Cardinality of set and its power set
Cardinality of a set A is strictly less than the cardinality of the power set of A
Cannot expand asterisk without proper permission How to know the motherboard BIOS version using the command line When should I use exposure compensation? When should I use exposure compensation, rather than ISO, shutter speed or aperture? Why can't helicopters reach mount everest? A spaceship crew stuck in a damaged spaceship that's losing power Earlier this year, I read a huge (1000+ pg) novel about a cloistered group of monks that sort of worshiped math. The story involved alternate dimensions or time travel or something. A big thing in the book was coordinates, containing data like for speed or something as an alternate axis. There were several rings of the monastery.   At some point the main character was called out of his monastery to never return. I'm hoping to figure out the name so I can discuss it with my book club. How to hide "internal" polygon boundaries? Why does flag marking as helpful/declined not always correlate with moderator action? I flagged a post and (choose one)   my flag was marked as helpful, yet nothing was done my flag was marked as declined, yet the post was deleted/edited/closed   Why? Are all multiplicative functions additive? Suppose $cf(x)=f(cx)$ and $f:\mathbb{R}\to\mathbb{R}$. I believe it follows that $f(x+y)=f(x)+f(y)$.   Proof: There is some $c$ such that $y=cx$. Then  $$f(x+y)=f\left((1+c)x\right)=(1+c)f(x)=f(x)+cf(x)=f(x)+f(cx)=f(x)+f(y)$$  QED.  I wonder if the same thing holds for when $f:\mathbb{R}^n\to\mathbb{R}^n$? I can't use the same trick, because all vectors are not scalar multiples.  I tried thinking of it in terms of basis units, but didn't get anywhere. How to prove that there exist no functions $f,g:\Bbb{R}\to\Bbb{R}$ such that $f(g(x))=x^{2018}$ and $g(f(x))=x^{2019}$? I have tried a little bit to solve the problem which goes as follows:   My intuition says that there exist no $f:\Bbb R\to\Bbb R$ such that $$f(g(x))=x^{2018}\text{ and }g(f(x))=x^{2019}.$$  Note that $$f(g(f(x)))=f(x)^{2018}\implies f(x^{2019})=f(x)^{2018}$$  Similarly, $$g(x^{2018})=g(x)^{2019}$$ Putting $x=1$ in $f(x^{2019})=f(x)^{2018}$, we get $f(1)=f(1)^{2018}$ and thus $f(1)=0$ or $f(1)=1$.  Similarly, putting $x=1$ in $g(x^{2018})=g(x)^{2019}$ we get $g(1)=g(1)^{2019}$ and thus $g(1)=0,1,-1$.  Now, I can't proceed further. Can anybody solve it? Thanks for assistance in advance. How to loop through a HashMap in JSP? What is the best way of searching the whole classpath for an annotated class?  I'm doing a library and I want to allow the users to annotate their classes, so when the Web application starts I need to scan the whole classpath for certain annotation.  Do you know a library or a Java facility to do this?  Edit: I'm thinking about something like the new functionality for Java EE 5 Web Services or EJB's. You annotate your class with @WebService or @EJB and the system finds these classes while loading so they are accessible remotely. Given a set of curved geodesics in 2D space, is there a way to determine the geometrical shape of said space? Is a function defined at a single point continuous? Is a function defined at a single point continuous?  For example $f:\{0\}\to\{0\}$ defined by $f(x)=\sqrt{x}+\sqrt{-x}$ is a sum of two continuous functions and is therefore continuous, however for $f$ to be continuous, it has to have the limit $\lim\limits_{x_0\to x}f(x_0)=f(x)$ but it dosen't seem to have that limit. Device : Moto X Play ( Rooted, Stock, Android 6.0.1, msm8916 Board, XT1562)  Why do I want to limit charging ?  ( Let's leave out whether my reasons are valid or not and accept that I want to, if you disagree with my reasons)  To improve the longevity of battery , I wish to limit charging when it reaches 90%, as explained here    Is it possible?  Yes, at least on my previous device Honor 6, as brought out here , with two different ways of doing it  I have tried  unsuccessfully to identify the file responsible in which changing values would limit charging  I am looking for the system file responsible for stopping charging, with it's true path (without symlinks)  (as I saw on on my previous device, locating the path of file was a big challenge after identifying it) Unsealed meat in freezer Below is my situation:  Is completely (vacuum) sealing leftover meat in the freezer absolutely necessary? I would always just use a plastic bag to cover the meat and fold it around couple of times so it's almost as good as sealed (usually grocery plastic bag is much bigger than the piece of meat) and have never had a problem with it last a decade or so I have been cooking.  But recently I got this new roommate who insists on me using special small plastic bags with sealers to use and he is being very paranoid, obsessive about it.  So my question is: Is he right? Does light pollution affect sleep quality while already asleep? Many articles on the internet write about melatonin and its effects, making you tired. A lot of articles also point out how the use of devices before going to bed in modern days likely decrease melatonin production. Therefore, making it harder to fall asleep. But this information is only about the process of getting to sleep, before we are actually sleeping.  A recent Harvard Medical article 1 states:     High exposure to bright, artificial outdoor lights during the night may result in sleepless nights for older adults.   Referring to a publication on Nov. 15, 2018, in the Journal of Clinical Sleep Medicine.  Considering that:    many people nowadays, use their mobile devices in bed prior to going to sleep; modern cities or houses have (street) lights that are always enabled at night; ignoring the phase of falling sleep; focusing only on the time we are asleep already, so during the actual sleep cycles.   Does a darker room with less or no (artificial) light pollution improve sleep quality, with all associated health benefits as a result? In other words: Does light pollution, while asleep decrease the sleep quality or does light pollution only negatively affect the phase of “falling asleep”?  I wasn't able to find any studies or information regarding sleep quality in relation to light pollution during sleep. Were such studies performed? Are there specific moments when light pollution affects sleep quality the most, for example during REM-sleep or in a specific sleeping phase?    1 I encountered The Squirrel at "A tree."              |   |                                                                         |   |                     "   Hello, I'm The                        |             |                        Squirrel. I can                        |             |                      provide you candies,                     |             |                      and lots of things.                                    |                      I know how much you                                    |        _.-'''-,      love candies. But I                                    |      .'        `\     feel alone in this                                    |     /           /          forest.       "                         |        |    /      .--^_^                                                   |        |    |     /  C ,,\                                                           |    |    |   \  _.)                           ___                            |     \   |   /  \                   ____.----'.-'                            |      '-, \./ \)\)               .-' --  __.-'                   |            \        `-/   );/           ___/  - _.-'                        |             \_________''--'-'________.-' --    /                            |              -   --        -  -     -     __.-'                                              -    --    -     --   --  /                                                 _________________________.-'                                 He asks you some questions. What are the correct answers, and what is the reward for getting them correct? How to fix "The system is running in low-graphics mode" error? Note: This is an attempt to create a canonical question that covers all instances of &quot;low-graphics mode&quot; error that occurs to a user, including but not limited to installation of wrong drivers, incorrect or invalid lightdm greeters, low disk space, incorrect installation of graphics card like ATI and Nvidia, incorrect configuration of xorg.conf file while setting up multiple monitors among others. If you are experiencing the &quot;low-graphics mode&quot; error when trying to login but none of the following answers work for you, please do ask a new question and then update the answers of this canonical question as and when your new question gets answered.   When I try to boot into my computer, I am getting this error:  The system is running in low-graphics mode Your screen, graphics cards, and input device settings could not be detected correctly. You will need to configure these yourself.   How do I fix the failsafe X mode and login into my computer?  Answer index: I couldn’t find any proof on the internet. I would appreciate an elementary proof , but any help would be appreciated!  Thank you.