SteveL27
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EonsNearby said:Okay...I hate abstracting
Gonna be a long semester in this class then :-) That's what this exercise is about. Learning to write abstract proofs.
Question: Is this your first proof-based class? And if so, is that normal for this class? Or are you expected to already know how to do abstract proofs? If the latter, you will have to work extra hard for a while till you get the hang of this. Something to talk about when you meet w/the teacher.
EonsNearby said:So A U B = {1, 2, 3, 4, 5, 6, 7, 8} and C U D = {9, 10, 11, a, b, c, d, e}.
First attempt: h(x) = f(x) if x is element 1, 2, 3 OR g(x) if x is element 4, 5, 6, 7, 8.
Why does x have to be in one or the other? Answer: Because the elements of A union B are exactly those elements that are either in A or in B.
But what if x is in both A and B? Answer: That can't be, because A intersect B is empty. That's a given.
Those are the types of things I'm looking for if I'm grading this proof. I'm looking for understanding; and I'm looking to see if you are using the givens.
EonsNearby said:This is one-to-one because every element of A U B is mapped to a unique element from C U D.
Why?
EonsNearby said:This is also onto because the range of h is every element in C U D.
How do you know? How do you know everything in C union D gets hit? It is essential to say WHY something is true based on the givens and what you're allowed to assume. Of course we HOPE your mapping is onto; but you have to say WHY it's onto. What if I say, "Hey, I don't think your mapping is onto." What is the answer?
EonsNearby said:(Just a quick question, could I have instead said h(x) = f(x) if x is an element that is from A OR g(x) if x is an element from B?)
Is this a possible bijection?
Yes; of course, that's the bijection. And you can see that it's very general. We didn't assume the sets are finite; we didn't assume there are any formulas.
But, once you say that, you have to prove that it's well-defined. What if x is in both A and B, then how should I defined h(x)?
Once you handle that issue, then show that h is 1-1 and onto. But you have to nail down every little detail.
That is what they're looking for in a proof like this. Everything you claim is true or hope is true must be shown logically to be true.