I was reading about the field of real numbers $\mathbb{R},$ and a basic question arose in my mind.  How one should prove that, for any numbers $a, b,$ and $c,$ $a + b = a + c$ if and only if $b = c?$  In order to prove this statement, we have to prove two things. I will start with the part usually known as the cancellation rule.  $\implies.$ Let $a, b,$ and $c$ be any numbers. Suppose that $a + b = a + c.$ Since $a$ is any number, we know from the filed axioms, that exists a number $-a$ such that $a + (-a) = (-a) + a= 0.$ Hence, adding $(-a)$ to both sides, we have $(-a) + (a + b) = (-a) + (a + c).$ By the Associative Law, we deduce that $((-a) + a) + b = ((-a) + a) + c.$ Since $(-a) + a = 0,$ we get $0 + b = 0 + c.$ Since $0$ is the additive identity, it follows that $b = c.$ $\square$  This part is really straightforward. Although, I have no idea what to do about the $\Longleftarrow$ part. I think of this last part as the statement that $+$ is well-defined in $\mathbb{R},$ which means that for all $x, y, z, w \in \mathbb{R},$ if $x = y$ and $z = w,$ then  $x + z = y + w,$ but I’m not sure if these two things are related to each other. Another thing that “bores” me is the fact that I use this fact to prove the $\implies$ part when I added $(-a)$ to both sides and stated that they still equal. In short: How can I prove that if $b = c,$ then $a + b = a + c?$ Is this related to the fact that $+$ is well-defined? Do we assume that $+$ is well-defined (i.e., is it an axiom), or can it be proved?
I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. What I do not understand is why making a new equation (adding or taking away from an expression) allows one to know what an unknown represents. What about equations lets us do this?
The one of solution of $x^4 - 2y^2 = -1$ is $x = 1$ and $y = 1$. However, the solution $(1, 1)$  of $x^4 - 2y^2 = 1$ is failed. We know $x = 1$ and $y = 1$ is small integers and we can check by trail method. In case more solutions are existing or not how to check? What are the solutions of $x^4 - 2y^2 = 1$? Determine if wire in switch box is safe to re-purpose? I need to pull a neutral wire to my light switch for a smart switch.  It's presently switching an outlet that I don't need switched, and the other side of the outlet already has unswitched power.  My plan is to use a smart switch, which requires neutral, to control a different device wirelessly.    I can't feasibly run a new wire, because the wires aren't run through conduit.  Instead, my plan is to repurpose the switched wire to get neutral from the outlet.    The switch, which controls an outlet has two wires, one black and one red.  The switched outlet has a white neutral wire, plus the red and black wires that I assume run directly to the switch.  I've completely disconnected the red and black wires from the outlet and the switch.  The black wire is measuring 120V AC when measured against ground using the box as ground, both at the outlet and at the switch. Using my DMM, the red wire is measuring 24V AC with respect to the box/ground at both sides.  I assume this is a  caused by induction from the charged black wire over to the floating red wire.    Our house is ~100 years old and this particular switch seems to be run using armored cable (note the paper wrap around each wire.)  There are literally only two wires in the switch box; no wire nuts.    If relevant, we do have some old knob and tube wiring active, but I'm pretty confident there's none on this circuit.  Is there some way I can determine with enough confidence that the red wire isn't connected elsewhere such that I can know it's safe to connect the red wire at the outlet to netural? When —the bartender at the Cantina at Mos Eisley spaceport in the first Star Wars—says:     “We don’t serve their kind here.”   In reference to C-3PO and R2-D2 entering the bar, it’s certainly a bit prejudicial, but are these commonly held views in this universe? We don’t see it happen anywhere else in the films but is it implied elsewhere? The difficulty is that it should be cross platform. Windows 2000, XP, Vista, OSX, Linux, other unix variants. I am looking for a snippet of code that can accomplish this for all platforms, and a way to detect the platform.  Now, you should be aware of  that user.home does not work correctly, so please do not provide me of textbook answers, I can find these myself in the manuals. Show there are infinitely many primes that are equivalent to 1 mod 8. Show there are infinitely many primes that are equivalent to $1 \pmod{8}$.  Hello there! I have been trying to do this problem for a pretty long time with no avail.    I noticed that this is really similar to Euclid's proof that there are infinitely many primes.  However, I can't find a way to use that here.  Can anyone help me? How are House points tabulated/tracked? I have a many-to many relationship between three objects. LGFProducts, PartsinProducts, and Parts. PartsinProducts is the junction object. I want to be able to get data like the names of the parts that are related to the selected LGFProduct.   I understand that parent-child relationship can only traverse one level down while grandchild-grandpa can travel 5 levels at most. What is the parent-child or child-parent in a many-to-many relationship?  I assumed that I should be traversing upwards from Parts to the junction object partsinproduct and then LGFProducts? But my friend who is not a Salesforce expert but has knowledge in CS said that I should do a parent-child query (of LGFProducts to partsinproduct) to child-parent relationship query (PartsinProduct to Parts). Or perhaps both of these options are wrong and another is appropriate.   Also, I am having a little trouble traversing for custom objects. I understand that I need to append the letter r instead of c but get confused. For example, this is my attempt at a three level upward traversal that fails.   SELECT Id, Name, Parts__r.PartsinProducts__r,LGFProducts__r.Name FROM Parts__c   Also if three levels of upward traversal are needed, could one provide an example with custom relationships? As seen above, I am having difficulty successfully querying a three level upward traversal statement for custom objects.  This is my attempt at the parent-child relationship query of LGFProduct to PartsinProducts:  SELECT Name,   (     SELECT Name     FROM PartsinProduct__r   ) FROM LGFProduct__c    I am then given this error: Change bookmark depth I want all bookmarks after "Anhang" (appendix) to be a level deeper, but I need them to be \chapter{} for the right numbering.  I write a scrbook document and that's what I got:    Is it possible to change the depth of the chapter to act like a section in the bookmarks?  UPDATE  MWE:  \documentclass[12pt,a4paper,oneside]{scrbook}  \usepackage[ngerman]{babel}     % deutsche Worttrennung \usepackage[T1]{fontenc}        % Ausgabe von Sonderzeichen und Akzenten in PDF verbessert \usepackage[utf8]{inputenc}         % Eingabe von Sonderzeichen möglich \usepackage{lipsum}  \setcounter{secnumdepth}{3}     % subsubsections nummerieren \setcounter{tocdepth}{3}        % subsubsections in TOC  \usepackage[ %     pdfproducer={Texmaker},     %      pdfcreator={pdfLaTeX},      %      hidelinks,          % versteckt die Boxen um die Links in der PDF     bookmarksnumbered,      % nummereiert Lesezeichen     ]{hyperref}         % PDF Metainformationen hinzufügen   \begin{document}  \appendix \pagenumbering{alph}  \thispagestyle{empty}  \vspace*{3em} \begin{center} \begin{Huge} \textbf{\textsf{Anhang}} \pdfbookmark{Anhang}{Anhang} \end{Huge} \end{center}  \chapter{Appendix 1}  \chapter{Appendix 2}  \section{Appendix 2.1}  \section{Appendix 2.2}  \chapter{Appendix 3}  \end{document} Open-source alternative to Adobe InDesign There is some open-source software, similar to Adobe InDesign? It would be awesome if it do exist...  I know all the others like Inkscape, Gimp, etc... but for InDesign i have none.  Could you give me one direction? Focal length of an ellipse and related results Word to describe the way cattle and buffalo consume fodder? Infix to postfix not working as expected Embedding ordinals in $\mathbb{Q}$ Received an email intended for another person I'm not really sure how email routing works, but someone ordered something on Amazon and I received the email instead of them.  Or maybe we both got it, I don't know.  The order doesn't show up in my account, so I'm certain I wasn't charged for it, but I shouldn't be getting other peoples' emails.  We'll say that my email is johndoe1@gmail.com, and somebody who's email is john.doe1@gmail.com places an order on Amazon.  The confirmation email is sent to me at johndoe1@gmail.com.  I checked the email header, and it did say To: john.doe1@gmail.com which is not my email address.  At first I thought that Google ignores periods in email addresses, but I tested the account setup and it doesn't give any error when you put a period in the address.  I didn't create the account; I just used the "check availability" function and the address I chose with a period was fine.  Maybe someone with knowledge about how Email works could tell me why this happened.  Is this a bug in the way Amazon sends emails?  Or is it a bug in how Google receives them?  Who should I report this issue to? "Unfortunately, ... has stopped" when Downloading file in Android Is 'acronymise/ze' a word? Is it used only colloquially, if at all? I just sent a text to a friend, who didn't understand an acronym I used for a game:      "Ah, I said it in a previous text so thought it was ok to abbreviate   it."   Though, since it was an acronym I had created, could I have used the word 'acronymise/ze' to be more specific?   I'm pretty sure it isn't a (formally used) word, but I wondered, as you often refer to acronyms specifically as acronyms, rather than abbreviations. Can you be more specific with a verb, too?  It sounds too odd to me, so I'm guessing generally no. But I wondered—so perhaps I'm wrong. Or half wrong. Is my US visa still valid after change in name? To debug a JACK/Pulseaudio issue, I want to understand when and why the pulseaudio daemon is started by systemd (on Fedora).  Using:  $ ps -o'pid,ppid,args' `pgrep pulse`   I see that the pulseaudio daemon is being started by systemd (pid=1)   PID  PPID COMMAND 2738     1 /usr/bin/pulseaudio --start   However, I was unable to find any unit-file on my system containing pulseaudio or even just pulse.  My specific questions are:  A) Is there a way to determine the systemd unit that caused the creation of a specific process (in my example output, process 2738, the PA daemon)?  B) Are there alternative approaches to find out which unit-dependency chain or other settings of systemd resulted in the invocation of /usr/bin/pulseaudio --start? Cartographic techniques for symbolizing routing data Make specific color in a texture transparent I am working with very old model files that have BMP textures. The designers did not use a transparency in the textures or a separate alpha b/w texture. They used a specific color value that was then removed by the engine to create a transparency, similar to a chroma key. In this case, all textures used a bright purple/pink color.   I cannot modifying the original texture images by converting them to PNG and add an alpha. I would like to figure out how to do it all in nodes.  The goal is to have this value keyed out and transparent. I have been messing around in the shader editor and cannot figure this out.  I've also been looking around online and in this forum, but the only search terms I can think of are chroma-keying which only results in Compositing results, which I don't believe is what I'm looking for.  Here is an example of the texture (its very small) and it's context:      Any help or direction would be much appreciated!