If B and C are subsets of Y, then prove $f^{-1}(C/B)=f^{-1}(C)/f^{-1}(B)$
Let $f:A\rightarrow B$ and $C\subset B,D\subset B$. Prove that $f^{-1}(C\backslash D)=f^{-1}(C)\backslash f^{-1}(D)$
How much web can Spider-Man produce? Especially if he is not using his  but .   Also what happens to the web that he leaves behind? Does it decay after certain time or lose it stickiness and drops off? For every twice differentiable function $f : \bf R \rightarrow [–2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statements are TRUE? For every twice differentiable function $f : \mathbf R \rightarrow [–2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statement(s) is (are) TRUE ?  (A) There exist $r, s\in \bf R$, where $r &lt; s$, such that $f$ is one-one on the open interval $(r, s)$  (B) There exists $x_0 \in (–4, 0)$ such that $|f'(x_0)| &lt; 1$   (C) $\lim\limits_{x\to\infty}f(x) = 1$   (D) There exists $\alpha\in(–4, 4)$ such that $f(\alpha) + f''(\alpha) = 0$ and $f '(\alpha)\ne0$  My attempt is as follows:-  $A)$ If function is continuous, then definitely in some part of the interval it will be increasing or decreasing hence will be one-one on the open interval $(r,s)$   $f(x)$ cannot be a constant function as in that case $f'(x)=0$ which means $f(0)=\sqrt{85}$, but it doesn't belong to co-domain of $f$  B) By mean value theorem we can say that there exists some $c\in(-4,0)$ for which $4f'(c)=f(0)-f(-4)$  Suppose if $f(0)=2$ and $f(-4)=-2$, then we can only say that there exists $c$ such that $f'(c)=1$. It will not be necessary that there exists a $c$ for which $|f'(c)|&lt;1$  C) $\lim\limits_{x\to\infty}f(x) = 1$, this is not at all necessary.  D) If function is not constant in $(-4,4)$, then we can safely say that at some point $\alpha$, $f(\alpha)\ne0$  But how to know $f(\alpha)+f''(\alpha)=0$  Actual answer is $A,B,D$ but I am able to ascertain only option $A)$. Please help me in this. Combine and Minify Multiple CSS / JS Files Do I need to pay in full for hostel reservation to show proof of accommodation for a Schengen visa? I am applying for a Schengen visa to travel to Germany and one of the requirements is showing proof of accommodation. I booked all my hostels via hostelworld.com, but only paid the non-refundable deposit (a couple of dollars for each place) instead of the entire amount. I intend to pay the rest of it on arrival at the hostel (or cancel it and go CouchSurfing, thus saving money).  But for the visa application, will the non-refundable deposit be enough to show proof of accommodation?  The reason I ask is, I emailed one of the hostels requesting a document showing my name and accommodation confirmation, and they replied saying they could only give me such a document once I pay in FULL.  Is this how I'm supposed to book for all places?  Have I been doing it wrong all this while (by only paying the tiny non-refundable deposit instead of full hostel fare)? How can I restart a Windows 7 computer without screen or mouse? How to change folders color? How do i change the folders colors? i am using this icon theme   and there are several folders. like black, gray, purple, orange(which is the default), yellow, green, red, blue, brown and lightblue  How can i use any of them? for example have a blue folder called "family" in images and another right beside that have a purple folder called "others".  Basically why have he created all those folders with different colors? What is time dilation really? Please will someone explain what time dilation really is and how it occurs? There are lots of questions and answers going into how to calculate time dilation, but none that give an intuitive feel for how it happens. I'm loading an XML document in my C# application with the following:  XDocument xd1 = new XDocument(); xd1 = XDocument.Load(myfile);   but before that, I do test to make sure the file exists with:  File.Exists(myfile);   But... is there an (easy) way to test the file before the XDocument.Load() to make sure it's a valid XML file?  In other words, my user can accidentally click on a different file in the file browser and trying to load, say, a .php file causes an exception.    The only way I can think of is to load it into a StreamWriter and simple do a text search on the first few characters to make sure they say "  Thanks!  -Adeena I have seen the word 'only' placed both before and after a noun. Where should the word 'only' go in this sentence? Is it the writer's choice?   If you have attended every lesson, only the songs need to be rehearsed at home. If you have attended every lesson, the songs only need to be rehearsed at home. ArcGIS ModelBuilder iterator reports only last input shapefile? Using Arc 10.3.1, if I manually enter the names of multiple shapefiles (all from the same folder) into the Check Geometry tool, the (single) output table lists the errors that exist in all of the shapefiles.    That is, only one output table is created, even if multiple input shapefiles have errors.  For example, if two shapefiles have errors, both will be listed in the single output table. Here's a screen print of the input:    However, if I iterate those same shapefiles (using the Feature Classes option), the output table only lists the last shapefile that went through the iteration.  Here's a screenshot of the model:     I suspect that the iterator is running Check Geometry not once, but three times, with a single input shapefile going through each iteration, overwriting the previous output.    How do I get the iterator to list all of the shapefiles with errors, either in separate output tables or aggregated in a single output table? What is the maximum number of pucture repairs an inner tube can take? What overpowered combinations would be available if I allow a bonus action to be used in place of a standard action? It has come up in game a couple of times that a player might want to cast a spell that has a casting time of 1 bonus action using their "main" action (if they have another bonus action they also want to take on that turn, such as giving bardic inspiration, or controlling a Bigby's hand, etc.)  On the face of it, it seems obvious that something (a bonus action) that is usually much faster than a full action could be done as your full action.  Although the question comes up most often with respect to spellcasting, if I house rule this, I would rule that any bonus action can be taken as a regular action instead; however, I would not allow the same type of bonus action to be taken twice (so no giving bardic inspiration to two allies on the same turn, for instance).   Are there any abusive or overpowered combinations I should be wary of if I were to allow a character to take 2 bonus actions instead of one regular action and one bonus action on a turn?   The issue of casting two bonus-action spells would not come up because the rule against casting 2 spells on your turn unless one of them is a cantrip with a casting time of 1 action would still be in effect:  PHB p. 203 (under Bonus Action casting time)     You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.   I know it's hard to prove/justify a negative answer to a question like this, but I'd be happy to get answers that say you don't think there would be any issues if you describe how you came to that conclusion. The following is an excerpt from a New York Times article of April 6.     “Tonight, I ordered a targeted military strike on the air base in Syria from where the chemical attack was launched,” Mr. Trump said in remarks at his Mar-a-Lago estate in Florida. “It is in this vital national security interest of the United States to prevent and deter the spread and use of deadly chemical weapons.”    Shouldn't that "from where" be either "from which" or else just "where" without "from?" I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In the complex projective line, antipodal points are identified, but in the Riemann sphere they are clearly distinct. Then how can the two be homeomorphic.   I found out the following map from the complex projective line to $S^2$. But how do I prove this is a homeomorphism? Finding the inverse of this functions seems complicated, and then how do I prove that the inverse map of a open subset is a open subset, and vice versa?  $f:\mathbb{C}\mathbb{P}^1 \to S^2 = [z:w] \to \frac{(2Re(w\bar{z}), 2Im(w\bar{z}),|w|^2-|z|^2)}{(|w|^2+|z|^2)} $ Rectifiability of a curve Let $f$ be a function defined on $[0,1]$ by   $$f(x) = { 0, \text{ if } x = 0} $$ $$f(x) = { x \sin \frac 1 x , \text{ if } 0 &lt; x \leq 1} $$  Prove that the curve $\{(x, f(x)) : x \in [0,1]\}$ is not rectifiable.   I'm not sure how to approach this. The general idea seems logical, we're proving that the length of the curve is infinite, but the method seems difficult to find. Simple as the title states: Can you use only Java commands to take a screenshot and save it?  Or, do I need to use an OS specific program to take the screenshot and then grab it off the clipboard? If the magnetic field doesn't polarize does it follow the electric field path of propagation? or does it vanish? Is it costly to do array.length or list.count in a loop I know that in JavaScript, creating a for loop like this: for(int i = 0; i &lt; arr.length; i++) is costly as it computes the array length each time. Is this behavior costly in c# for lists and arrays as well. Or at compile-time is it optimized?  Also what about other languages such as Java, how is this handled? There are two syntaxes for command substitution: with dollar-parentheses and with backticks. Running top -p $(pidof init) and top -p `pidof init` gives the same output. Are these two ways of doing the same thing, or are there differences? How do high traffic sites service more than 65535 TCP connections?