The next writing assignment

1) The Topic
Pick a natural number m>0. As discussed in class, every natural number n can be uniquely expanded as n=b*m+r,  where b is an integer and . Consider the relation    with domain , range [0,m-1] and elements (n,r) where r is the remainder of n mod m. Notice that is an equivalence relation, for which each of the equivalence classes consists of numbers congruent to each other mod m,i.e. for  consists of all numbers with remainder r when divided by m.   These equivalence classes form a partition  into m pieces; call this partition  (the upside down capital pi means disjoint union).  Notice that that problems 17 and 18 in section 4.6 of the textbook, discuss a way of getting a new partition from two old ones P,Q : the new partition has sets that are non-null intersections of the pieces of the old one.  Given two integers x and y, we say that the partition P separates x and y, if [x]≠[y].  For example x and y are separated mod 2 if one is even and the other odd, but not separated if they are both even or both odd.

2) To prove:
   a)    (Note: your classmate Chelsey Anderson points out that this problem can be solved using the Chinese Remainder theorem)
   b)  If separates x and y then so does for every integer k>0, and in particular so does .
   c)  If , then x,y are separated in every  with
   d) if  then x,y are separated in  for the least m such that

3) The first draft is due to me by email in PDF format on Sunday Dec1, by 5pm.  They should *NOT* have your name, but instead your posting id, instructions to find your posting id are below in this blog.   I will redistribute the papers on Sunday night; your edits will be due by Tuesday Dec 3,  at 5pm by email to me in PDF format.  You may print your editing paper, edit by pen, then photograph and save as pdf to email it to me.

4) you should research your paper, on the web, in the journals, and by asking anyone you can pin down.  Be sure to cite all sources, both formal and informal.  Use inline citations. Your references should list title, date and journal for published articles, should use title, author if available, url and date accessed for online articles, and should list the name and date of any personal communications.