I know this is a duplicate but the other two haven't helped me much.    Fist attempt: Tried proving through double inclusion, but wasn't sure of how to convey being an element of one implied being an element of the other in either direction, although I suspect from left to right would be the easier of the two.  Second attempt: Tried proving equality directly, using the fact that the boundary of a set is equal to the set difference of its closure and interior, but struggled proving closure of union is a union of closures, or the interior of a union is a union of interiors.   I'm getting pretty frustrated and any help would be greatly appreciated!
Assume $\bar A\cap\bar B=\emptyset$. Is $\partial (A \cup B)=\partial A\cup\partial B$, where $\partial A$ and $\bar A$ mean the boundary set and closure of set $A$?   I can prove that $\partial (A \cup B)\subset \partial A\cup\partial B$ but for proving $\partial A\cup\partial B\subset \partial (A \cup B)$ it seems not trivial. I tried to show that for $x\in \partial A\cup\partial B$ WLOG, $x\in \partial A$ so $B(x)\cap A$ and $B(x)\cap A^c$ not equal to $\emptyset$ but it seems not enough to show the result.
I have an ASUS N550JK laptop (BIOS version 204) running Ubuntu 14.04 (3.13.0-43-generic kernel).  My issue is that the battery indicator ALWAYS shows that it's using battery power - even when plugged in and fully charged. So even though I have selected to show the battery status in the menu bar only when charging or in use (in power settings), it is ALWAYS displayed. When it reaches a full charge, the indicator icon will either show just the percentage (usually 99% or 100%), or sometimes a very big number for time (like 22:14) along with the percentage. If I go to the power settings menu, it always says "Using battery power", with something like 40 hours remaining. This is all while plugged in and fully charged.  Are there any fixes less drastic than trying to update BIOS, or is a BIOS update necessary to fix this? The latest BIOS version is 208 according to the ASUS website, but the (very brief) descriptions of the updates don't mention anything about battery issues. If at all possible I'd like to avoid updating BIOS because of the risk involved.  I've already tried letting the battery drain all the way and then charging to full, with no luck.  Thanks in advance! It appears that in anticipation of D&amp;D Next Wizards of the Coast has removed much of the Dungeons and Dragons, 3rd Edition Web content. I didn't perform a thorough search to determine if all the content is removed (e.g. I know the Three Dragon Ante Web enhancement  and the Mind's Eye column about psionic tattoos  are both gone--better not to ask why those are my go-to articles), but it all might be.  What's the best way--and by best I mean both legal and convenient--to continue to access this content? How to monitor who is pinging me? Is there a way to know who is pinging my PC on my network? Is it possible to monitor all ICMP packets? How can I do that? Verbatim Environment in Figure Caption I've been trying to wrap my head around the basic concepts of category theory, and I thought I would attempt to illustrate what I understand with the category of sets, probably the easiest example. Particularly, I've been trying to prove that $id_A$ (the identity morphism on $A$, for all $A \in Obj(\mathbf{Set})$) is $1_A \colon A \rightarrow A, x \mapsto x$.  This is a very intuitive and reasonable statement, and it's trivial to prove that $1_A$ is indeed an identity morphism on $A$, and I suppose uniqueness of $id_A$ can be demonstrated analogously to uniqueness of the identity element in a monoid (considering the subcategory which has $A$ as its only object, and endofunctions on $A$ as its only morphisms).  In this manner, it is not hard to prove that the proposition in the title is true, but this demonstration requires to make an assumption or guess as to what could $id_A$ be. Specifically, the scheme of the proof is: assume $id_A$ = $1_A$, see that it works with the definition of an identity morphism, show that the identity morphism is unique, and, in conclusion, $id_A$ can only be $1_A$. What I'm looking for, nonetheless, is a somehow more direct proof, that doesn't assume $id_A$ = $1_A$ at the start. I want to place myself in a state of little or no knowledge about sets and functions, and under this assumption, why would I assume $id_A$ = $1_A$ at first? Why not try with $id_\mathbb{Z}$ = $f \colon \mathbb{Z} \to \mathbb{Z}, x \mapsto x^2 + 1$, for example? It wouldn't work, but I don't have any reason to think that $1_\mathbb{Z}$ is a better guess for $id_\mathbb{Z}$.  I suppose that the proof for which I'm asking would work for categories of sets with additional structure, and probably for posets as well, although I'm not clear as to what modifications it would require to work.  Thanks. Differences between no child vs no children a) They have no child. b) They have no children.  Are both a) and b) correct? Though b) seems more common. Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$ My problem is how to somehow relate the the gcd and congruence. I know that $(a,b) = ax + by$. I also know that $a \equiv b \pmod n$ means $n\mid a-b$. Any hints?  Thanks! I am using pdflatex on a fresh (and current) MikTex 2.9 install with the following MWE:  \documentclass{article} \usepackage{animate} \begin{document}  Test \end{document}   Pdflatex throws the error:  Package: pdfbase 2017/03/23 v0.14 driver independent access to low-level PDF fe atures \g_pbs_page_int=\count175  ! Undefined control sequence. &lt;argument&gt; \luatexversion                            &lt;95 l.924 }   I have read the other posts containing the offending control sequence, especially  but the solutions offered there didn't help in my case.  This problem might have arisen with a recent MikTex update. I have now completely uninstalled, redownloaded, reinstalled and updates MikTex -- with no success. Rigorous proof that surjectivity implies injectivity for finite sets How to accumulate values of each two rows across a line in linux? Many researchers list out their submitted manuscripts on their website. Some specify the journal name, like     Smith J, Oliver J (under review) My awesome paper. Nature.   while others don't, like     Smith J, Oliver J (under review) My awesome paper.   One reason I can think of as to why some don't mention the journal name is that they may get embarrassed in the case where the current journal rejects the paper. Then, the "circle" will know this paper gets rejected from Nature first and then ends up in an inferior journal.  Besides this, are there any other considerations preventing people from mentioning to which journal they submit a paper to?  P.S.: I am asking because my advisor is a researcher who only mentions accepted  papers on his webpage. As a student hunting jobs, I feel it is advantageous for me to mention my submitted papers on my webpage. I hesitate to do so, as I worry I may make my advisor unhappy by announcing in which journal our papers are currently under review. Remove Meta Privilege for MSE (only) What I am asking here is to please remove the privilege to participate in Meta for MSE.  This privilege is literally useless. You can't really have a meta site for a meta site. There's no Meta.Meta.SE here or any other SE site out there: Meta.SE covers for every site in SE, including itself.     When users find out that you don't really get anything new, it's kinda sad for those people. There is no use for it except just to be there and fill up the space between the first and third privilege. Privileges should be shown only if you earn something new, and this privilege to participate in the meta isn't new because it's where you report problems in all of SE, including Meta.SE.    So please remove the privilege to participate in meta just for Meta.SE.    This is not a dupe as this is a feature-request to remove it, not support on why it is there. The Follow Questions and Answers feature is now live across the Network How to make DNS server propagate faster if you are a site owner? It's often the case that I want to change name servers because my site is messed up.  Well, it takes a very long time for the name server to propagate.  Is there something in DNS, or registrar or anything I can do to speed that up.  Flushing my dns server using  and it rarely works Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$ Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$   We see that the only poles are at $x=\pm bi$.  Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ because the integral over the circular arc is $0$.  So now I just need to calculate the residue at $+bi$. Can anyone give hints on how to do this?  I was thinking about multiplying by $(x-bi)^2$ and then evaluating at $bi$ but I'm not sure why this even makes sense. Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$ Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x&gt;0$ $$ P\left( \sup_{t \geq 0 } M_t &gt; x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}. $$ My approach: I thought that rewriting the conditional probability to an expectation would help so we obtain that we must prove: $$ \mathbb{E} \left[ 1_{\sup_{t \geq 0 } M_t &gt; x}  \mid \mathcal{F}_0 \right] = 1 \wedge \frac{M_0}{x}.$$ A hint to me was given that I should consider stopping the process when it gets above $x$. Thus a stopping time that would do this is $\tau = \inf\{t\geq 0 : M_t&gt;x\}$. But now I'm stuck as I want to apply optional sampling results but we have an indicator which complicates things. How could I proceed from this? Any help is appreciated! The ideal generated by $2$ and $ x$ in $\mathbb{Z}[x]$ What are the differences in targets, and when should I switch to one? So I get the 4 options for target selection are:   Attackers Badges Randoms K.O.s   What I don't understand is the pros/cons for selecting each one, and what situations would make each one useful.  Things like Badges and K.O. makes no sense to start off with because nobody starts off with badges, and nobody is going to start off near death unless they are purposely trying to lose (by hard dropping int he same position). I've tried the strategy then of just sticking with Attackers, but then I all of sudden get wave after wave of garbage late into the game.    So it begs the question -- what are the differences in target choices, and how can I use that to my advantage to become a Tetris 99 champion? Why cant we replace diode connected BJTs by simple diodes in a current mirror circuit?