Re: an odd ambiguity

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
> 
> 
>