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.

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 […]

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See MR0332477 (48 #1080 […]

I give an example (perhaps the best-known example) below, but let me first discuss equiconsistency rather than straight equivalence. Usually an equiconsistency is really the sort of result you are after anyway: You want to establish that certain statements in the universe where choice holds correspond to determinacy, which implies the failure of choice. The […]

The other answers have correctly identified the issue. Let me highlight the difficulty: it is relatively consistent with the axioms of set theory except for the axiom of choice that there are infinite sets which do not contain a copy of the natural numbers (that is, there are infinite sets $X$ such that there is no injection $f\!:\mathbb N\to X$). This means […]

This is $\aleph_\omega^{\aleph_0}$. First of all, this cardinal is an obvious upper bound. Second, if $A\subseteq\omega$ is infinite, $\prod_{i\in A}\aleph_i$ is clearly at least $\aleph_\omega$. The result follows, by splitting $\omega$ into countably many infinite sets. In general, the rules governing infinite products and exponentials are far from being w […]

If $\lambda$ and $\kappa$ are cardinals, $\lambda^\kappa$ represents the cardinality of the set of functions $f\!:A\to B$ where $A,B$ are fixed sets of cardinality $\kappa,\lambda$ respectively. (One needs to check this is independent of which specific sets $A,B$ we pick, of course.) At least for finite numbers, this is something you may have encountered in […]

R. Solovay proved that the provably $\mathbf\Delta^1_2$ sets are Lebesgue measurable (and have the property of Baire). A set $A$ is provably $\mathbf\Delta^1_2$ iff there is a real $a$, a $\Sigma^1_2$ formula $\phi(x,y)$ and a $\Pi^1_2$ formula $\psi(x,y)$ such that $$A=\{t\mid \phi(t,a)\}=\{t\mid\psi(t,a)\},$$ and $\mathsf{ZFC}$ proves that $\phi$ and $\psi […]

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.