Re: Empty set

On Thu, Sep 12, 2013 at 09:06:54PM +0300, Jukka K. Korpela wrote:
> >And below the university level Germans
> >write { }, which I like better.

> The notation { } is quite correct.

IMO, in math texts the correctness is significantly less important
than being not ambiguous. (It is practically impossible to produce a
text without misprints and/or minor problems, so the readers are quite
accustomed to making corrections — as far as the context allows
EXACTLY ONE correction.)

The situation with {} is very similar to the situation with 0̸ for
the empty set and with \ for set subtraction. The Knuth's version of TeX
was designed for typesetting his books, and he (probably) did not
encounter situations where the meaning of these symbols is ambiguous.

When AMS was designing AmSTeX, the editors had experience with more
types of text, and knew solutions for these ambiguity problems: use
visually unique symbols \varnothing and \smallsetminus. (Frankly
speaking, this is my reconstruction; the symbols may have been
introduced earlier than AmSTeX.)

So:

  Is {} empty set, a Poisson bracket, or the function of taking the
  fractional part?

  Is 0̸ the empty set, or ϕ?

  Is \ the set-difference, or left-quotient?

So as far as I understand, the mathematicians who would use 0̸ and \ for ∅ and
∖ are the same people who would use DOT DOT DOT for ellipsis and won't use
\mid in {x∈ℝ||x|<1}. (\mid would typeset it similar to
  {x∈ℝ | |x|<1}
.) In short: people who should know better.

> It just isn’t an atomic symbol
> for the empty set but an expression consisting of the two characters
> “{” and “}”, with a list (here, an empty list) of elements between
> them.

In my book, atomicity is a secondary consideration. As far as an
OpenType font typesets⁽¹⁾ "DOT DOT DOT" the same as "…", and the line
breaking algorithm won't break inside "DOT DOT DOT", I would be happy
with the non-atomic version.

⁽¹⁾ Well, one good application of … not covered by OpenType is to
    visually distinguish "… ." from ". …". (Ellipsis inside a
    sentence vs ellipsis inside a paragraph.)

Hope this helps,
Ilya
Received on Thu Sep 12 2013 - 15:47:14 CDT