Let M be any set with multiplication. Given and only using the following axioms $1. \forall a,b\in M : a\cdot b=b\cdot a \\ 2. \forall a,b,c\in M: (a\cdot b)\cdot c=a\cdot (b\cdot c) \\3. \exists 1\in M\ ~ \forall a \in M:a\cdot 1=a \\4. \forall a \in M \setminus \{0\} \exists a^{-1} \in M : a\cdot a^{-1}=1$ Show that: $\forall a,b \in M: a\neq0$ the equations $a\cdot x=b$ and $x\cdot a=b$ has an unambigous/unique solution x. My solution: Separate the equation into two cases and solve for x respectively case 1: $a\cdot x=b \\ a^{-1}\cdot (a\cdot x)=(b)\cdot a^{-1}\\ (a^{-1}\cdot a) \cdot x= b\cdot a^{-1}\\1\cdot x=b\cdot a^{-1}\\x=b\cdot a^{-1}\\$ case 2: $x\cdot a=b\\a^{-1}\cdot (x\cdot a)=(b)\cdot a^{-1}\\x\cdot (a^{-1}\cdot a)=b\cdot a^{-1}\\x\cdot 1=b\cdot a^{-1}\\ x=b\cdot a^{-1}$ Both cases gave the same solution for x, therefore x is an unambigous/unique solution to the equations. I'm new to latex so feel free to edit or comment on that as well!  thanks:)
I just started real analysis.  I don't have a background in proofs or logic, simply calculus.  So I'm trying to learn more about proofs--so forgive the basic question, please.  How do you go about proving this theorem: If $a$ and $b$ are any numbers, then there is one and only one number $x$ such that $a + x=b$.  This number is given by $x=b+(-a)$.  First part of the proof I understand--it's simple.  We simply use some axioms and do the following:  $a+b-a=b$, which of course is true.  This is just true from plugging in the $b+(-a)$ for $x$.  But how about the uniqueness issue?  In my text, it says:  $$(a+x)+(-a)=b+(-a) x=b+(-a)$$  How does this prove uniqueness?  Is this equation essentially saying that the ONLY possible value of $x$ is expressed by $b+(-a)$?  How would I have known to set this up as part of the proof if I was asked to prove something is unique?  Sorry for the basic question--have to start somewhere.
Difference between == and === in JavaScript Do photons bend spacetime or not? I have read this question:     where Safesphere says in a comment:     Actually, photons themselves don't bend spacetime. Intuitively, this is because photons can't emit gravitons, because, as any massless particles not experiencing time, photons can't decay by emitting anything. The latest theoretical results show that the gravitational field of a photon is not static, but a gravitational wave emanating from the events of the emission and absorption of the photon. Thus the spacetime is bent by the charged particles emitting or absorbing photons, but not by the photons themselves.        where John Rennie says:     as far as I know there has been no experimental evidence that light curves spacetime. We know that if GR is correct it must do, and all the experiments we've done have (so far) confirmed the predictions made by GR, so it seems very likely that light does indeed curve spacetime.   Now this cannot be right. One of them says photons do bend spacetime, since they do have stress-energy, but it is hard to measure it since the energy they carry is little compared to astronomical body's stress-energy. So they do bend spacetime, it is just that it is hard to measure it with our currently available devices.  Now the other one says that photons do not bend spacetime at all. It is only the emitting charge (fermion) that bends spacetime.  Which one is right? Do photons bend spacetime themselves because they do have stress-energy or do they not? Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet? Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet."  I have a (not very mathematically intelligent) friend who presented it as if it were a mathematical fact, which got me thinking... Is this really true?  Of course, I've learned that dealing with infinity can be tricky, but my intuition says that time is countably infinite while the number of works the monkeys could produce is uncountably infinite.  Therefore, it isn't necessarily given that the monkeys would write Hamlet.  Could someone who's better at this kind of math than me tell me if this is correct?  Or is there more to it than I'm thinking? What does “pull sb. out of the hat” mean? I found the phrase, “pull her out of the hat” in the following sentence of the quote from Frank Bailey, the most relied-upon former aide of Ms.Sarah Palin, whose memoir, “Blind Allegiance to Sarah Palin: A Memoir of Our Tumultuous Years” is due out on Tuesday this week.  “I doubt if there's anything new in this book, just more detailed views of the gruesome story of her rise and fall. What would really be interesting is an account of who made the decisions in the GOP to pull her out of the hat. It certainly wasn't McCain.”  I don’t find the idiom, “pull one out of the hat” in any dictionaries at hand. Does it mean to “pick up” (Sarah Palinfor (vice) presidential candidate) or “remove her from (the candidate seat)? Can somebody teach me the meaning of this phrase? MacBook Pro detects syncmaster (monitor) but doesn't display Do we need to unbind event listeners in directives when angular starts to destroy? I notice that in a pdf file compiled from Latex, "fi" such as in "field" cannot be separated as "f" and "i" when copying text out of the pdf file. I wonder why and if this can be changed? Thanks and regards! Need help with a question regarding the Big Oh In class we are currently covering upper/lower bounds, big Oh and omega and the like. I am pretty good on the "typical" functions one would do, but at a complete loss at "general" statements. This ons function, for example:  $$\forall f\in\mathcal{F}: \lfloor \sqrt{\lfloor f(n)\rfloor }\rfloor \in O(\sqrt{f(n)}).$$  the domain F by the same is all natural numbers that return a real positive number when inputted. My first instinct with something like this is to remove the floors, the general strategy I though to solve this is remove the outer floor, then get rid of the square root, and then the inner floor. But I have no idea how to go about this or if this is even the right way to do it.  Help please? Thanks in advance. Is there any problem if the last name prints first than the first name on an air ticket? Is there any problem if the last name prints first then the first name in international air ticket? I run windows 7. I run a Minecraft server, but I can only host around 9 players before lag appears.   I heard that Ubuntu can run a server with much better performance than Windows 7. I kind of like the whole idea of Ubuntu, but I don't want to change my OS to Unix or anything else.  Basically, my question is, does running a server (specifically and Minecraft server) run easier and faster on a Ubuntu server? On top of that, how would one set that up? Is there a video that explains how Stack Overflow works and can be used? Is there a video that explains how Stack Overflow works and can be used? I.e. a short tutorial to get new users started. If there isn't, then the people behind this site or others who know how it works should make one, because it's overwhelmingly different from other sites and thus difficult for beginners. PGFPlots: Understanding how to draw plots on different layers and arrange them I have gone through previous questions [,,], but I still couldn't figure out how to draw plots on different layers and change their order of drawing.  For this MWE, I would be grateful if I could figure out how to change the order of appearance of the three plots in such a way that the legend entries have the same order while keeping the plots commands/definitions in the same order as much as possible.  \RequirePackage{luatex85} \documentclass{standalone} \usepackage{pgfplots} \pgfplotsset{compat=newest}  \begin{document}  \begin{tikzpicture}     \begin{axis}[             legend style={                 at={([yshift=30pt]0.5,1)},                 anchor=north},             legend columns=-1,             xmin=0, xmax=10,             ymin=0, ymax=10,         ]         %         \addplot [red,      line width = 5pt,   domain=0:10]    {x}; \addlegendentry{$x$}            \addplot [black,    line width = 5pt,   domain=0:10]    {(x-5)^2}; \addlegendentry{$(x-5)^2$}         \addplot [yellow,   line width = 5pt,   domain=0:10]    {10-(x-5)^2}; \addlegendentry{$10-(x-5)^2$}     \end{axis} \end{tikzpicture}  \end{document} How do I allow a non-default user to use serial device ttyUSB0? I have an Ubuntu 11.10 system with 2 users:   The first was created during the installation The second instead was created after. It belongs to the sudoers group.   Now the problem is that when the second tries to use a device ttyUSB0 the following error is returned:  "Could not open serial port /dev/ttyUSB0"   I was able to fix it by using:  sudo chown :second_user /dev/ttyUSB0   However when I disconnect the device and reconnect it the problem comes back.  Is there a way to allow different users to access the devices? I suppose I have to add the user to a specific group. Currently the owner is root and the group is dialout. However I'm not sure about the group and I don't know how to add the user.  Thanks! We are developing a C# application for a web-service client. This will run on Windows XP PC's.  One of the fields returned by the web service is a DateTime field. The server returns a field in GMT format i.e. with a "Z" at the end.  However, we found that .NET seems to do some kind of implicit conversion and the time was always 12 hours out.  The following code sample resolves this to some extent in that the 12 hour difference has gone but it makes no allowance for NZ daylight saving.  CultureInfo ci = new CultureInfo("en-NZ"); string date = "Web service date".ToString("R", ci); DateTime convertedDate = DateTime.Parse(date);               As per :     UTC/GMT Offset       Standard time zone: UTC/GMT +12 hours   Daylight saving time: +1 hour   Current time zone offset: UTC/GMT +13 hours     How do we adjust for the extra hour? Can this be done programmatically or is this some kind of setting on the PC's? Check Constraint with Case Statement Proving $a^2$ is even $\implies$ $a$ is even by contradiction I know that if you held an infrared remote in front of a digital camera, it'll flash a blue/purplish light when you press the buttons. Why?    (Image by  .) Quadrilateral problem with midpoints of diagonals For an arbitrary convex quadrialteral $ABCD$:   $$AF=FC, BE=ED, GI=IH$$  Prove that points $F$, $E$ and $I$ are collinear.  I was able to solve the problem by using vectors and I think that it can be also solved analytically but I wanted some more elegant proof, more in the spirit of Euclid. My bookmarks are not shown anymore after upgrading to 13.04 Raring ringtails and I can't add any new bookmarks. The function is not shown and the shortcut Ctrl+D isn't working either. Is this functionality just gone for some reason (as the split screen with F3) or is there something wrong with my upgrade?  Thanks for any hints. Server login attempts NOT blocked after configuring fail2ban So, I installed (and configured) fail2ban on three Ubuntu VPS servers. I configured (imo) a maximum of 3 failed login attempts (via Putty as root) within 2 minutes. Then I restarted the fail2ban service (according to ).    But when I look at my auth.log files in /var/log/ I still see 4, 5 or more failed attempts from certain IP's. So obviously I did something wrong as the IP is not blocked after 3 failed attempts.   It might have to do something with iptables, but that's where it gets a bit "foggy" for me..   Not sure how to setup a max-three-failed-login-attempts rule..  Additional: I see that the login attempts mostly have different ports for every attempt. Does that influence the count to max-three?  Because I see a port being "poked" twice and then no more attempts. Then the IP tries a different port (again 2 times). (all bots of course)