined by the
> range of the second map.
> >So the result would in fact lack simplicity and
> symmetry.
>
> OK, this makes more sense. (Though, as was already
> pointed out,
> you probably meant "domain of g" in (1).)
>
The categorical importance of consider all functions and not just surjective ones, goes beyond sipmle book keeping. For example in the category of sets the coproduct of two sets is given by their disjoint union and the canonical injections are not surjective. If you restrcit to the category of sets with surjective functions, there is no coproduct. There are many other properties that Sets have but sets with surjective functions will not have. Categoricaly that is the mais reason to considering all functions and not just the surjective ones.
> Thanks!
>
> kj
>
> --
> NOTE: In my address everything before the first
> period is backwards;
> and the last period, and everything after it, should
> be discarded.
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