My one true weakness in math. I guess that explains why I enjoy computer science more than math and logic. With computer science you generally understand all the instruments you have at your disposal when solving problems. As long you understand the commands you can give in your programming language, there won't be any crucial method or tool that you can miss that will prevent you from solving your problem. Computer science is pretty flexible in terms of solutions; while it not be super efficient at least your program can still work. Math for me at least not so much. Sometimes when an answer in given in class, I think to myself "there is no way I could have ever thought of that solution on my own". I just probably haven't even considered thinking that way in terms of solving the problem. For example regarding question 2 on the test, my mind was not analyzing the problem the way that the answer was given (the great Heap has tested me, and I have failed). Assigning delta as a function that uses epsilon never struck me, but I'll definitely catch that in the future. As I've said before, I really need to get into a mathematical mindset and estrange myself from interpreting symbols through language when they only just represent symbols. My main frustration with myself is when I am posed with something new and abstract, I just fall into this state of mind that I can't do it. I can't understand the problem because I'm not viewing from different perspectives or can't remember a useful equation. It's not until I actually see an answer that I perceive I can keep that mentality when dealing with a problem.
On the bright side I do believe that I am getting the hang of proofs, there's just a few things that I need to keep reminding myself. Like if the statement of a variable is actually posed in the statement that needs to be proved, one must prove somewhere that the variable is in fact an element of its set. Also it is very beneficial sometimes to bring in extra variables (I don't really understand where it would be useful though), setting up y' can help with proving a statement with y in it. But there are still some things I need help clearing up. If you're supposed to prove something such as P(x) ^ Q(x), are you supposed to reformat the statement so it becomes an implication? Or are there other ways to directly or indirectly prove such statements?
You don't have to reformat it as an implication (in fact, you might not be able to). If you have disjunction it might be easiest to reformat it as an implication though.
ReplyDeleteFor conjunctions (say Forall x in X, P(x) ^ Q(x)) simply prove all parts individually, that is, do something like:
1. Assume x in X
2. [proof]
3. Then P(x)
4. [proof]
5. Then Q(X)
6. Then P(X) ^ Q(X) #by lines 3 and 5
7. Then for all x in X, P(X) ^ Q(X)
Indentation appears to be lost... indent lines 2-6
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