I was wondering what a number sign (i.e. "#") means when put in front
of a designation of a set of elements. For example, what does #S mean
where S is a finite set of integers? Is it just the number of elements
of the set? If so, then why not just use |S|, the cardinality (=
number of elements) of the finite set S?


Please try to get back to me ASAP. It's 5am already, am writing a
paper due around noon, and would like to get some sleep.

Hi!!

The symbol # is used to denote the number of elements of a set, for example 
#S = 4 means that the set S has four elements.
This is one of the several different notations used nowadays.
See the following pages to see how this notation is used:
"Cardinal numbers":
http://mathcircle.berkeley.edu/BMC3/infinity/node2.html

"The Arithmetic of Cardinal Numbers":
http://www2.sjsu.edu/faculty/watkins/cardinals.htm

"Transfinite Numbers and Set Theory":
http://www.math.utah.edu/~alfeld/math/sets.html


Other common notations used are:
|S| 
"Cardinal number":
http://en.wikipedia.org/wiki/Cardinal_number

"Cardinal Number":
http://mathworld.wolfram.com/CardinalNumber.html


card(S)
"A Hierarchy of Infinity - Cardinal Numbers":
http://www.shu.edu/projects/reals/infinity/answers/cardalg.html

"CARDINAL NUMBERS":
http://www.ii.com/math/cardinals/


C(S)  (less common than the others)
"CARDINAL NUMBERS":
At page 4 you can read "Notation: The cardinality of a set, a, is
indicated by C(a)."
http://www.uwec.edu/andersrn/SETSIV.pdf


You can use this definition:
m is a cardinal number if and only if there is a set s such that |s| =
m (or #S = m, or card(s) = m, or C(s) = m).


I hope this helps you. If you find something unclear or incomplete in
this answer please let me know by using the clarification feature
before rate this answer, I will gladly respond all your requests.

Best regards.
livioflores-ga