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#1 2007-04-10 07:02:09

clooneyisagenius
Member
Registered: 2007-03-25
Posts: 56

Test Coming Up.... Help?

Some things I'm confused about:
1. How to show that the union of countable sets is also countable. For example, let A1, A2, ... be a sequence of sets, each countable. How to prove the union is a countable set.

2. What the use of lattice paths is. I understand how to make/get one... by I'm not sure how, on the test for example, it could be used in a problem.

3. How to count something in 2 ways

4. Cardinality.

and
5. Binary coding / n-tuples. For example, our teacher emailed us today with a question he says we should be able to answer easily: The 51st State of the union is going to be the State of Tom.  I will be
issuing license plates for cars using 3 letters from the alphabet.  How many plates can I make?  In essence we are counting ?-ary ?-tuples.  Finally, what is the probability that a randomly made plate will spell Tom?

ANY ADVICE/HELP WOULD BE WONDERFUL.

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#2 2007-04-10 07:32:47

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Test Coming Up.... Help?

1. You asked that here, and we responded.  If there is something which you don't understand in the response, then please ask.  But I know of nothing else to say in regards to this problem.

2. A lattice of what?  Just any partially ordered set?

3. Thats a pretty vague question.  It takes at least half a dozen examples of practice to get used to counting in different ways.  Take the number of permutations of n select k objects.  This is by definition P(n, k).  But we could also first choose k objects, which is C(n, k), and then there are k! different permutations of these k objects.  Since we are counting the same thing, they must be equal.  So P(n, k) = C(n, k) * k!

4. What about it?

5.  There are 26 different letters in the alphabet.  So this is like "base" 26.  3 digits in base b have b^3 different permutations with repeats.  So this is 26^3.

Only one combination spells Torn out of 26^3 of them.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2007-04-10 07:44:51

clooneyisagenius
Member
Registered: 2007-03-25
Posts: 56

Re: Test Coming Up.... Help?

Ricky - I didn't have another example to ask the question for. Sorry, but I still dont understand anyone's answers.

2. Lattice path - as my book says - A lattice path in the plan is a path joining together integer points via steps of unit legth rightward and upward. Typically the rightward is given a value of 1 and the upward is given a value of 0.  If given a path from (0,0) to (0,1) to (1,1) to (2,1) to (2,2) I understand the path will be (1,0,0,1) but after that I'm not sure what good they are? What's their use?

Cardinatlity - I guess what the point is?

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