So the above is the problem and my idea of how to approach it. This problem comes from the section on the Countable Additivity of Integration and the Continuity of Integration, but I was not sure how to incorporate those into the prove, if you even need them for the result.
I had no idea what...
I am a senior math major and my goal is to get into a top 20 PhD program. Right now I do not feel competitive enough to get into those schools, so I am pursuing a masters degree first.
I'm applying to these schools:
-University of Washington
-Texas A&M - College Station
-University...
I'm not allowed to do that in this class. I have to do a line by line prove and I can't use an implication. I see what you're doing.
I have to do something like this:
1. P v P...Premise
2. P v P v ~P...Addition
3. P v ~(~P v P) De Morgans
etc.
That seems like a valid argument for showing that \phi_n converges to f, but I'm not sure how to show it's increasing. And as far \psi_n, converges, well I imagine that I'd use a similar argument, but I'm still not sure how to show it's decreasing.
Ok, so it's very easy to show P v P = P (where = is logically equivalent) using a truth table as well as using a conditional proof.
P v P Premise
~p Assumption
p Disjunctive Syllogism (1, 2)
p & ~p Conjunction (3, 4)
~p --> (p & ~p) Conditional Proof (2--4)...
Ok, I don't think I'm on the right track here. I ASSUMED that the set of all countable collections \{I_k\}_{k = 1}^\infty of nonempty open, bounded intervals such that E \subseteq \bigcup_{k = 1}^\infty I_k is a countable set itself, which it probably isn't.
I'm not even sure where to start...
of a countable collection of open intervals.
I'm having a hard time seeing how this could be true. For instance, take the open set (0, 10). I'm having a hard time seeing how one could make this into a union of countable open intervals.
For instance, (0,1) U (1, 10) or (0, 3) U (3, 6) U (6, 10)...
I've prove everything except for the fact that E is bounded below. It would appear that you would need to know something about the functions taking a minimum value as well to show this using my method, so perhaps there is another way of thinking about things to show a lower bound?
I could probably find the answer to this problem easily by a quick google search, but I don't want to spoil it. Instead, could someone give me a hint in the right direction?
Ok, so it seems to me like a contradiction would work here. It seems like directly proving the existence of an...
The question I had, which didn't work, is this: Given positive integers m and n, does there exist a tiling such that for ANY two-coloring of the tiling, every tile has m friends OR n strangers. (In this case two tiles are friends if they are adjacent and share a color and strangers if they are...
I have this really cool idea of asking Ramsey Theory questions on tilings (tessellations).
Classical Ramsey theory asks what is the minimum number of people one needs to invite to a party in order to have that every person knows m mutual friends or n mutual strangers.
I was thinking about...
The only Spivak book that might be useful would be his differential geometry / calculus on manifolds books, but those topics aren't covered at all on the preliminary exams, so I don't really need to worry about them.
Yes, I'm aware of that. But does that mean I should whip open my analysis and algebra textbooks and just drill problems everyday for a few hours? What's the most efficient way?
How do I take my mathematics to the "next level?"
Let me explain what I mean:
I am finishing my junior year in undergraduate. I consider myself "good" at math, in the sense that I make A's in my classes. But that's not enough.
I want to be a world class math student because my dream is to...