Solve the following problems from Chapter 3 of the textbook: 2, 4, 5, 7, 9, 10, 26.

In addition, solve the following problem:

Let be a linear map between finite dimensional vector spaces over For we denote by the preimage of under This is the set of all vectors such that

Recall that a subset of a vector space over is convex iff whenever and then also Show that is convex iff is convex. Is it necessary that is linear for this to hold, or does it suffice that is continuous?

Assume that is a subspace of and show that is a subspace of

43.614000-116.202000

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When you ask the question “Is it necessary that is linear for this to hold, or does it suffice that is continuous?”, did you want simply our opinion or did you want a proof/counterexample. If a proof/counterexample is desired, can we clarify what topology we have on and . My guess is since they are finite dimensional and isomorphic to for some and is a metric space, we would use that.

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

Consider any club subset of $\kappa $. Check that it has order type $\kappa>\lambda $, and that its $\lambda $th element (in its increasing enumeration) has cofinality $\lambda $.

A very nice introduction to this area is MR0891258(88g:03084). Simpson, Stephen G. Unprovable theorems and fast-growing functions. In Logic and combinatorics (Arcata, Calif., 1985), 359–394, Contemp. Math., 65, Amer. Math. Soc., Providence, RI, 1987. Simpson describes the paper as inspired by the question of whether there could be "a comprehensive, self […]

There are continuum many (i.e., $|\mathbb R|$) such functions. First of all, there are only $|\mathbb R|$ many continuous functions, so this is an upper bound. On the other hand, for any real $r$, $f(x)=x+r$ satisfies the requrements, so there are at least $|\mathbb R|$ many such functions.

I'm posting an answer based on Asaf's comments. The following reference addresses this question to some extent: MR0525577 (80g:01021). Dauben, Joseph Warren. Georg Cantor. His mathematics and philosophy of the infinite. Harvard University Press, Cambridge, Mass.-London, 1979. xii+404 pp. ISBN: 0-674-34871-0. Reprinted: Princeton University Press, P […]

When you ask the question “Is it necessary that is linear for this to hold, or does it suffice that is continuous?”, did you want simply our opinion or did you want a proof/counterexample. If a proof/counterexample is desired, can we clarify what topology we have on and . My guess is since they are finite dimensional and isomorphic to for some and is a metric space, we would use that.

Yes, I am thinking of as and I would like a proof or a counterexample in this context.