Homework this week

Sections 5.1, 5.2

makeup exam in the testing center

The makeup exam will take place in the testing center Wednesday Dec 4 and Thursday Dec 5.  This is a very busy time in the testing center.  You will need to be in and out before 3pm.

Another writing assignment question

.....I cannot seem to get the proof of if m and k are not co-prime numbers. If you could explain to me how to start or give me any insight it would be appreciated!

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Well, you should carefully read the Theorem Statement section in wikipedia article on the Chinese Remainder Theorem, particularly the part following the fact that simultaneous congruences can be solved even when the divisors are not pairwise coprime.  Then I would reflect on what the existence and/or nonexistence of solutions to the simultaneous congruences says about the elements in the partition Pm.Pk

Writing Assignment Question

Hello Professor Taylor,

I am writing to get a bit of clarity of what we are proving in this paper. From the blog instructions, I gather that we are trying to prove that:

"Given two integers x and y, we say that the partition P separates x and y, if [x]≠[y]."

and then from 2)

given a)

we are to prove b), c), and d)   

Again, I am just writing to see if I am understanding this correctly or if I am off base here. Thank you for your time and have a good afternoon.

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Well, no.  Everything you are to prove is in item 2), and the four parts are somewhat independent. The statement

"Given two integers x and y, we say that the partition P separates x and y, if [x]≠[y]."

is the definition of what it means for a partition to separate two elements x,y. 

By the way, note that I made an additional comment inline to item  2a) on the writing assignment announcement below.

question and answer

Hello

Im having trouble with problem #17 in section 4.4, the book gives a hint so i thought in the back so I thought to prove that R is reflexive, antisymmetric and transitive on A. Is there an another route that might not take as long or Im I going about it ok??

Thank you
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17.  If a subset of a partially ordered set has exactly one minimal element, must that element be a smallest element?  Give either a proof or a counter example to justify your answer.

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As you say, the back of the book gives a relation, a subset and a point in that subset that they claim is acounter example. The trick is to show that the relation is a partial order, and that the point and subset  have the needed properties. 

Online LaTeX editor

Hey, this link is a useful equation editor for creating image files of mathematical text

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.