week # 7

It's been seven weeks since this semester started, which means we are already in half way through the whole term. Every time I look at the stuff learned before, I feel like that the time in university goes so fast. This week, we are still continuing on proofs but more complicated than before, which is to prove by cases. This kind of proof is not hard but we have to separate it into several cases and prove every case in order to make a good proof. What's more, I think the most useful part during this week is to introduce some rules, which can also be seen as the conclusion of some basic and necessary rules of proof.
Elimination:
conjunction elimination: If you know A ^ B, you can conclude A separately (or B separately).
existential instantiation: If you know that there exists k in X, P(k), then you can certainly pick an element with that property, let k' in X, P(k').
disjunction elimination: If you know A or B, the additional information :A allows you to conclude B.
implication elimination: If you know A implies B, the additional information A allows you to conclude B. On the other hand, the additional information :B allows you to conclude :A.
universal elimination: If you know for all x in X, P(x ), the additional information a in X allows you to conclude P(a).
Introduction:
implication introduction:If you assume A and, under that assumption, B follows, than you can conclude A implies B.
universal introduction: If you assume that a is a generic element of D and, under that assumption, derive P(a), then you can conclude for all a in D, P(a).
existential introduction: If you show x in X and you show P(x ), then you can conclude not x in X, P(x ).
conjunction introduction: If you know A and you know B, then you can conclude A ^ B.
disjunction introduction: If you know A you can conclude A or B.
The most difficult part this week, which I think is the worst case by introducing two functions which are the upper bound O(U) and the lower bound. The formal definition was pretty complicated and confused when I first saw it. However, after understanding the actual meaning of them, it is much clearer to me. The issue is that I can understand it when I'm looking at the definition but can hardly write it down by myself. Therefore, it is probably a good idea by practicing related problems in order to get familiar with it.