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Re: on reading ``o(1)''

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Ben Bacarisse <ben.usenet@bsb.me.uk> writes:

> Daniel Bastos <dbastos@toledo.com> writes:
>
>> Ben Bacarisse <ben.usenet@bsb.me.uk> writes:
>>
>>> Daniel Bastos <dbastos@toledo.com> writes:
>>>
>>>> Suppose f in o(1).  By definition, there's a natural number n(0) such
>>>> that for any c > 0
>>>>
>>>>   0 <= f < c * 1 = c
>>>>
>>>> for any n >= n(0).  (AFAIK, c is a real number.)  That seems to imply
>>>> that f = 0 because that's the only thing that could be less than any 
>>>> c > 0.
>>>
>>> Not exactly.  First, I would not write n(0) so it looks like a function.
>>
>> This notation is great!  Let me defend it.  Consider n as a function
>> that takes an argument 0 and gives you the constant ``n0'' --- as you
>> like to write.  Functions can give anything you prefers to read. :-)
>
> I respectfully disagree.  By all means make n a function but make it a
> function of something useful.  As you use the notation, it is a
> single-point function.  What are n(1), n(-e)?  

These are not defined at this point.  There are various ways to define a
function.  I can define a function by giving you a table of values.
Such function is defined on those values and nowhere else.  

> What value does making it a function have when there is only one point
> in the co-domain?

Well, one value is in using it as USENET notation. :-)

> Secondly (and this is more important) you use n as not a function at the
> very same time:  "for any n >= n(0)".  Now it's possible to unravel
> that but, formally, the quantifier binds n in both locations.  At the
> very least, use two names:  "there is an m(0) such that for all n >=
> m(0) ...".

I think you're correct here, but you're being pedantic, as a professor
of mine would put it.  Speaking of him, let tell you a little story.

One time I called this professor of mine's attention because he was
writing f^-1 to mean the preimage of some set element.  I remarked that
our textbook was wisely using a notation such as f^-| --- imagine a
little symbol that looks like a T rotated 90 degress clockwise.  The
author himself commented that that notation was to distinguish preimages
from inverse functions.

Looking back now in retrospective, I agree with my professor, without
disagreeing with the textbook author.  He explained that being pedantic
is fine, but it's also kind of pointless when you're talking to people
who understand what you're saying.  His words were something like this.
``If you don't know what's going on, it doesn't quite matter how I'm
saying.  You don't understand it anyway.  If you do know what's going
on, then there's no problem in reusing notation because you know what I
mean anyhow.''

I think he's right.  Obviously, if you write f^-1(a) and someone gets
confused or something, asks what you mean, then you can just explain
it's not the value of applying the inverse function of f on a, but
rather the subset of the domain of f for which any element is mapped to
a --- when f is applied to it.

Brian M. Scott himself would actually make a distinction between
preimage and inverse functions.  For preimages, he'd write f^-1[a],
saving f^-1(a) for the inverse function.

My professor would write f^-1(a) for both.

I think I'd go with Brian M. Scott, but I certainly wouldn't call my
professor's attention again.  Because he's right.

(*) Back to ``n''

When n is a dummy variable, I think anything is fine, even overloading n
--- because it's clear that while n(0) is a fixed object, n is a dummy
variable.  I wouldn't write for all n > n.  But I think n > n(0) is
unambiguous.  I agree this might confuse some people.  If it does, I'll
change it right away.  When it does not, I could go back to it.  I can
go back and forth.  (For instance, I can switch languages speaking to a
foreigner, if I can speak his language.)

> And to return to the first point, if you want to make it a function,
> make it a useful one.  The n0 in question depends on c so make it a
> function of c:
>
>   For any c > 0 there is a natural number n(c) such that, for all k >=
>   n(c), 0 <= f(k) < ck.

Here you're perfectly correct.  This is really how it should be.  That
does inform the reader.  It's what I do when I write epsilon-delta
proofs, because a delta is usually dependent on the epsilon.  (And that
I picked up with Paul Sperry.)

Of course, if I use two constants there, I'd have to write n(c,0) and
then n(c, 1).  To me that's fine.

> <snip>
>> I picked this notation up with Brian M. Scott, an admirable
>> mathematician who used to hang out at alt.algebra.help.
>
> I strongly suspect he'd agree on this.  Making things into functions is
> a great idea (and too few mathematician do it, relying instead on
> subscripts and context to make it clear that this or that quantity is
> linked to some other) but n is not a function when writing "for all n >
> ..." whatever goes in the ...

After Brian M. Scott, when I look at subscripts now in texts, what I
actually imagine is a function with the subscript being its argument. :-)

I think this is about communication.  My way exchanging information is
to adapt myself to whom I'm speaking to.

Thank you!

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Thread

on reading ``o(1)'' Daniel Bastos <dbastos@toledo.com> - 2017-03-24 21:29 -0300
  Re: on reading ``o(1)'' Kaz Kylheku <336-986-7787@kylheku.com> - 2017-03-25 02:40 +0000
  Re: on reading ``o(1)'' Michael Forsythe <forsythe@example.com> - 2017-03-25 05:51 +0000
    Re: on reading ``o(1)'' Ben Bacarisse <ben.usenet@bsb.me.uk> - 2017-03-26 22:14 +0100
      Re: on reading ``o(1)'' Michael Forsythe <forsythe@example.com> - 2017-03-28 10:32 +0000
      Re: on reading ``o(1)'' Michael Forsythe <forsythe@example.com> - 2017-03-28 10:33 +0000
  Re: on reading ``o(1)'' Ben Bacarisse <ben.usenet@bsb.me.uk> - 2017-03-26 22:11 +0100
    Re: on reading ``o(1)'' Daniel Bastos <dbastos@toledo.com> - 2017-03-28 21:05 -0300
      Re: on reading ``o(1)'' Ben Bacarisse <ben.usenet@bsb.me.uk> - 2017-03-29 14:23 +0100
        Re: on reading ``o(1)'' Daniel Bastos <dbastos@toledo.com> - 2017-04-03 09:40 -0300

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