I have an infinite set $X$, I need to take a proper (countable or not) infinite subset of $X$ so there are theorems that guarantee its existence but it's not clear what set of axiom I need to take, maybe ZFC. Besides, how do I know the complement of that subset is also infinite?
Let $X$ be a set and consider the collection $\mathcal{A}(X)$ of countable or cocountable subsets of $X$, that is, $E \in \mathcal{A}(X)$ if $E$ is countable or $X-E$ is countable. If $X$ is countable, then $\mathcal{A}(X)$ coincides with the power set $\mathcal{P}(X)$ of $X$. Now suppose that $X$ is uncountable. Assuming the axiom of choice, we can conclude that $\mathcal{A}(X) \ne \mathcal{P}(X)$, since $|X| = |X| + |X|$. So the question is:     Can we prove in ZF that $\mathcal{A}(X) \ne \mathcal{P}(X)$ for every uncountable set $X$?   I'm assuming that a set $X$ is uncountable if there is no injective function $f : X \rightarrow \mathbb{N}$.
Test to see when table is candidate for compression (row or page) Does anyone know a tool like this page  where one can determine if a table is a good candidate for row or page compression. Also I am trying to understand the benefits of compression. I have been reading about it, but still have these questions.  Will compression make more 8KB fit in the buffer cache? When does data become uncompressed i.e. in a select? will backups become faster? Is the reason that compression ads cpu overhead because e.g. inserts need to be compressed before the insert? Is there any good reason to compress tables without investigating how the tables is being used (i.e. static vs. volatile).  A colleague has started to compress many tables (page). I don't know the analysis behind this decision. I know we sometimes have resource semaphore waits during some peak hours. I actually don't think he knows what he is doing. I can see that our sos scheduler yield waits have gone slightly up. I think the compression is causing more harm and it seems that maintenance tasks overshoot more now (e.g. backups, checkdb...). I have Ubuntu 14.04 installed on my laptop. Can I clone the entire system with apps, settings, etc on a USB 3 pendrive, and then boot from that on other computers as a portable Ubuntu installation to take around with me?  I am trying a function that Timeshift has that's called "Clone the current system on another device". I've selected the pendrive as the target device. Timeshift was talking about this being bootable. But is it going to bootable on only this laptop, or on any system? For example, on an Intel Mac, or a Windows computer without the grub bootloader? Is that Timeshift function what I am looking for, or do I have to do something else?  Edit: The Timeshift clone failed to boot on the first system I tested it on. (A Dell Hackintosh that usually boots from USB just fine.) Is there a password manager for Ubuntu What is the most popular open source password manager which I can use on a Ubuntu machine or preferably across Ubuntu/Windows machines with some form of data compatibility. If there is a free web based version, that would be even more preferable. How to check if a 2D point is in a polygon? In my textbook it is given that      'The wave function describes the position and state of the electron and its square gives the probability density of electrons.'    Can someone give me a very simple example of a wave function with explanation?  (Note: This question is not a duplicate. I have searched for other questions of this type but the answers were overwhelmingly difficult to understand.) How do I remove snap? apt pretends it's not there but I can still call commands with it.  Trying to rm it says "read only file system"  root@tunnel:/# apt remove snap Reading package lists... Done Building dependency tree Reading state information... Done Package 'snap' is not installed, so not removed 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. root@tunnel:/# snap whoami email: - The system cannot find the file specified I have just downloaded WinEdt and written my first code, but it will not run because I keep getting the following error.  I am at a loss as to how to fix this.  Error launching Console Application PDFLaTeX ... Command Line: pdflatex.exe --interaction=errorstopmode --synctex=-1 &quot;practice.tex&quot; Startup Folder: C:\Users\shahand\WinEdt Team\WinEdt 8 The system cannot find the file specified. $e^{mx}$ in solving second order differential equations In a book I am reading on differential equations, the author writes about the solution to a homogenous, linear, second order differential equation with constant coefficients. The author says something like, "Let us suppose that the solution is of the form $y=e^{mx}$ " . After this the author introduces the characteristic equation of a differential equation of the form mentioned above, and proceeds to describe how to solve it w/ undetermined coefficients, variation of parameters, etc.   How did mathematicians first come up with this "assumption" that the solution was of the form $y=e^{mx}$? And, are there any other forms of solutions for these types of equations? Launch apps full screen Turning separate lines into a comma separated list with quoted entries I have the following data (a list of R packages parsed from a Rmarkdown file), that I want to turn into a list I can pass to R to install:  d3heatmap data.table ggplot2 htmltools htmlwidgets metricsgraphics networkD3 plotly reshape2 scales stringr   I want to turn the list into a list of the form:  'd3heatmap', 'data.table', 'ggplot2', 'htmltools', 'htmlwidgets', 'metricsgraphics', 'networkD3', 'plotly', 'reshape2', 'scales', 'stringr'   I currently have a bash pipeline that goes from the raw file to the list above:  grep 'library(' Presentation.Rmd \ | grep -v '#' \ | cut -f2 -d\( \ | tr -d ')'  \ | sort | uniq   I want to add a step on to turn the new lines into the comma separated list. I've tried adding tr '\n' '","', which fails. I've also tried a number of the following Stack Overflow answers, which also fail:      This produces library(stringr)))phics) as the result.      This produces ,% as the result.      This answer (with the -i flag removed), produces output identical to the input. WinForms: How to programmatically fire an event handler? The identity cannot be a commutator in a Banach algebra? The Wikipedia article on  claims, without a proof or reference, that there does not exist a (unital) Banach algebra $B$ and elements $x, y \in B$ such that $xy - yx = 1$. This is surprising to me, but maybe the proof is straightforward; anyone have a proof and/or a reference?   More generally, I would have naively thought that I could embed any ring into a Banach algebra. I guess there are actually serious restrictions to doing this; are these issues discussed anywhere? Are there any rules on the positioning adverbs should take in a sentence? For example:     Ever wish you could share information broadly   Could it be rewritten to:     Ever wish you could broadly share information   Are there any rules for the position of the adverbs. Let $(X,\mathcal{M},\mu)$ be a measure space and $\{f\}$ be a sequence of functions on $X$, each of which is integrable over $X$. Show that $\{f_n\}$ is uniformly integrable if and only if for each $\varepsilon \gt 0$, there is a $\delta \gt 0$ such that for any natural number $n$ and measurable subset $E$ of $X$, if $\mu(E) \lt \delta$, $$ \left|\int_E f_n~d\mu\right| \lt \epsilon.$$     I think one direction $(\Rightarrow)$ is clear. Since $f_n$ being uniformly integrable imply that for every $\varepsilon\gt 0,~\exists \delta \gt 0$ such that for any $E\in \mathcal{M},~\mu(E)\lt \delta$, $\int_E |f_n|~d\mu$ for every natural $n$. But then   $$ \left|\int_E f_n~d\mu\right|\le \int_E|f_n|~d\mu \lt \varepsilon.$$    Any suggestions for the other direction?   Edit:  Following Davide's suggestions, I have   $$ \begin{align*} \int_E |f_n|~d\mu &amp; \le \int_{E\cap [f_n \ge 0]} f_n~d\mu + \int_{E\cap [f_n \lt 0]} (-f_n)~d\mu\\ &amp; = \left| \int_{E\cap [f_n \ge 0]} f_n~d\mu \right| + \left| \int_{E\cap [f_n \lt 0]}f_n~d\mu \right| \\ &amp; \lt \varepsilon + \varepsilon = 2\varepsilon. \end{align*} $$ Example of modules that are projective but not free; torsion-free but not free Copying protected files between servers in one line? I'd like to copy squid.conf from one server to another.   The servers don't talk to each other. I'd like to go through my workstation. Both servers have the file, so it will be overwritten on the target. The files have 600 permission and are owned by root. root login via ssh is disabled (PermitRootLogin no). I'd like to do it in one line, if possible, since it will be a part of a setup guide.   I know to do  ssh source 'tar czpf - -C /etc/squid/ squid.conf' | \     ssh target 'tar xzpf - -C /etc/squid/'   to copy files between servers and preserve permissions. However, in this case I will get "Permission denied".  I also know I can do this:  ssh -t source 'sudo cat /etc/squid/squid.conf'   This way the -t allows sudo to ask for the admin password before outputing the content of the file.  The problem is, I don't know how to combine those techniques into something that will ask for the sudo password on each server, and transfer the file to its destination. Is this possible?  UPDATE: Here's the best I could come up with:  ssh -t source 'sudo tar czf /tmp/squid.tgz -C /etc/squid squid.conf' &amp;&amp; \ ssh source 'cat /tmp/squid.tgz' | \     ssh target 'cat &gt;/tmp/squid.tgz' &amp;&amp; \ ssh -t source 'sudo rm /tmp/squid.tgz' &amp;&amp; \ ssh -t target \     'sudo tar xzf /tmp/squid.tgz -C /etc/squid &amp;&amp; sudo rm /tmp/squid.tgz'   Calling this a one-liner seems like a stretch. I think I'll just break it down to separate steps in the setup guide. If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$ If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$  Really speaking, I don't have any clue ... Continuing my work through Dummit &amp; Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in ):     Prove that if $G/Z(G)$ is cyclic, then $G$ is abelian. [If $G/Z(G)$ is cyclic with generator $xZ(G)$, show that every element of $G$ can be written in the form $x^az$ for some $a \in \mathbb{Z}$ and some element $z \in Z(G)$]      The hint is actually the hardest part for me, as the quotient groups are somewhat abstract. But once I have the hint, I can write:  $g, h \in G$ implies that $g = x^{a_1}z_1$ and $h = x^{a_2}z_2$, so \begin{align*}gh &amp;= (x^{a_1}z_1)(x^{a_2}z_2)\\\  &amp;= x^{a_1}x^{a_2}z_1z_2\\\ &amp; = x^{a_1 + a_2}z_2z_1\\\ &amp;= \ldots = (x^{a_2}z_2)(x^{a_1}z_1) = hg. \end{align*} Therefore, $G$ is abelian.  1) Is this right so far?  2) How can I prove the "hint"? Is there a software utility to adjust screen gamma/brightness/contrast? My system is not dual-boot, I run a standard Ubuntu desktop system "on the metal" (I think running it in a VM is the same).  Pressing c while booting does not cause the GRUB menu to appear (which I believe it does for a dual-boot system).  How can I get the GRUB-menu to present itself on a single-boot system?