If a set theory allowed the existence of $x\cup y$, would that imply the existence of $\cup S$?
Can the union axiom be weakened?
Parser for C# Which parsers are available for parsing C# code?  I'm looking for a C# parser that can be used in C# and give me access to line and file informations about each artefact of the analysed code. Is the following proof correct, or are certain steps illegal?  Suppose $f:G_1\to G_2$ is a surjective group isomorphism, $N_1\lhd G_1, N_2\lhd G_2$ normal subgroups, with $N_1=f^{-1}(N_2)$. Prove that $(G_1/N_1)\cong (G_2/N_2)$.  Proof:  As $f$ is surjective, we an use the first isomorphism theorem to get $G_1/\ker(f)\cong G_2$, which implies that $$(G_1/\ker f)/(N_1/\ker f)\cong G_2/(N_1/\ker f)$$  Using the first isomorphism theorem we also know that $$N_1/\ker (f)\cong f(N_1)=f(f^{-1}(N_2))=N_2$$  Combining this we can see that $(G_1/\ker f)/(N_1/\ker f)\cong G_2/(N_1/\ker f)\cong G_2/N_2$.  Using the third isomorphism theorem we have that: $$ (G_1/\ker f)/(N_1/\ker f)\cong G_1/N_1$$  And from the last two lines we have $G_1/N_1\cong G_2/N_2$ $\tag*{$\Box$}$ I dug a large room to specifications, 3 blocks tall and 16 blocks wide as well as long. I dug it at 10 levels above bedrock, in a location {(-240, 0) -> (-224, 16)} specified by  slime finder app. I am definitely more than 24 blocks away and less than 32 away from the spawn room, as specified by the wiki .  Yet, no slimes. I have been sitting in my location patiently listening for the moist slapping sounds of slime. I have gone in to the room every time I heard a phantom slap but still nothing.  Am I just being dumb somehow or do the minecraft lords just hate me?  If it helps/anyone cares here's my seed: 6405460532872068998 I have seen many camera review sites illustrate the effect of using different focal lengths on the same photograph using frames. There are also similar illustrations to compare different sensors (). I find this kind of visualization to be very useful in comparing the actual effect of the longer focal length\different sensor size. Of course the ability to do a wider focal length visualization would also be equally useful, but that's not possible.  Is there any ready-made software or plugin that enables this? If not, is there any simple technique to visualize the effect of longer focal lengths on an image (one would of course have to use the zoom information in the image EXIF data to accurately represent this)? All I can think of is to use some trigonometry to do the necessary calculations for the frame sizes for cropping. Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing singleton sets? If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group. Assume that $\pi$ is an automorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|&gt;\frac{3|G|}{4}$, then $G$ is an abelian group.    Anyone has any idea on how to solve this ? I have no idea how to start. Why is there no option to login as root? Why does Ubuntu not allow the GUI users to login as root at the time of system start up like administrator in Windows?  Why they are restricting us to login only through the terminal? Is there any simple (IE: right click in Nautilus) way to password protect a particular folder/file in Ubuntu? I've got a few files containing sensitive info and I'd much prefer that if/when I leave my computer alone, they aren't accidentally accessed by someone else.  The secruty does not have to be extremely tight. My only concern is that when family/friends come over, I don't really like the idea of them looking at my bank details, accounts or, you guessed it, porn collection.  A simple, effective way to let me put my machine in the hands of someone else knowing that said machine can not cause me embarresment is the sole reason why I'd like to see this in Ubuntu. Why can "find" not be used in PowerShell? What does find.exe find objectionable about the parameters when it is used in a PowerShell console shell?  These commands work as expected in a cmd.exe shell:  PS C:\Windows\System32\WindowsPowerShell\v1.0&gt; find /i "System.Diagnostics.Process" *.ps1xml FIND: Parameter format not correct PS C:\Windows\System32\WindowsPowerShell\v1.0&gt; find /i "System.Diagnostics.Process" *.ps1xml FIND: Parameter format not correct PS C:\Windows\System32\WindowsPowerShell\v1.0&gt; C:\Windows\System32\find.exe /i "System.Diagnostics.Process" *.ps1xml FIND: Parameter format not correct PS C:\Windows\System32\WindowsPowerShell\v1.0&gt; C:\Windows\System32\find.exe /i "System.Diagnostics.Process" .\DotNetTypes.format.ps1xml FIND: Parameter format not correct How do I hide the System Reserved partition from the Windows Explorer?    On the above screenshot, I want to get rid of System Reserved (D:). How do I do that?  I will never need to access it from there, why would I need the entry then?  PS: I want to hide it, not delete it. Why are my sauces splitting? To find minimum distance between two curves What is *Cell standby* and how can I keep it from eating my battery? Looking up my battery statistics, I almost always find Cell standby amongst the top consumers:    I never thought that "idle" would use so much energy. I saw that other people encounter similar problems:       Answers on those questions just give some raw ideas. I would like to know what exactly stands behind this, an whether there are ways to keep this "standby" from eating my battery? Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups? I think the answer is yes.   Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then $\{(e_{\lambda_1},e_{\lambda_2}):\lambda_1,\lambda_2\in\Lambda\}\subset\mathbb{R}^2$ is a Hamel basis of $\mathbb{R}^2$. Consider set theoretic bijection $i:\Lambda\to\Lambda\times\Lambda$ define map $\varphi:\mathbb{R}\to\mathbb{R}^2$ by its action on element of Hamel basis of $\mathbb{R}$ by equality $\varphi(e_\lambda)=(e_{i(\lambda)_1},e_{i(\lambda)_2})$. This is an isomorphism between $\mathbb{Q}$-vector spaces $\mathbb{R}$ and $\mathbb{R}^2$.  Question Could you tell me is this proof correct, and if it is not, where is the mistake?  EDIT: Thanks to GAJO we found that this proof is wrong. The correct one basis is  $$ \{(e_\lambda,0):\lambda\in \Lambda\}\cup\{(0,e_\lambda):\lambda\in\Lambda\} $$ So we had to look at bijections of the form $i:\Lambda\to\Lambda\coprod\Lambda$ and the desired linear operator is defined by equalities $\varphi(e_\lambda)=(e_{i(\lambda)},0)$ if $i(\lambda)$ lies in the first copy $\Lambda$ and $\varphi(e_\lambda)=(0,e_{i(\lambda)})$ otherwise. I have to prove that:     $$\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$$   I tried to open up the right side with Pascal's definition that: $$ { n \choose k} = {n-1 \choose {k}} + {n-1 \choose {k-1}}$$  Here is what I came up with, and I am sure it is wrong because it does not equal the left side:  $$ {m+n+1 \choose m+1} = {m+n \choose m+1} + {m+n \choose m} = ... ={m+n \choose m+1} + {m+n-1 \choose m} + {m+n-2 \choose m-1} + ... + {m \choose m+1} = \sum_{k=0}^{n} {m+k \choose m+k-n+1 }  $$  Which, again, probably is wrong because it is not equal $\sum_{k=0}^{n} { m+k \choose m}$. Any help is appreciated Explicitly finding the sum of $\arctan(1/(n^2+n+1))$ This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right).$$ How to calculate the sum of sequence $$\frac{1}{\binom{n}{1}}+\frac{1}{\binom{n}{2}}+\frac{1}{\binom{n}{3}}+\cdots+\frac{1}{\binom{n}{n}}=？$$ How about its limit? How can I change the registered name in Windows 7 to my own? how do I change the registered name in Windows 7 to my own name? Keyboard layout/language changes in every boot try+gerund/infinitive (Difference in meaning) I tried to open the door. I tried opening the door   We can use both infinitive and gerund with verb "try" as used in above sentences. Does it change the meaning of the  sentence if use gerund or infinitive?.(here i don't know if the meaning of sentences is changed.)     The same confusion arise in these two sentences.  1.I stopped seeing him. 2. I stopped to see him.(But in case of these two sentence i know the meaning is changed but don't know exactly what the change is).