Let $A$ and B be two subsets of the universal set. Give a rigorous proof of the following theorem. $A$ is a subset of $B$ if and only if $A \cup B = B$.  I'm not sure how to prove this. I'm not sure how you do a rigorous proof.
I am trying to define a proof for  $A \cup B = B$ if and only if $A \subset B$  I started with:  $$A \cup B$$ $$x \in A \lor x \in B$$   but I am unsure on how to continue.
I have played and tried to GM games since I was in elementary school, beginning with games like Shadowrun, Earthdawn, and Champions. Recently, I've gotten dack into gaming with a variety of other games (newer Shadowrun and Earthdawn, plus TOON! and Outbreak:Undead).  I've found examples of actual play and advice on how to actually GM a game in games to be lacking. I can only help but feel I am not alone as someone who isn't already a LITERAL dungeon MASTER.  I fully realize that there are an unlimited number of GM styles, personalities, etc. I do not think I am a very good GM though. No one else wants to be the GM, and few care enough to learn the complex rules that I often have to assist with and explain AS WE PLAY. Being left with the responsibility is quite overwhelming unless I create dynamic encounters or make it all up on the spot. I'm often left having no idea what to do.  Examples of Why I Think I Need Help  This question isn't about these examples so please don't dig into them in the answers - they're here to explain the basic things that leave me feeling like I don't know how to proceed as a GM.  I've read preparation is important, but last time I tried to make a story, the players immediately killed the primary character (a friendly person.) within about 2 minutes of starting the game. While I realize now that I let the dice take control over the story, it was rather difficult to continue a story that relied on this friendly character, with the players making enemies with the character's faction. It is still discouraging when my only other attempt at a prepared story ended with the players wanting to go in a complete opposite direction of where I planned. Both times, the hours of work I put into it were completely wasted, leaving me as ignorant as I always was before.  My group often wants to play Shadowrun, Earthdawn, or L5R- but I feel as though I am a horrible GM. The last time we played Shadowrun, I was bored out of my mind as we rolled hundreds of dice to do an obvious straight-forward "kick down the door" mission.   If I make security as I believe it should or as the forum threads describe their security or a few example missions I can scrounge up (examples of stories, not actual gameplay), my players literally feel completely hopeless as to how to achieve their mission without total failure.   One time when I was a teenager, I had players blow up at me saying "We have no idea what would work. You're telling us that our character's experience or the ally operative says that would get us killed, but what else can we do?" Since there are no examples of what players can do - I am left with the same feeling as the players. As an adult, I've had a similar occurrence where a player said to me, "I don't know what to do. I don't think we can even begin this mission." I had to make something up and just go with it, creating an NPC group (that fit in with the story) that goes in before the players to do something stupid so they don't have to. Mainly because I ALSO didn't know how to accomplish the mission, so I acted as the players would using my NPC group. Extremely boring and IMO a failed dungeon run.  I've had successful stories, but I've had more boring encounters than successful ones. Seeing as how roleplaying takes endless hours to complete a story, I do not see benefit in an activity that wastes more time than is enjoyable.  How Do You Learn To GM?  Where are the resources? Books, blogs, gameplay examples? Am I missing them? How did YOU learn to be a good GM? Or do you feel as bad at this as I do, even when it's a successful, fun encounter? i have a scenario where i want to know in which position number the column name comes   like i have sample data below   ID|NAME|S1|BQ|S2|VN|D1 1|vimal|10.12|12.4|56.2|12.2|11.22 2|vilas|10.12|12.4|56.2|12.2|11.22 3|viky|10.12|12.4|56.2|12.2|11.22   Colum name : S1 S2 D1   output should be :  3 5 7 Minkowski's complex Euclidean space vs. the real pseudo-Euclidean version Invalid selectOptions found. Use SelectOption type in Apex Quantum mechanics scattering theory I can describe "IMHO" as an acronym for "in my humble opinion."  How can I describe the reverse processes of translating the acronym into the actual phrase?  That is to say, if "IMHO" is the acronym, what do we call "in my humble opinion"? What is the function __construct used for? I have been noticing __construct a lot with classes. I did a little reading and surfing the web, but I couldn't find an explanation I could understand. I am just beginning with OOP.  I was wondering if someone could give me a general idea of what it is, and then a simple example of how it is used with PHP? It's easy to check that for any natural $n$ $$\frac{n+1}{n}=\cfrac{1}{2-\cfrac{n+2}{n+1}}.$$  Now,   $$1=\frac{1}{2-1}=\frac{1}{2-\cfrac{1}{2-1}}=\frac{1}{2-\cfrac{1}{2-\cfrac{1}{2-1}}}=\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{1}{2-1}}}}=\ldots =\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{1}{2-\frac{1}{2-\dots}}}}},$$  $$2=\cfrac{1}{2-\cfrac{3}{2}}=\cfrac{1}{2-\cfrac{1}{2-\cfrac{4}{3}}}=\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{5}{4}}}}=\cfrac{1}{2-\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{6}{5}}}}}=\ldots =\cfrac{1}{2-\frac{1}{2-\frac{1}{2-\frac{1}{2-\frac{1}{2-\ldots}}}}}.$$  Since the right hand sides are the same, hence $1=2$. How to deal with travel excitement? How does one usually deal with excitement about travel, prior to travel ? A lot of people feel them butterflies in their stomach before a big travel is coming up. Is there an effective way to put these feelings away ? Or how can one best deal with such excitement/feelings (which almost cause interference with day to day activities) ? Customized prompt does not hold when a child bash is ran I have been learning bash, and while doing so I played around with the .bash_profile file to be able to put greeting messages etc. Later I just removed these changes. But after doing that my default configuration of prompt got lost: originally, the prompt was all in bold, and \w part was blue so that when I was in a deeply nested directory the prompt would not seem so entangled. I think that was the default for Ubuntu (correct me if i'm wrong), however like  I said it got lost and my prompt was non-bold and monochromatic resulting in a mess when I'm in long named directories.  So, last night I tried customizing my prompt in the fashion I mentioned above. It worked well after I added the code to .bash_profile  Code was like this:  export PS1="\[$(tput bold)\]\u@\h:\[$(tput sgr 0)\]\e[1;34m\w$ \e[m"   However if I try to run another bash shell in the current one, the prompt becomes unformatted:    Is that normal? Isn't the whole point of exporting a variable to make it accessible from all the child processes forked from that shell? Why it doesn't that apply to this situation? Sum $\frac{1}{6} + \frac{5}{6\cdot 12} + \frac{5\cdot 8}{6\cdot 12\cdot 18} + \frac{5\cdot 8\cdot 11}{6\cdot 12\cdot 18\cdot 24}+\ldots$ A series is given as follows   $$\frac{1}{6} + \frac{5}{6\cdot 12} + \frac{5\cdot 8}{6\cdot 12\cdot 18} + \frac{5\cdot 8\cdot 11}{6\cdot 12\cdot 18\cdot 24}+\ldots$$  Can you give me hints to get started finding its value? Thanks. How to handle users who don't select answers for their questions? What is "#!' in a script file? $cat shebang.sed  #! /bin/sed -f s/red/xxx/ s/BLUE/1234/  $./shebang.sed rgb lower (#1): "xxx green blue" UPPER (#2): "RED GREEN 1234"  $sed -f shebang.sed rgb lower (#1): "xxx green blue" UPPER (#2): "RED GREEN 1234"   I may guess what #! is doing. However, I don't say I know it. what is "#!"? Why does the multi-paragraph quotation rule exist? My friend installed Windows 7 on his computer (upgrading from XP).  He wasn't sure if his processor supported x64 or not, so he installed the x86 version, just to be on the safe side.  After I looked it up, it turns out his processor does support x64, and now we want to upgrade.  The question is: is it possible to upgrade without completely re-installing the OS?  Can I? Or do I have to format the C: partition again? Is the product of symmetric positive semidefinite matrices positive definite? The MathJax code in Mathematics Stack Exchange doesn't render after I updated to the new version (It used to):     App Version: 1.6.2.5 Device: iPhone 6s OS Version: Version 9.0 (Build 13A342) How to tile textures in Cycles? I have often heard it recommended that a user account should be disabled by setting its shell to /bin/false. But, on my existing Linux systems, I see that a great number of existing accounts (all of them service accounts) have a shell of /sbin/nologin instead.  I see from the man page that /sbin/nologin prints a message to the user saying the account is disabled, and then exits. Presumably /bin/false would not print anything.  I also see that /sbin/nologin is listed in /etc/shells, while /bin/false is not.  The man page says that FTP will disable access for users with a shell not listed in /etc/shells and implies that other programs may do the same. Does that mean that somebody could FTP in with an account that has /sbin/nologin as its shell?  What is the difference here? Which one of these should I use to disable a user account, and in what circumstances? What other effects does a listing in /etc/shells have? The man page for GNU find states:    -exec command ;     [...] The  string `{}'  is  replaced  by the current      file name being processed everywhere it occurs in the      arguments to the command, not just in arguments where      it is alone, as in some  versions  of  find.     Both  of  these  constructions might need to be escaped      (with a `\') or quoted to protect them from expansion      by the shell.     That's from the man to find (GNU findutils) 4.4.2.   Now I tested this with bash and dash, and both don't need to have the {} being masked. Here is a simple test:  find /etc -name "hosts" -exec md5sum {} \;    Is there a shell, for which I really need to mask the braces? Note, that it doesn't depend upon whether the file found contains a blank (invoked from bash):   find ~ -maxdepth 1 -type d -name "U*" -exec ls -d {} \;  /home/stefan/Ubuntu One   This changes if the found file is passed to a subshell:  find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d {}' \;  ls: cannot access /home/stefan/Ubuntu: No such file or directory ls: cannot access One: No such file or directory   which can be solved by:  find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d "$0"' {} \;   in contrast to:   find ~ -maxdepth 3 -type d -name "U*" -exec bash -c 'ls -d "{}"' \;  /home/stefan/Ubuntu One   but that's not what the man page is talking about, is it? So which shell treats {} in a different way?