RFN (Request for Notation)

RFN (Request for Notation)

[David B. Benson]

Dear Colleagues,

I am preparing notes for a sophomore (second year)
course, and I am finding several places wherein
I can find no standard notation.  Your suggestions
will be most appreciated.

(1) Everybody knows that {(n,n+1) | n \in Z}
    is the <<successor>> relation.  What to call
    the corresponding idea when the underlying
    order is only a partial order?
      I am currently using <<followers>> as in
    phrases such as
    ``...the followers relation for the subsets
    of the finite set {a, b, c}...''
    in which ({a},{a,b}) is in the followers
    relation, but {{a},{a,b,c}) is not, anymore
    that (n,n+2) is in the successor relation.

      Is there a better word than <<followers>>?

(2) I need a snappy name for an order pair in
    a relation R.  The books I have seem to just
    say ``...the ordered pair (x,y) in relation R...''
    The problem is that there are many uses of
    ordered pairs, and this is a specific use,
    a description of the fact that 
       x is R-related to y
    by the fact that (x,y) \in R.
      The word ``association'' will not do as this
    has other meaning in computer science.  I am
    considered <<relator>> for an order pair in
    a relation, but have the impression that this
    word has been used for other purposes in the
    literature.

(3) I badly need a good name for the sets
        Nat_k = {n \in Nat | n < k }
    These are widely used and I am surprised that
    there is no satisfactory name in wide-spread use.
      These are NOT the sets Z_k = Z mod k,
    although the Nat_k form a system of distinct,
    canonical representatives for the Z_k.
      These are the set of array indices in computer
    languages such as C and SML.  In this use, the
    Nat_k have nothing whatsoever to do with Z_k
    and I certainly do not want to confuse the students!

Thank you in advance for any and all suggestions,
David
--
Professor David B. Benson                                (509) 335-2706
School of EE and Computer Science (EME 102A)             (509) 335-3818 fax
PO Box 642752, Washington State University               dbenson@eecs.wsu.edu
Pullman WA 99164-2752   U.S.A.

Re: RFN (Request for Notation)

[Justin Pearson]

David B. Benson wrote:
> Dear Colleagues,
> 
> I am preparing notes for a sophomore (second year)
> course, and I am finding several places wherein
> I can find no standard notation.  Your suggestions
> will be most appreciated.
> 
> 
> (2) I need a snappy name for an order pair in
>     a relation R.  The books I have seem to just
>     say ``...the ordered pair (x,y) in relation R...''

Depends what you want to talk about. If you want to talk about R as
simple a set then I would normally  refer to an
element of R a tuple of R where the context tells you that
the tuple is of length two. 



Regards



Justin

Re: RFN (Request for Notation)

[Paul Levy]

> 
> (3) I badly need a good name for the sets
>         Nat_k = {n \in Nat | n < k }
>     These are widely used and I am surprised that
>     there is no satisfactory name in wide-spread use.
>       These are NOT the sets Z_k = Z mod k,
>     although the Nat_k form a system of distinct,
>     canonical representatives for the Z_k.
>       These are the set of array indices in computer
>     languages such as C and SML.  In this use, the
>     Nat_k have nothing whatsoever to do with Z_k
>     and I certainly do not want to confuse the students!
> 

I agree that there is a need for a standard name, and that Nat_k would 
be a confusing name.  I have been calling
this set $k for some time but I am happy to change my macro if there
is some other accepted name or if $ has some other mathematical meaning. 
More generally, if k is an ordinal, one writes $k for the set of
ordinals less than k, the canonical well-ordered set of order-type k.
(The traditional ZF definition of ordinal makes k equal to $k, but
that is just an implementation.)   

A similarly useful terminology for arrays and the like is
"obaz", which indicates the use of the "ordinals begin at zero"
convention.  Thus you can refer to the cell with index 7 in your array 
 as the obaz 7th cell instead of as the 8th cell.
You can't call it the 7th cell, without qualification, because in
English the obao ("ordinals begin at one") convention is the 
established default, unfortunately.

As an example, today is the obaz 28th day of the obaz 10th month of the
year obaz 1999, which is the final year of the obaz 19th century and not 
the obaz zeroth year of the obaz 20th century as many obaoists mistakenly
believe.   Though I'm hardly the zeroth person to point this out (obaz).

Warning: this usage may alienate the less tolerant of your friends.

Regards
Paul


-- 
Paul Blain Levy
Computer Science Department, Boston University
http://www.dcs.qmw.ac.uk/~pbl/
If language were arbitrary, it wouldn't be interesting.