Can we use  to figure this out?   Intuitively the domain seems to be a subset of the codomain, but where do powersets come in?   Thanks in advance!
The collection of all subsets of $\mathbb{Z}$ is uncountable, due to   But how can I prove that the collection of all finite subsets of $\mathbb{Z}$ is countable?
Blending colors in photoshop and illustrator I am working on a print for my wall and I am trying to create something like this:      I have already created the letters in Illustrator and now I am adding the colors in Illustrator and then exporting everything to Photoshop and then using the liquify tool to blend the colors to create a similar effect. I also used the healing patch tool to blend in the hard edges.  Is that the right way of doing it? Or is there another way? The  mentions that it's possible to post one more question after a six-months waiting period and if the question is good enough, the question ban can be lifted:     If you're unable to improve your existing questions, you'll get the chance to ask one new one 6 months after your last question. If that question is positively received, you may be able to continue asking questions; if not, then the ban will be reinstated.   What is positively received in this context? Is it enough if the question has a zero score and is not closed? Or does it have to have a positive score?  How long does the system wait after posting this new question to evaluate this criteria and possibly lift (or reinstate) the question ban? Remove rogue app from menu bar I have a rogue item on my menu bar from a terrible piece of printer/scanner software. I long ago removed the app, but the button for the app remains in the menu bar.   Is there any central way for me to remove an item from the menu bar? Any plist I can hack? command line trick, etc?  It's the one to the left of the Facebook icon:    (I tried holding Command and dragging it away, but it doesn't seem to respect that convention.) Make a failed SSH forward a fatal error I have a script which connects to a remote host to replicate some files via SSH.  I must connect to an rsync daemon listening on port 873.  To accomplish this I have a ssh configuration like this.  Host blah     LocalForward 10873 127.0.0.1:873     Hostname 10.2.4.1   Unfortunately because of the timing of this script, there is a small chance that on occasion when it attempts to establish the SSH session another similar script will have the port open to a different host.  If I connect while another ssh session is open I get the obvious message Address already in use.  me@local:~$ ssh blah bind: Address already in use channel_setup_fwd_listener: cannot listen to port: 10873 Could not request local forwarding. Linux remote 2.6.26-2-amd64 #1 SMP Tue Jan 25 05:59:43 UTC 2011 x86_64 motd... remote:~$    What I want to have happen is to have the SSH client immediately exit with a exit code so I can put some logic into my script to report a failure instead of trying to run any commands against the port forwarded to the incorrect host. How do I get %LocalAppData% in c#? Colors in Photoshop CS6/CC look different than in browser? One of my friends, a graphic designer told me recently that sometimes colors in Photshop look slightly different than the ones rendered in browser.  I know it is possible when you save a file and then use it on a website due to palettes, but he uses given color in PS, for example #b81e1e, then he creates a DOM element and sets its background to #b81e1e and the color is rendered differently in browser than in PS (I guess it's not that subtle as he can easily spot it). It's not a regular thing and it doesn't happen always, but he states it happened to him a few times, although he can't really tell me with witch colors.    Is it even possible? Because I have never really heard about anything like this &amp; nothing like this ever happened to me... Examples and further results about the order of the product of two elements in a group How to restore a system after accidentally removing all kernels? I was trying to delete old kernels, but I must have deleted all of the kernels on my Ubuntu 11.04 laptop. Is there any way to fix this via USB boot or mounting hard drive on another system? Installing Maya on Ubuntu Linux Installing Autodesk Maya on Ubuntu is possible, but can be a bit challenging.  Assuming that one already has the required specs and a solid graphics card driver installed, how can it be done? Though tangentially related to another post on MathOverflow (), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...    Suppose $\Sigma$ is a sigma-algebra over a set, $X$. For any given topology, $\tau$, on $X$ denote by $\mathfrak{B}_X(\tau)$ the Borel algebra over $X$ generated by $\tau$.     Question 1. Does there exist a topology, $\tau$, on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$?   If the answer to the previous question is affirmative, it makes sense to ask for the following too:     Question 2. Denote by ${\frak{T}}_X(\Sigma)$ the family of all topologies $\tau$ on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$ and let $\tau_X(\Sigma) := \bigcap_{\tau \in {\frak{T}}_X(\Sigma)} \tau$. Is $\Sigma = \mathfrak{B}_X({\frak{T}}_X(\Sigma))$?   Updates. Q2 was answered in the negative by Mike (). Graphing y=x^2 and y=x Brown sugar instead of white sugar How to deal with variable income in the budget We have a budget planned out for various categories, and in order to make ends meet we count on child support for my wife's child from a previous marriage.  However, lately we have not been receiving this amount.  How do we deal with this in the budget?    My wife has just been "borrowing" it from our savings, but I don't think that is reasonable given that I find it highly unlikely we will ever be able to recover it.  On the other hand, she doesn't think it reasonable to cut out of her budget and cutting it out of the budget for the kids (which is where it really should be going!) isn't reasonable either. Function return object Return vs. Enter? I noticed that my MacBook's keyboard has the word enter above the return key. I also noticed that on Mac OS X, there's a special symbol for return ↩, and another for enter ⌤. How would I send the enter key if my MacBook has no numpad?  Is there any difference between these two keys? Please tell me which choice below (or rather a better choice that I need to know) works better in my sentences in informal English:     I got myself out on a limb. I need to clear my check by tomorrow. No matter how, I must ............ 2 million bucks by tomorrow.   a) line up [to prepare, organize, or arrange something]  b) hook up [hook somebody up with something to help someone get something that they need or want]  c) fix up [to arrange for someone to have something that they need]  For me, all three options work in this sense with the same meaning and informality degree. Difference between “with”, “at” and “for” with a business title and a company name Avoiding jumping frames in beamer Recovering Home folder after failure to boot I have a small issue. I was having some issues in getting some libraries to work and to resolve it, i deleted some shared libraries by mistake(sudo ...).Now ubuntu fails to boot up but i can access the files from recovery mode. I can see my home folder intact but i cannot even run sudo nautilus as it says it shared libraries not found. The data in my home folder is quite important and i want retrieving it is important. I am running a windows dual boot, if it helps. If anyone could help copy the home folder to a windows drive from the terminal itself(as i can access it at recovery mode) it would be of immense help. Thanks.:-) Would someone give a thorough explanation of 1.14 Minecraft's /data modify?