The ordered pair $(a,b)$ is defined in formal set theory as the set $\{\{a\},\{a,b\}\}$. Then ordered triples $(a,b,c)$ are defined in this way: $((a,b),c)$. However, can we define ordered triples as the set $\{\{a\},\{a,b\},\{a,b,c\}\}$, and analogously for higher n-tuples. In other words, does such a definition uniquely determine the components of the ordered triple?
releasing a second question concerning ordered stuff and set theory, which is very similar to my first question () which I'm putting here for reference.  The next problem that I'm stuck at in my book is this: we have an ordered triple $(a,b,c)$ and we want to define it using set theory. Why would $\{\{ a\} ,\{ a,b\} ,\{ a,b,c\}\}$ not work? The book requires from me to prove why it is not valid.  Additional info: I don't know how this matters but I'm gonna put it up anyhow. It is written within the book that the correct way to show it is $\{\{ a\} ,\{\{ a\} ,\{\{ b\} ,\{ b,c\}\}\}\}$  EDIT: Now, I consider that the correct case should matter. Besides pointing out why the first case is wrong I'm curious as to why the second one is correct
Altering layer's CRS to correct Layer Misalignment in QGIS I've got a vector layer in my project which isn't aligned with the rest of the project (see attached JPEG).  These are work files so I'm not sure when they were created or how they might have been messed with since they were created.  We moved from ArcGIS to QGIS a couple of years ago.  The infilled blue vector layer is the one I need to move, slightly north and east, so that it correlates to the blue line (also a vector).  All layers (and the project) are currently in EPSG:27700 - OSGB 1936.   If I need to create a new CRS based on OSGB 1936 and give it some new parameters, then how would I go about this?  Further information:  This is on Exmoor in SW England, and the data is uniformly almost exactly 8m out to the SW point to point - i.e I need to shift it 8m NE. Proof that total derivative is the only function that can be added to Lagrangian without changing the EOM So I was reading this:  and while the answers for the first question are good, nobody gave much attention to the second one. In fact, people only said that it can be proved without giving any proof or any.  So, if I have a Lagrangian and ADD an arbitrary function of $\dot{q}$, $q$ and $t$ in such a way that the equations of motion are the same, does this extra function MUST be a total time derivative?   EDIT Ok, I changed my question a little bit:  Question: If I have a function that obeys the Euler-Lagrange equation off-shell, this implies that my function is a time derivative? (This was used in Qmechanic's answer of this other question: , equation 14.)  Also, why people only talk about things that change the Lagrangian only by a total derivative? If this is not always the case that keeps the equation of motion the same, so why is it so important? And why in the two questions I posted about the same statement on Landau&amp;Lifshitz's mechanics book only consider this kind of change in the Lagrangian? How can I distribute a native executable for a Python program? Lately all of my work has been in Python (and Ruby), and I'm in love with those two languages. There are many libraries for game development on Python but the problem I see is that many of those frameworks don't cover the "distribution" aspect of game creation. I'd like to code my game on Python (and any other pure Python extra libraries I need) and be able to distribute an executable for the main PC platforms (Windows, Mac and Linux).  Any options out there or what are the best combinations for this? (maybe pyglet + cx_freeze) Noticing that Newtonian gravity and electrostatics are equivalent, is there also a relationship between the general relativity and electrodynamics? How do I get the result of a command in a variable in windows? I'm looking to get the result of a command as a variable in a Windows batch script (see  for the bash scripting equivalent).  A solution that will work in a .bat file is preferred, but other common windows scripting solutions are also welcome. Text formatting now allowed in comments (List of proven and disproven abilities) Can the software be downloaded on an Android device?  If yes, then what is the process? How do users navigate to Meta? Size commands do not work in mathmode MWE:  \documentclass{article} \begin{document} $\wedge{\small\wedge}{\tiny\wedge}$ \end{document}   All the \wedges have the same size... Mirror modifier. Mirrored half is not aligned, but rotated I was working on my character's hand when I realized half of the body was suddenly deleted. I am trying to recreate the other half, but it just won't mirror properly in the center. I tried moving the origin to the 3D cursor while it was completely centered, but the mirror modifier is still applied at an angle. Creating intuition about Laplace & Fourier transforms How to set class weights for imbalanced classes in Keras? Rigor in quantum field theory Before I start, YES, I have done my own research - I know how annoying a repeat question is. Yet each answer is so specific to others...  When I tried the beta 2.8 64 bit, it says "Blender requires a driver with at least OpenGL 3.3 support. The program will now close."    The question: I know that it is true that 2.8 requires OpenGL 3.3. But does my PC not have it, doesn't support it, or do I have it and blender won't detect it? I'm not sure what info to include or what to look for, so here's my PC's specs:  Windows 10 (64 bit) Core -        Intel Core i5-2400 CPU @ 3.10GHz RAM -         8GB (7.89GB usable) System type - 64-bit Operating System, x64-based processor  Thanks!  EDIT: Graphics properties: This is an attempt to gather together the various questions about time that have been asked on this site and provide a single set of hopefully authoritative answers. Specifically we attempt to address issues such as:   What do physicists mean by time? How does time flow? Why is there an arrow of time? Formatting a bootable USB that thinks it's a CD-ROM After upgrading to Windows 10, I remember that I had an old bootable recovery usb drive from the manufacturer that I no longer needed. I want to format it but it's showing up as a CD-ROM and format options are not available.  I also tried formatting it in a live linux mint environment, but linux cannot even mount the cd-rom.  How can I format this recovery usb that is showing up as a cd-rom? What's the protocol for requesting a tag warning? On a particular site, RPG Stack Exchange, we've decided on our meta on a couple of tags that should have tag warnings, and on what those tag warnings should say. I am  on the site in question.  What should we do to get the tag warnings implemented? Do we do any/all of the following?:   start a [feature-request] on our meta for the tag warnings? raise a CM escalation? contact staff via the contact form? ping a staff member in the Teacher's Lounge? anything else? The Megahertz Myth became a promotional tactic due to differences between the PC's INTEL 8086 processor and Apple's Rockwell 6502 processor.  The 8086 ran at 4.77MHz while the 6502 ran at 1MHz.  However, instructions on the 6502 needed fewer cycles; so many fewer, in fact, that it ran faster than the 8086.  Why do some instructions need fewer cycles?  And why can't the instructions of the 6502, needing fewer cycles, be combined with a fast cycling processor of the 8086?  Wikipedia's article for instructions per cycle (IPC) says     Factors governing IPC   A given level of instructions per second can be achieved with a high IPC and a low clock speed...or from a low IPC and high clock speed.   Why can't you have both high instructions per cycle and high clock speed?  Maybe this has to do with what a clock cycle is?  Wikipedia mentions synchronization of circuits?  Not sure what that means.  Or maybe this has to do with how a pipeline works?  I'm not sure why instructions in a short pipeline are different from instructions in a long pipeline.  Any insight would be great!  Just trying to understand the architecture behind the myth.  Thanks!  References: New messages flagged blue bubble does not go away When someone flags a message in chat all 10k+ users in chat get a little blue bubble to vote on. It shows up in the upper left hand corner of your gravatar (which is in the bottom left of the page).  When someone flags a message and it shows up, if I wait for a second to click on it, and then click on it, and it has already been handled, it shows the message  "There are no flags to display."  In the past, after showing this message, the blue (1) would disappear. However, lately it has not been going away and requires a page refresh in order to remove. Can this be fixed?  This issue is addressed here:  from September 12th 2012. But since it is still occurring 10 months later, I believe the cause is different. I am a freelancer programmer and so far I have only worked on projects on Windows platform. I don't have much idea about OS X hardware. I am not looking for a very high end configuration but integrated development environment and one database server should just run fine. I have a Dell laptop i7 processor, 8 GB RAM, 1 TB HDD; I am fairly satisfied with the performance. I am mainly going to use it for app development and not for any rich graphics app development.  What are the minimum and recommended hardware requirements for Xcode development on OS X?