Is it possible for any set to be an element of itself? Is it possible for any set to be an element of itself? I want brief explanation for whether it's possible or not.
Example of set which contains itself I am trying to understand   How can a set contain itself? Can you show example of set which is not a set of all sets and it contains itself.
At what voltage is a NiMH battery empty? I have a battery pack of NiMH batteries. It is ten cells with 1.2V, 4000mAh each, put together in series. So rated voltage is 12V. After charging, i.e. when the charging device says that it is finished, I measure 14.3V.   When I now discharge, at what voltage should I consider my battery as "empty"? Is this the lowest voltage that I can discharge it to without damaging it? Which voltage is that? Shouldn't rep gains from old questions & answers eventually stop (or at-least reduce)? Prove that a finite union of closed sets is also closed Let $X$ be a metric space.  If $F_i \subset X$ is closed for $1 \leq i \leq n$, prove that $\bigcup_{i=1}^n F_i$ is also closed.  I'm looking for a direct proof of this theorem.  (I already know a proof which first shows that a finite intersection of open sets is also open, and then applies De Morgan's law and the theorem "the complement of an open set is closed.")  Note that the theorem is not necessarily true for an infinite collection of closed $\{F_\alpha\}$.  Here are the definitions I'm using:  Let $X$ be a metric space with distance function $d(p, q)$.  For any $p \in X$, the neighborhood $N_r(p)$ is the set $\{x \in X \,|\, d(p, x) &lt; r\}$.  Any $p \in X$ is a limit point of $E$ if $\forall r &gt; 0$, $N_r(p) \cap E \neq \{p\}$ and $\neq \emptyset$.  Any subset $E$ of $X$ is closed if it contains all of its limit points. Aborting evaluation when the memory exceeds a certain limit While I was reading the book Exploratory examples for real analysis, I came across this:     to show that : $$\lim_{x\rightarrow x_o}\frac{1}{h(x)} = \frac{1}{\lim_{x\rightarrow x_o}{h(x)}} = \frac{1}{M}$$      Discussion:Let $\epsilon \gt 0$.we want to find $\delta \gt 0$ such that       if  $0 \lt |x-x_0| \lt \delta $,then $\Big|\frac{1}{h(x)}-\frac{1}{M}\Big|\lt \epsilon$.      In order to get expression involving $|h(x)-M|$,we first find the common denominator:        $\Big|\frac{1}{h(x)}-\frac{1}{M}\Big|=\Big|\frac{M-h(x)}{M~h(x)}\Big|$      $~=\Big|\frac{h(x)-M}{|M|~|h(x)|}\Big|$.      As we have to deal with variable term $|h(x)|$ in denominator.We can construct bound .Let $\epsilon =\frac{|M|}{2}$ ,since lim$_{x\to x_o}h(x)=M$,there exists $\delta_1\gt 0~~$ s.t.       if $0 \lt |x-x_0| \lt \delta_1 $,then $|h(x)-{M}|\lt \frac{|M|}{2}$.      As you study steps below,identify that step where $h(x)\neq 0$ for all $0 \lt |x-x_0| \lt \delta_1 $ plays a critical role:      $|M|=|M-h(x)+h(x)|$      $~~~~~~\leq |M-h(x)|+|h(x)|$      $~~~~~~= |h(x)-M|+|h(x)|$      $~~~~~~\lt \frac{M}{2}+|h(x)| \implies$      $\frac{|M|}{2} \lt |h(x)| \implies$      $\frac{1}{h(x)}\lt \frac{2}{|M|}$,where $0 \lt |x-x_0| \lt \delta_1 $.      Now that we've bound for $1/h(x)$,we can apply the definition to term $|h(x)-M|$.      Since     lim$_{x\rightarrow x_o}h(x)=M$,we can find $\delta_2 \gt 0$ such that,       $~~~~~~~~~~~~~~~~~~~~~~~~~~~~$if  $0 \lt |x-x_0| \lt \delta_2$, then $|h(x)-M|\lt \frac{|M|^2 \epsilon }{2}$   The thing I can't understand is for what purpose did the author work with such complicated expression involving $\epsilon $ in the end of the discussion? Does Thunderbolt support chaining multiple monitors to a MacBook Pro? This has always been a big want for me. I love using multi-monitor setups but only want a single laptop. Does the 2011 MBP Thunderbolt support chaining multiple monitors together? I know that you can add a monitor and say, a few hard drives, but can you connect it to 2 27" monitors and your flat screen TV (for example)? Thanks! I recently installed Ubuntu 18.04 and I'm not keen on the desktop environment. I want the desktop env. I used to have in Ubuntu 14.04. How can I get it?  I've attempted to use , but when I try to install the Unity's 7.2.5 version using the command:  apt install unity=7.2.5   it gives me   E: Version '7.2.5' for 'unity' was not found   For sure I need to add a PPA for this, but which one!? If $x_1,x_2,\ldots,x_n$ are the roots for $1+x+x^2+\ldots+x^n=0$, find the value of $\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$ Let $x_1,x_2,\ldots,x_n$ be the roots for $1+x+x^2+\ldots+x^n=0$. Find the value of   $$P(1)=\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$$   Source: IME entrance exam (Military Institute of Engineering, Brazil), date not provided (possibly from the 1950s)  My attempt: Developing expression $P(1)$, replacing the 1 by $x$, follows $$P(x)=\frac{(x_2-x)\cdots (x_n-x)+\ldots+(x_1-x)\cdots (x_{n-1}-x)}{(x_1-x)(x_2-x)\cdots (x_n-x)}$$    As $x_1,x_2,\ldots,x_n$ are the roots, it must be true that   $$Q(x)=(x-x_1)\cdots(x-x_n)=1+x+x^2+\ldots+x^n$$ and  $$Q(1)=(1-x_1)\cdots(1-x_n)=n+1$$ Therefore the denominator of $P(1)$ is  $$(-1)^{n} (n+1).$$ But I could not find a way to simplify the numerator.  Another fact that is probably useful is that  $$1+x^{n+1}=(1-x)(x^n+x^{n-1}+\ldots+x+1)$$ with roots that are 1 in addition of the given roots $x_1,x_2,\ldots,x_n$ for the original equation, that is  $$x_k=\text{cis}(\frac{2k\pi}{n+1}),\ \ k=1,\ldots,n.$$  This is as far as I could go...    Hints and full answers are welcomed. Is it possible to log a program's network activity in Windows 7? Does anyone know of a program or way to log all network activity to a specific program on one's computer (program and/or service)?  I know I can see this "live" in Resource Monitor, but I'm wondering if anyone knows of a program or way to track and log all network activity to/from a specific process and/or service.  I'm finding a lot of "live"-type monitors, like Windows Resource Monitor, but nothing that can log. Is it possible for test error to be lower than training error Is it possible to have test error lower than training error?  I have a classification problem with 2000 samples, 500 of which are positives, 1500 are negatives. I split my data into 70% training data, 30% test data.  Run random forest with 200 estimators and cv=10. I did this several times and compared the recall and precision score and notice the scores for my test set is significantly better. Is this possible? With java.sql.ResultSet is there a way to get a column's name as a String by using the column's index? I had a look through the API doc but I can't find anything. biblatex and new line for DOI, URL and Eprint Hitting arrow keys adds characters in vi editor When I tried to use arrow keys in insert mode in vi editor the following characters are being inserted in the editor:   for ↓ I get B, for ↑ I get A, for ← I get D, for → I get C.   Please help me in resolve this problem. Can a self-complementary graph on 100 vertices have precisely one vertex of degree 50? Listing only directories in UNIX Can you please explain this to me?          Could you please explain this to me?   I am unable to figure out which to use which situation. I did google, and some posts say they are both the same, even if the second one is more formal. Other posts say could is the past tense of can, but in the above example, I don't think could is used as past tense of can.  What is correct way to use these words? Calculate Euler equations of fluid dynamics without division? I read this short story about 20 years ago, and I'd love to find the title and author. This is what I can recall about the story:   It takes place in a classroom full of kids (around 10 years old?) The planet always gets rain The kids lock the main character (a girl, I think) in the closet during the very rare occurrence of sunlight and she misses the entire event.    I know it's not much to go on, but I hope it rings a bell for someone! Can a willing lycanthrope be True Polymorphed? Can a lycanthrope or other shapeshifter be willingly affected by the spell True Polymorph? I know that it can be affected by regular Polymorph, but the text for the enhanced version seems to differ. I'm unable to update nodejs, and I can't see why.  If I were to make a guess, I'd say that I have two repositories listed in apt-get, that both contain a nodejs package.  Here's what I've done:  Remove old version:  sudo apt-get remove --purge nodejs npm   Then, going by , I've run:  curl -sL https://deb.nodesource.com/setup_5.x | sudo -E bash -   And then:  sudo apt-get install -y nodejs   I then run:  nodejs -v   Which returns:  v0.10.25   Can anyone explain what's going wrong? And how I can remedy this situation?  Many thanks.