Hello,

I have a few discrete math questions that need to be answered.  They
are questions for a practice exam.  I need them to be answered and
explained well so understand them and can use them to prepare for the
actual exam.  The practice exam as well as the notes required for
question eight are in Adobe Acrobat format, so I decided to create a
web site so you can easily view them.  I also included the answers
that I have so far in a Microsoft Word document.  Some of the
questions are answered, but are missing an explanation.  I need all of
the questions (except number seven) to be answered and explained. 
Also, please notice that some questions require drawing a diagram.

Here is the URL to the site:  http://members.cox.net/520/

* Remember: Answering question number seven is not required.

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Request for Question Clarification by answerguru-ga on 24 Jul 2003 08:15 PDT

Hi student79,

If you don't get any takers for this question you might consider
re-evaluating your list price. Personally, I feel that $30-40 is a
fair price for this type of question.

answerguru-ga

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Clarification of Question by student79-ga on 24 Jul 2003 09:31 PDT

answerguru,

I am new to using this service, so I wasn't sure about the list price.
 I would like to open the question to bids.  I take it that you are
willing to do it for $30.  If I don't get a better price, then would
you be interested?

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Request for Question Clarification by answerguru-ga on 24 Jul 2003 11:15 PDT

Yes I would be interested....for your reference you may want to read
the link which should help you price questions more effectively in the
future:

http://answers.google.com/answers/faq.html#pricing

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Clarification of Question by student79-ga on 26 Jul 2003 03:23 PDT

answerguru,

It looks like you are the only bidder on this project.  We agreed that
$30 was a fair price, so go ahead and work your magic.

Without getting into the detailed explanations, I noticed a few
mistakes in the answers you have so far:

1 c. Don't forget that 0 is an even number too. This adds another bit
string, making the total 64 - as we'd expect from symmetry.

2 b. (iii) - we must have at least two women, since there are only
five men available.
  c. We can also choose groups that don't contain either of the two
people, so you need to add (10C7) = 120 to your answer.

5 a. Note that you were asked to write the definition of each
property; make sure you do this. You also need to prove the properties
of this relation, not just assert them. For instance, to prove the
transitive property, suppose a|b and b|c. By definition this means
that c = kb and b = la for some integers k and l. Then c = k(la) =
(kl)a, so a|c and the relation is transitive.

Your definition of the reflexive property seems very strange. The
correct definition is that R is reflexive iff for every a in A, a R a.

To prove R antisymmetric, suppose a | b and b | a. Then there exist
integers k, l such that a = kb and b = la. Then a = (kl)a and kl must
be 1. So k = l = 1 or -1. But since all elements of A are positive, we
must have k = l = 1 and hence a = b. (Note that the correct definition
of antisymmetric is that for all a, b in A, a R b and b R a ==> a =
b.)

Since R is reflexive, antisymmetric and transitive it is a partial
order. The Hasse diagram and boolean matrix should be pretty obvious.

Question 6 looks pretty simple. You have the direct links, then follow
them to get the transitive ones; don't forget to include the pairs of
form (a, a).