275: Section 12.3, exercises 63, 66, 68, 74-77.
Section 12.4, exercises 4, 10, 32, 42-44, 48, 50.
This homework will be graded out of 10 points. Each exercise is worth 1 point. You can turn in as many exercises as you want. Indicate the ones you want to be extra credit problems. Of those, 2 will be chosen randomly to be graded, so you can have up to 2 extra credit points.

I would find it helpful when showing proofs in class that you spend a little bit more time showing every step on the board. The reason I ask for every step to be shown is that I find it hard to follow letters rather than numbers. Thanks I hope this helps.

I found values for and in all the examples you asked for. I solved them like we would any other problem in the book which is beside the point, however there is a pattern if there is anything to be deduced from that.

When the the numerator is ; ; When ; ; When ; ; .

I’ve been searching the internet trying to find a solution that is more what I believe you are looking for however I haven’t found anything suitable yet. I can’t wait to see the answer to this!

Hi! So, I have been working pretty much all day on the last two problems in seciton 7.4. I seeem to get stuck. The last problem in particular, I have evaluated by parts so that I can appropriately integrate for M but when I am looking for x hat my numbers are so far out of the x range. Also, I am once again getting stuck trying to find y hat. I was wondering if you could at least go over it in class? Any hints would be helpful! Thanks

Hi Aubrey,
Problem 7.4.38 is just partial fractions after a change of variables.
I think you are trying the wrong approach with 7.4.46, since you are not being asked to find the center of mass of the shaded region. Maybe you meant a different problem?

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${ […]

Marginalia to a theorem of Silver (see also this link) by Keith I. Devlin and R. B. Jensen, 1975. A humble title and yet, undoubtedly, one of the most important papers of all time in set theory.

Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know e […]

Equality is part of the background (first-order) logic, so it is included, but there is no need to mention it. The situation is the same in many other theories. If you want to work in a language without equality, on the other hand, then this is mentioned explicitly. It is true that from extensionality (and logical axioms), one can prove that two sets are equ […]

$L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZF}$) well-orders all of $L$, so that its restriction to any specific set $A$ in $L$ is a set well-ordering of $A$. The well-ordering $\varphi$ you are asking about can be obtained as the restriction […]

Gödel sentences are by construction $\Pi^0_1$ statements, that is, they have the form "for all $n$ ...", where ... is a recursive statement (think "a statement that a computer can decide"). For instance, the typical Gödel sentence for a system $T$ coming from the second incompleteness theorem says that "for all $n$ that code a proof […]

When I first saw the question, I remembered there was a proof on MO using Ramsey theory, but couldn't remember how the argument went, so I came up with the following, that I first posted as a comment: A cute proof using Schur's theorem: Fix $a$ in your semigroup $S$, and color $n$ and $m$ with the same color whenever $a^n=a^m$. By Schur's theo […]

It depends on what you are doing. I assume by lower level you really mean high level, or general, or 2-digit class. In that case, 54 is general topology, 26 is real functions, 03 is mathematical logic and foundations. "Point-set topology" most likely refers to the stuff in 54, or to the theory of Baire functions, as in 26A21, or to descriptive set […]

I would find it helpful when showing proofs in class that you spend a little bit more time showing every step on the board. The reason I ask for every step to be shown is that I find it hard to follow letters rather than numbers. Thanks I hope this helps.

Hello Dr. Caicedo,

I found values for and in all the examples you asked for. I solved them like we would any other problem in the book which is beside the point, however there is a pattern if there is anything to be deduced from that.

When the the numerator is ; ; When ; ; When ; ; .

I’ve been searching the internet trying to find a solution that is more what I believe you are looking for however I haven’t found anything suitable yet. I can’t wait to see the answer to this!

-Will

Hi! So, I have been working pretty much all day on the last two problems in seciton 7.4. I seeem to get stuck. The last problem in particular, I have evaluated by parts so that I can appropriately integrate for M but when I am looking for x hat my numbers are so far out of the x range. Also, I am once again getting stuck trying to find y hat. I was wondering if you could at least go over it in class? Any hints would be helpful! Thanks

Hi Aubrey,

Problem 7.4.38 is just partial fractions after a change of variables.

I think you are trying the wrong approach with 7.4.46, since you are not being asked to find the center of mass of the shaded region. Maybe you meant a different problem?