What is the importance of the concept of probability space?
Why do we need sigma-algebras to define probability spaces?
Kernel of a substitution map Why is disk usage greater than the size of all files on it? How to add a matrix to a LaTeX document How do we add a matrix to a LaTeX document? More homework help.  Given the function $f:A \to B$.  Let $C$ be a subset of $A$ and let $D$ be a subset of $B$.  Prove that:  $C$ is a subset of $f^{-1}[f(C)]$  So I have to show that every element of $C$ is in the set $f^{-1}[f(C)]$  I know that $f(C)$ is the image of $C$ in $B$ and that $f^{-1}[f(C)]$ is the pre-image of $f(C)$ into $A$.  Where I'm stuck is how to use all of this information to show/prove that $C$ is indeed a subset.    Do I start with an arbitrary element (hey, let's call it $x$) of $C$? and then show that $f^{-1}[f(x)]$ is $x$?  I could use a little direction here...  Thanks. Story about workers repeatedly doing trivial work I'm trying to remember a (short) story I read in school some time ago. I don't have much, but here it is:   Numbers of workers are assigned to trivial task, like changing broken lightbulbs (or something similar). Once they finished their daily work, another group of workers are putting the broken lightbulbs in again, allowing for an endless, trivlal working circle.   I know that isn't much, but maybe someone has an idea, thanks. How to replace an old version of Ubuntu with a newer one from USB/DVD? I dual booted ubuntu 11.04/11.10 a while ago with a USB stick and i want to uninstall it.   I would like to completely uninstall the old version and get 12.04.  I have a 64 bit laptop. Please help! Why does camera this tracking fail completely even when the solution value is very low? Restarting with new account on iOS/GameCenter Add something to crontab programmatically (over ssh) I have a deployment script, it must add something to a user crontab (trigger a script that clean logs every XXX days), however this must only be done during the first deployment, or when it needs to be updated.  (I can run xxx.py deploy env or xxx.py update env)  so I have to do this:  Check if my cronJob already exist Put my cronJob if it does not already exist or update my cronjob if one of the parameter of the command is different   I don't see how to add/check/remove something to the crontab without using crontab -e or editing the crontab file (download it, rewrite it, re-upload it)  PS : this is a user specific cronjob, "webadmin" is going to do it and he should not use sudo to do it. Dual boot. How do I reinstall the Ubuntu partition? Could the Horcruxes be destroyed with Avada Kedavra? In The Deathly Hallows Hermione says this about destroying Horcruxes:     “It doesn’t have to be a basilisk fang,” said Hermione patiently.   “It has to be something so destructive that the Horcrux can’t repair   itself. Basilisk venom only has one antidote, and it’s incredibly   rare—"      "That’s a problem we’re going to have to solve though, because ripping, smashing, or  crushing a Horcrux won’t do the trick. You’ve got to put it beyond magical repair.”   Mad-Eye Moody from The Goblet of Fire tells us this about Avada Kedavra,     ...there's no countercurse. There's no blocking it.   but also that,      Avada Kedavra's a curse that needs a powerful bit of magic behind it - you could   all get your wands out now and point them at me and say the words, and I doubt   I'd get so much as a nosebleed.    Harry is also seen once attempting to use Crucio on Bellatrix Lestrange in The Order of the Phoenix out of anger at the death of Sirius, but he fails. She tells him,     "You need to mean [Unforgivable Curses], Potter! You need to really want to cause pain - to enjoy it - righteous anger won't hurt me for long - I'll show you how it is done, shall I? I'll give you a lesson -"   Even if Harry and company are not powerful enough to perform the curse themselves, they know several powerful wizards (Kingsley, Lupin, et al.) who could likely perform the curse.  Lastly we know that Voldemort himself destroyed the last bit of his soul that was inside Harry with Avada Kedavra.  Could the Horcruxes be destroyed with Avada Kedavra as well? How much Valyrian steel is there that is unused for link making and why aren't people making weapons out of it? I was reading  that stated Maesters can have Valyrian steel links:     Valyrian Steel links (for the study of Magic + the Occult) are exceedingly rare and held by less than one Maester in a hundred. Ironically, they aren't seen as especially prestigious but rather viewed with suspicion.   This made me wonder how much Valyrian steel there is lying around for Maesters to be adding them to their chains. It seems like such  resource like this would be used for crafting weapons not being kept as ingots ready for Maesters to make links.  Indeed the Wiki I linked above claims:     the amount of Valyrian steel in the world is finite and extremely rare   How much of his extremely rare resource is being kept in reserve for link making and why hasn't it been used for more useful stuff like making weapons? How to shift EXIF date/time created by time in days, hours, minutes? I have two camera bodies, I was shooting while on vacation, when I downloaded the photos to lightroom, one body was set to the correct date/time, the other was off by 722 days, 4 hours, 32 minutes.  Is there a tool available that can help me fix the date/time so that they appear in sequence while browsing my lightroom library?  I'd like for them to be as close to the right time.     I was able to adjust created year, month, date, but it doesn't appear lightroom will allow me to adjust increments (only hour increments).  Many thanks if you can help me in this task! Remap Caps Lock to Ctrl-b I would like to align the view straight on to a face, perpendicular to the normal. How do I align the 3D viewport to the face of a mesh? In the Ubuntu dock on Ubuntu 18.04, multiple windows of the same application are grouped and orange dots are shown to tell you how many windows of the application are opened. When you click on the icon, a small screen capture of each of the windows is shown.  However, these screen captures are shown in an order that's based on which window was last active on the desktop. This is something that clashes with the way I use the dock. I'm used to mentally mapping the windows to their dock positions. As such, I will often be trying to get to "the top one" and then I end up needing several clicks before I actually get to the terminal (or some other application) that I was looking for.  Is there some way to make sure that the order of these screen captures does not change whenever a window becomes active? How can I best describe "making a new discovery"? Let $T$ and $T^{'}$ be topologies on set $X$ such that $T \leq T^{'}$. If $(X,T^{'})$ is path-connected, is $(X,T)$ path connected? Let $T$ and $T^{'}$ be topologies on set $X$ such that $T \leq T^{'}$. If $(X,T^{'})$ is path-connected, is $(X,T)$ path-connected?  I think it is, and I tried to prove it by contradiction, but I got stuck. Any hint helps! I have snippets of Html stored in a table. Not entire pages, no  tags or the like, just basic formatting.  I would like to be able to display that Html as text only, no formatting, on a given page (actually just the first 30 - 50 characters but that's the easy bit).  How do I place the "text" within that Html into a string as straight text?  So this piece of code.  &lt;b&gt;Hello World.&lt;/b&gt;&lt;br/&gt;&lt;p&gt;&lt;i&gt;Is there anyone out there?&lt;/i&gt;&lt;p&gt;   Becomes:  Hello World. Is there anyone out there? Play a Sound with Python