I'm studying the proof that path-connectedness implies connectedness. However there is one assertion that I don't understand. Let $f:[0,1] \rightarrow X$ be a continuous mapping where $X$ is a metric space. Let $U$ and $V$ be two disjoint subsets of X such that $X = U \cup V$. I don't understand why $f^{-1}(U) \cap f^{-1}(V) = \emptyset $. Is it because: $$ \emptyset = f^{-1}(\emptyset) = f^{-1}(U \cap V)=f^{-1}(U) \cap f^{-1}(V)$$ That makes sense but I thought the last equality $f^{-1}(U \cap V)=f^{-1}(U) \cap f^{-1}(V)$ only holds iif $f^{-1}$ is injective. Does that mean $f^{-1}$ is injective or am I missing something? Or is it because $f$ is continuous? For a reference to the proof:
Assume $f:A\to A'$ is a function, $B\subset A'$, $C\subset A'$, and $f^{-1}(B\cap C)=\emptyset$  How can we see that $f^{-1}(B)\cap f^{-1}(C)=\emptyset$?
I know how to CALCULATE dot product, but still don't understand MEANING of it.  I mean, If I have to explain vector addition I would say that "Sum of two vectors give you the third vector that has the same effect as other two combined"  But I don't know how to explain dot product. For example we have two vectors: v=(3,1) and w=(2,4). Dot product of these two is 3*2 + 1*4 = 10. What does this result (10) tell us? I have been getting this question lately from students and although I have a lot of information to offer, I have not found a source that I can point people to where they can read an update answer (I have found a lot of misinformation and obsolete information). So, some of the questions I have for package formats like snap, appimage, flatpak and others in this evolution of universal packaging systems are:   Who created the package format? What features does it offer? What features are unique to it? (That the others do not yet have) Who supports it? What Distributions use it? What focus does the package have? (For Desktop, Clouds, Mobile, etc..) Which are more actively developed? After the therapy, eight children (43%) became able to crawl/move on their back.   Or should I use "on their backs"?   Singular because each child only has one back, or plural because we're dealing with eight backs? How to create a Java application which can be run by a click? Adding DC voltages of PSU I need to add two voltages together that share a common ground (power supply from computer). I tried using bridge rectifiers and Op-Amps. I am trying to hide Authoring information, Revision information, Publishing options tabs in create/edit node for any content type. I tried different modules with no luck. Please give me suggestions how to achieve this. Thanks a lot Using \footnote in a figure's \caption Maybe this is an easy one, but I struggled with this now too long :)  I want to have a footnote in a caption of a figure, see the example.   \begin{figure}[!ht]   \caption{a figure caption\footnote{where i got it from}}   \label{somelabel}   \begin{center}        \pgfuseimage{...}   \end{center} \end{figure}   The compilation error reads as follows  ! Argument of \@caption has an extra }. &lt;inserted text&gt;              \par  l.192 ...i got it from}}   The actual tex code for my figure with the answer of Leo  \pgfdeclareimage[width=6cm]{aba.medcenter}{aba.medcenter}  \begin{figure}[!ht]   \begin{minipage}{\textwidth}     \caption[Medcenter Monthly Medication System]{Medcenter\textsuperscript\textregistered Monthly Medication System\footnote{Quelle Bild: http://www.amazon.com/dp/B000RZPL0M}}     \label{aba.medcenter}     \begin{center}       \pgfuseimage{aba.medcenter}     \end{center}   \end{minipage} \end{figure}   leads to the error  ! LaTeX Error: Command \itshape invalid in math mode.   in the same line. If I comment out the foot note, everything compiles fine. \textsuperscript\texttrademark isn't the problem, too. Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem. Proof/Derivation of Closed form of Binomial Expression $\sum\limits_{k=0}^{2n}(-1)^k\binom{2n}{k}^2$ The binomial expression given as follows: $$\sum_{k=0}^{2n}\left(-1\right)^{k}\binom{2n}{k}^{2}$$ results nicely into the following closed form: $$(-1)^{n}\binom{2n}{n}$$ I wish to know how exactly is it being done? I haven't been able to make much progress in solving it. My approach: \begin{align} \sum_{k=0}^{2n}(-1)^{k}\binom{2n}{k}^{2} = \binom{2n}{0}^{2} - \binom{2n}{1}^{2} + \binom{2n}{2}^{2} - ... -\binom{2n}{2n-1}^{2} + \binom{2n}{2n}^{2} \\ = \binom{2n}{0}.\binom{2n}{0} - \binom{2n}{1}.\binom{2n}{1} + \binom{2n}{2}.\binom{2n}{2} - ... -\binom{2n}{2n-1}.\binom{2n}{2n-1} + \binom{2n}{2n}.\binom{2n}{2n} \\ \text{By Symmetry of binomial coefficients} \\ = \binom{2n}{2n}.\binom{2n}{0} - \binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n-2}.\binom{2n}{2} - ... -\binom{2n}{1}.\binom{2n}{2n-1} + \binom{2n}{0}.\binom{2n}{2n} \\ = \binom{2n}{2n}.\binom{2n}{0} + \binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n-2}.\binom{2n}{2} + ... +\binom{2n}{1}.\binom{2n}{2n-1} + \binom{2n}{0}.\binom{2n}{2n} - 2.\left(\binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n -3}.\binom{2n}{3} + ... + \binom{2n}{1}.\binom{2n}{2n-1}\right) \\ \text{By Vandermond's identity, first component, i.e.not enclosed within -2.(...) evaluates to C(4n, 2n)} \\ \binom{4n}{2n} - 2.(\binom{2n}{2n -1}.\binom{2n}{1} + \binom{2n}{2n -3}.\binom{2n}{3} + ... + \binom{2n}{1}.\binom{2n}{2n-1}) \end{align} I'm lost beyond this point. It will be extremely helpful if someone can direct me in the right direction or provide the answer to this perplexing and challenging problem. Thank you. $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ for distinct prime $p,q$ How to train students to write high-quality research papers? As someone supervising postgrad students, I wonder what is the best way to train these students to write high-quality research papers?    The best way seems to be through coauthorship but ...  On the one hand I can be hands-off and rather give them feedback as best as I can but let them do most of the work themselves, sort of like a "sink or swim" strategy.  On the other hand I can be hands-on and rewrite the paper (or large parts of it) and hope that they learn from the example I give.  On the one hand I want them to learn to write papers without me some day and I want them to learn as quickly as possible to take load off me.  On the other hand I want us to publish in the best place possible with as little stress as possible.  On the one hand I want them to see the peer review comments from what they themselves have written.  On the other hand I do not want them to be discouraged by the peer review system early on and do not want to submit papers I know we could have written better.  On the one hand I want them to get confidence by being published and presenting work at good venues.  On the other hand I wonder if they need to get there themselves to get that confidence boost.    At the moment I'm very hands on. There's a few factors for this: I don't like submitting papers that are not up to a certain standard, I'm a native English speaker in a non-English speaking country with non-native speaking students, publications are a high priority for me, etc.  But recently a friend/colleague warned me that I should not be so hands-on with the writing of the papers of my students. He pointed to recently graduated researchers in my area that published good work as students but who struggled after leaving their PhD supervisor saying their supervisor did them no favours in the long term ... that by being so hands on, while the students have success getting published and so forth with me, I'm not really doing the most important thing a good supervisor should do: making myself obsolete.    My question is how to train students to write good research papers and in what ways can I balance being hands-on and hands-off?  I'm also interested in methods not directly involving coauthorship. VMware won't work after Ubuntu Upgrade I upgraded to Ubuntu 15.10 and now if I want to open VMware Player it doesn't do anything. I also uninstalled and installed it again, still nothing.  What can I do? The interaction of TeX and Lua can be quite delicate at times.  While the former has category codes and tokens, the latter has escape sequences and datatypes.  Mixing the disparate concepts of these two worlds can lead to unexpected behaviour which favours common errors, known as gotchas. In the community wiki answer below you can find an overview of the common mistakes that people make when getting to grips with LuaTeX. Getting file names without extensions Lost reputation after answering my own question with bounty Timeline:  Asked a question, no (interesting) answers for two weeks  Set bounty on it this morning  Found the answer to my question two hours later. Answered my own question.  Bounty reputation is lost. It was not awarded to me, it was not returned to me (the 100 I invested).   Right now, I feel really stupid for answering my own bounty question.   If this is accepted behavior, then user should be warned - you are answering your own bounty question, your bounty reputation will be LOST. Focus existing terminal with `Ctrl-Alt-T` shortcut I'm writing a technical documentation and I want to explain step-by-step how to go through various menus.  For example I used to write Open with &gt; VLC media player.  That's how it looks in Windows 8:     What I want in Latex should looks similar to this: light grey box, dark grey border and black windows-like font.    I've seen , suggesting the package  - but it looks very old fashioned - are there any more modern approaches? How to return only the Date from a SQL Server DateTime datatype SELECT GETDATE()   Returns: 2008-09-22 15:24:13.790  I want that date part without the time part: 2008-09-22 00:00:00.000  How can I get that? Xpath expression with multiple predicates When to use a linked list over an array/array list? I use a lot of lists and arrays but I have yet to come across a scenario in which the array list couldn't be used just as easily as, if not easier than, the linked list.  I was hoping someone could give me some examples of when the linked list is notably better.