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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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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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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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