Re: an odd ambiguity

> d(n) = |{ d | d||n| }|

You definitely trump my example, even without the absolute value.

    Neil


On Wed, Aug 4, 2021 at 4:07 AM David Farmer <farmer@aimath.org> wrote:

>
> The number theoretic function  d(n)  equals the number of
> positive divisors of the absolute value of  n .  In symbols:
>
> d(n) = |{ d | d||n| }|
>
> Moving left-right and  outside-in:
>
> d is the name of the function on the left
> (that is a common name for that function,  \tau is the other common name)
>
> |A|  cardinality of the set  A
>
> d is a bound variable on the right side
> (it actually is common to do that in this context
>
> | "such that" separator
>
> d|  "divides" relation
>
> |n|  absolute value of  n
> (That is the only silly part of this example, because the
> divisors of  n  are the same as the divisors of  -n ,
> so the absolute value is not actually needed.)
>
>
>
> On Wed, 4 Aug 2021, Paul Libbrecht wrote:
>
> >
> > One more interpretation is the use of the pipe-character as separator
> for coordinates in
> > German’s schoolbooks.
> >
> > E.g.: http://oriesen.ch/doku/Raumgeometrie1S.pdf gives an example of
> that (on the second page):
> > copied from the PDF: Zeichne in der linken Figur die Punkte P ( 6 |3 |3
> ) , Q ( 0 |4 |6 ) und R
> > ( 1 |5 |3 ) ein. (Which has quite a different spacing than in the PDF).
> In this usage and in
> > this work, it seems that the spacing should be the same on both sides.
> >
> > As an “intent”, the whole notation made of “(“, the “|” and “)” should
> be considered. Is this
> > planned in our intents syntax? Should the intent be on the mrow and can
> we avoid to pronounce
> > the pipes?
> >
> > paul
> >
> > On 4 Aug 2021, at 6:27, Neil Soiffer wrote:
> >
> >       We've mentioned how ambiguous "|" can be, but I don't remember
> seeing anyone
> >       mentioning this example:
> > { x   ∣ x  ∣ 10}
> > The set of all x such that x divides 10.
> >
> > In one expression are both the low priority separator "such that" and
> the medium priority
> > relational operator "divides" (both are infix). There are two characters
> that could be
> > used: vertical bar (U+007C) and divides (U+2223).  The Unicode Standard
> indicates that
> > both should be U+2223 (I'm not sure that equivalence is correct)
> >
> > In TeX, there seems to be agreement that the first bar is be \mid.
> However, there seems to
> > be disagreement for what to use for the second bar. Some people suggest
> \mid, others "|",
> > and still others \divides (which only exists in the MnSymbol package
> AFAIK). There are
> > spacing differences and maybe height differences. Using different macros
> means there is a
> > potential semantic distinction if the author actually uses them as
> opposed to using the
> > ASCII "|". A reason TeX distinguishes them is that the spacing around
> the vertical bar
> > differs a little. Someone will surely correct me on this if I'm wrong,
> but the spacing of
> > these two uses is opposite their contextual meaning. TeX considers \mid
> to be a relational
> > operator, but relational operators return boolean values -- \mid is
> really a
> > separator/punctuation. On the other hand, \divides really is a relation
> (m divides n is
> > either true or false), but it is spaced as a binary operator (at least
> in this context).
> > Typographically, this is what is supposed to happen, but it seems
> counter-intuitive. Very
> > strange.
> >
> > What does this mean for MathML? One thing is that in practice, software
> can't be sure the
> > correct symbol is used in MathML (I leave it to someone else to report
> what TeX,
> > ASCIIMath, and WYSIWYG editors use). The other issue is what the
> operator dictionary
> > should say about the spacing and priority for these two symbols.
> Currently they both have
> > the same spacing and priority, but that seems wrong.
> >
> > Thoughts?
> >
> >     Neil
> >
> >
> >