I am working with the following set of axioms:  1) There is a set.  2) Two sets are equal iff they contain equal elements.  3) if $A$ is a set and $p(x)$ is a set property (e.g. $x = x$), then $\{a \in A : p(a)\}$ is also a set.  4) if $A$ and $B$ are sets, then there is a set which contains only elements of $A$ and $B$.  5) if $\mathbb A$ a set, then $\bigcup \mathbb A :=\{a | \exists A \in \mathbb A : a \in A\}$ is also a set.  6) For each set there is its power set.   I my lecture it was claimed that if $\mathbb A \ne \emptyset$, then $\{a \in \bigcup \mathbb A | \forall A \in \mathbb A\ : a \in A \}$ is also a set.   I am not sure why do we require that $\mathbb A \ne \emptyset$.    Indeed, the fact that $\{a \in \bigcup \mathbb A | \forall A \in \mathbb A\ : a \in A \}$ is a set follows directly from 3 and 5 without the requirement.
In MK (Morse-Kelley) set theory life is easy: $\forall X\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)$. If $X=\left\{\right\}$ then $\bigcap X=U$, where $U$ is the universal class. So the (unary) intersection of the empty set is the class that contains all sets as elements. In ZF (Zermelo-Fraenkel) set theory, instead, proper classes are not allowed. So, how can I define $\bigcap X$ in ZF? I tried with the following definitions:   $\forall X\left(X\not=\left\{\right\}\rightarrow\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)\right)$. This means that $\bigcap\left\{\right\}$ is undefined, which is not that good. $\forall X\forall y\left(y\in\bigcap X\leftrightarrow X\not=\left\{\right\}\land\forall x\left(x\in X\rightarrow y\in x\right)\right)$. This means that $\bigcap\left\{\right\}=\left\{\right\}$, which is the opposite of MK.   I couldn't find any other valuable definition. Any ideas? Thank you.
Where can I see my full list of Questions and Answers in my profile page of Stack Exchange Android App? I want to see all my Questions and Answers in a particular SE site -  - profile page through Stack Exchange Android App. How can I see that? When I tried to see that I found only Top Posts(5) and Top Tags. WUBI installed Ubuntu not showing up at boot menu after Windows reinstallation I just installed Ubuntu 13.10, and I can't configure my wi-fi Epson TX430W printer.  Using the Settings-&gt;Printers interface, I can detect the printer. When I click add, I briefly see a "Searching for drivers" dialog. It dissapears, and then I'm left with the printer forever in "Installing" status.  If I close the window and reopen Printers, it doesn't show up on the list.  How can I solve, or at least diagnose the problem? Can someone confirm that its not possible to change the height of a dropdown that is shown when you click on a select box.   The size attribute of the select makes it look like a list, the height attribute in the CSS doesnt do much good either. How to get the current user in ASP.NET MVC How to link dataframes in ArcGIS or QGIS? The below image shows a layout in ArcGIS 10.1 with four dataframes.  Each dataframe is of the same geographic area, although they are different images. For example, the top left image is a 1989 DOQ, the top right image is a reversed 1989 DOQ, the bottom left image is a 2012 DOQ, and the bottom right image is a reversed 2012 DOQ.   Is it possible to link the dataframes so that panning the image in one dataframe pans the images in all of the other dataframes (i.e. similar to how you can "Link Views" in Erdas Imagine)? Is this possible in QGIS? What is the minimum RAM required for Pokemon GO? While poking around on Pokemon Go's , it lists some specs required for Android and iOS devices. I'm particularly interested in Android devices, but I do not see a minimum/recommended RAM listed on their site.        Android 4.4 to Android 6.0.1  (Android N will not supported until the   official Android release)    Preferred resolution of 720x1280 pixels   (Not optimized for tablet)    Strong internet connection (Wi-Fi, 3G, or   4G)    GPS and Location Services    Intel CPUs are not supported      My device is a , which has a 1.5GB RAM and Quad-Core processor. It runs fine for the most part, but a majority of the features heavily lags and are sluggish while I've seen the same features run smooth on other devices.  What is the minimum/recommended RAM for Pokemon Go to run more smoothly? Do all accepted students in top-tier engineering programs get invited to interview before getting admitted? Why is '.' a hard link in Unix? How do I resize/change resolution of pictures easily? I have a Multisite Wordpress installation with 7 different blogs. I want to show an aggregate of the summaries of posts from those 7 blogs in the main blog. Do you know a good way to do this? Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$ Today, we had a math class, where we had to show, that $a_{100} &gt; 14$ for  $$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$  Apart from this task, I asked myself: Is there a closed form for this sequence? Since I didn't find an answer by myself, can somebody tell me, whether such a closed form exists, and if yes what it is? The differences between TeX engines I am confused about what TeX and its offspring (LaTeX, pdfTeX/pdfLaTeX, e-TeX, ConTeXt, LuaTeX, etc.) are.  Some, like LaTeX, seem like add ons to TeX (maybe a change to the eyes and mouth, but not stomach). Others, like pdfTeX, seem like changes to the underlying system (i.e., a change to the stomach).  Is there a good reference that explains the differences and the evolution? After enabling hibernate grub menu appears on start-up with 30s timeout My system is not dual boot just single OS UBUNTU 16.04. Prior to making the following changes, the Grub screen would not appear at start-up (OK).  After following these steps from  to enable hibernate in the system menu:   Open terminal and enter;   sudo nano /etc/polkit-1/localauthority/50-local.d/com.ubuntu.enable-hibernate.pkla  add the lines into the empty file;  [Re-enable hibernate by default in upower] Identity=unix-user:* Action=org.freedesktop.upower.hibernate ResultActive=yes  [Re-enable hibernate by default in logind] Identity=unix-user:* Action=org.freedesktop.login1.hibernate;org.freedesktop.login1.handle-hibernate-key;org.freedesktop.login1;org.freedesktop.login1.hibernate-multiple-sessions;org.freedesktop.login1.hibernate-ignore-inhibit ResultActive=yes  ^X then Y to save (a filename was suggested and I accepted without taking note).  Reboot for the changes to take effect.   hibernate now works and appears in the menu (OK) but whenever I reboot I now get a Grub menu at start up with a full 30 second timeout (NOK). I tried playing around with the Grub menu timeout settings but still get the full 30 sec timeout.  How do I stop the Grub menu from appearing and reduce my boot time to what it was before I made these changes.   Current Grub settings;   GRUB_DEFAULT="0" GRUB_HIDDEN_TIMEOUT="0" GRUB_HIDDEN_TIMEOUT_QUIET="true" GRUB_TIMEOUT="1" GRUB_DISTRIBUTOR="`lsb_release -i -s 2&gt; /dev/null || echo Debian`" GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX="" I am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible I would prefer it if this could be kept relatively simple, but I imagine it is entirely possible there is no real answer to my query anyway.   As I said above, I want to gain a little deeper understanding of what the n-th homology group actually means: I can happily calculate away using Mayer-Vietoris but it doesn't really give me a great deal of intuition about what the n-th homology group actually means. For example, with homotopy groups, the fundamental group is in some sense a description of how loops behave on the object in question, and it is obvious to me why that is what it is for say, the torus or the circle. However, I have no idea what, if anything, I am actually saying about a triangulable object when I talk about it having 0-th homology group this or 1st homology group that.  The best I have been able to find online or in my limited book selection is the brief description "intuitively, the zeroth homology group counts how many disjoint pieces make up the shape and gives that many copies of $\Bbb Z$, while the other homology groups count different types of holes". What 'different types of holes' are there, roughly speaking? I appreciate that it may often be completely non-obvious what the low-order homology groups are for some complicated construction, but perhaps in simpler examples it might be more explicable. Are there (simple) cases where I could say, just from looking something like e.g. the torus, what its zero-th or first or second etc. homology group was based on the nature of the object? I guess in the zero-th case it is, as my source () above says, related to the number of disjoint pieces. Can we delve deeper than this for the other homology groups?   Any book/website suggestions would be welcomed (preferably websites as I am nowhere near a library!) - I have Hatcher but not a great deal else, and I haven't gleaned as much as I wish to from that alone. Of course I know that there is a great deal we don't know about homology groups even today, so I don't expect some magical all-encompassing answer, but any thoughts you could provide would be appreciated. I hope this question is appropriate for SE Mathematics, apologies if not! -M I have asked this question on SO but perhaps that was the wrong location. Apologies for double posting but the question may have a better audience here.  The original question is:   And the text is below. Thanks.    When running the postgresql [9.3] "COPY TO" functionality, the postgres user creates a file. I have told it to put the file in the tmp directory (although this can be changed if needed).  This is run as part of a cron job, the user is an "export" user set up especially for this task. The end goal is for the exported file to end up in /home/export/created_file.csv.  1) postgres can't stash the exported file straight into the home directory of another user without setting up the folder to be overly permissive.  2) the export user can copy the file from /tmp to it's home directory, but then can not clean up (delete) the postgres owned file in /tmp.  This is a daily process and so old files will need to be deleted in some way. I can add a cron job to clean up the files afterwards but I was hoping for a cleaner solution - either a way to change the permissions on the file that postgres creates, or some way to give the export user the additional permission of being able to delete that file in advance of it's creation.  Giving the export user the same permissions as postgres in general however isn't an option, as it would then have free reign over the database.  Any advice, be it unix based or postgres based would be very welcome, thanks. What is the difference between operator "=" and "==" in Bash? Prove that $Ax=0$ has infinite solutions if and only if $ \text{rank}(A)&lt;n$   Here is one way of my if proof, but I don't know how to proof the other part.  Let $r$ be the rank of $A$. Then $ r\leq  n$. If $ r&lt;n$, then there are $n-r$ linearly independent solutions. Furthermore any linear combination of these solutions will also be a solution of $Ax=0$. Hence, in this case, the equation $Ax=0$ has infinite number of solutions Why did Aragorn and Theoden ride out at the Hornburg? In the Battle of the Hornburg from The Two Towers, Aragorn and Theoden say "We must ride out." Why did they do this?  Was it just a matter of honor? Did they ride out on their horses in the middle of battle because they thought it was the end, as they were outnumbered? Was it because Aragorn knew Gandalf was coming so they were confident they were about to win? Given that $X,Y$ are independent $N(0,1)$ , show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $N(0,\frac{1}{4})$ It is given that $X,Y \overset{\text{i.i.d.}}{\sim} N(0,1)$  Show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}} \overset{\text{i.i.d.}}{\sim} N(0,\frac{1}{4})$  I was thinking of making polar transformations $X=r \cos \theta, Y=r \sin \theta$  Then I am getting stuck at ranges of $\theta$