notation: sideways U means subset, proper or not?

[Gary Russel /ADVISOR L. FAUSETT]

As long as all participants agree, "any" notation will do,
but who started the practice of using a sideways U (open to
the right) to mean "is a subset of" rather than the more
restrictive "is a proper subset of"?

If one draws a line under the "is less than" symbol (<), one
has the "is less than or equal to" symbol.
Therefore, if one draws a line under the "is a proper subset of"
symbol (sideways U, open to the right), one has the
"is a proper subset of or equal to" symbol.
"Is a proper subset of or equal to" means "is a subset of."

I've also heard of nonnegative numbers being considered positive,
with the numbers greater than zero being called "strictly positive."

[Paul Tanenbaum]

Even the names of the LaTeX commands agree with you that using
\subseteq to mean (1) any subset (possibly proper) and \subset to mean
(2) only a proper subset is a better way to go. Unfortunately, there
are many authors who stil use \subset for meaning (1). So while I
always use the above-described better notation, I also explicitly
add a notice that \subset for me implies proper.
Paul

[Herman Rubin]

In article <4omqi6$o...@condor.cs.jhu.edu>,

There is a substantial amount of arbitrariness in the notation.
Some use what you have stated, and some use the subset sign
with a not-equal below it for proper subset. Most people
writing in set theory do not use the equal sign.

The names given in LaTeX are rather arbitrary, as is the whole
shebang. I have complained about not having a plain multifont
WYSIWYG setup; nobody should have to type that many characters
to get a mathematical symbol in.

--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558

[G. A. Edgar]

In article <4omqi6$o...@condor.cs.jhu.edu>, p...@condor.cs.jhu.edu (Paul
Tanenbaum) wrote:

> Even the names of the LaTeX commands agree with you that using
> \subseteq to mean (1) any subset (possibly proper) and \subset to mean
> (2) only a proper subset is a better way to go.

Then there are \subsetneq and \subsetneqq for "subset but not equal"

[D. J. Bernstein]

Herman Rubin <hru...@b.stat.purdue.edu> wrote:
> There is a substantial amount of arbitrariness in the notation.

Yes, but some notations are more arbitrary than others.

< and <= are consistent with the established notations for real numbers.
</= and < are not.

> Most people writing in set theory do not use the equal sign.

Have you collected statistics on this? I just checked a few set theory
books (not yours), and they all use < and <=. In fact, I'd guess that
the inconsistent notations are more popular in number theory than they
are in foundations.

This reminds me of the blackboard-bold issue. People who use Bbb claim
that it's the de facto standard, but a survey of 21 books showed that
bold is twice as popular as Bbb.

---Dan

[Toby Bartels]

Gary Russel <ram8...@zach.fit.edu> wrote:

>If one draws a line under the "is less than" symbol (<), one
>has the "is less than or equal to" symbol.
>Therefore, if one draws a line under the "is a proper subset of"
>symbol (sideways U, open to the right), one has the
>"is a proper subset of or equal to" symbol.
>"Is a proper subset of or equal to" means "is a subset of."

I like this. We should have two symbols, since there are two concepts.
An underscore is simpler than an underscore with a slash through it.
Thus, the adorned symbol should take an underscore and the loose meaning,
leaving the unadorned symbol to take the strict meaning.

>I've also heard of nonnegative numbers being considered positive,
>with the numbers greater than zero being called "strictly positive."

I like this too. We should have two phrases, since there are two concepts.
`strictly' is simpler than `or equal to'.
Thus, the adorned phrase should take `strictly' and the strict meaning,
levaing the unadorned phrase to take the loose meaning.

That's how I like it, at least.

-- Toby
to...@ugcs.caltech.edu