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Another test! I knew all the answers but I did not have enough time to spell them all out. I was so sure of the answer of one question that I forgot I needed to write it down. Uh. I have to train up my speed...
I've just realized that it should be numerical equivalence rather than denumerability. I must be very tired from my part-time job. Yea that's very beautiful. I still remember the thrill when I was in the algebra lecture. Analysis is less interesting in school's syllabus, but the book you recommended contains a few topics that make the whole thing just very interesting. I just need time to consolidate - make those stuff mine.
It is beautiful, isn't it? Exciting, too. You never know where some familiar idea will pop up next. Mathematics is all about connections like this, and there are very few things discovered in one area that have no application to many others!
Denumerability is another equivalece relation! I'm afraid there are infinitely many equivalence classes for denumerability. I like this kind of wrapping of mathematics. Theories of one level are often applicable to the higher levels. I really want to know how far things can go... Maths won't stop surprising me.
The 6th is particularly hard to prove. I may have a vague idea on what it's getting on, but both of them have infinite points, and this means I have to make one infinity more than the other infinity, and my brain starts to get messed up at this point. lol No clue.
The factorial zero is a surprise. I should check it out soon.
Ouch. I finally got my time to read your proof in the D-nile question in detail after finishing recruitment of participants for my cog. psy. experiment. I first printed out the part not hidden, and get to the right direction to complete the proof, but rejected it out of useless self-doubt, and later discovered that I was correct after reading what's hidden! And I did exactly the same for several questions in my second math test. I'm going to kill myself.
lol It's alright. I'm still working on the test and I have just successfully solved a problem. I just knew the prof. intended to make things hard for us. I simply need more time to work on the problems. I insist to do everyone on my own!!
I failed the test today, very likely. I can only answer half. I know the professor intentionally set up the test this difficult. A lot of people swore after the test, including me. I think I need at least 3 hours to fully answer the whole paper. And again I knew how to do one question RIGHT AFTER I handed in the paper.