I need to prove $(-\vec{v})+\vec{v}=\vec{0}$ using only the axioms of vector space addition and multiplication, except the commutativity axiom.   I have tried using this approach $\vec{0}=\vec{v}+(-\vec{v})$ but I get stuck and can't find a way to get the $(-\vec{v})$ first.
I'm trying to prove commutativity of addition for vector spaces, using the axioms for vector spaces. Apparently commutativity can be proven! Im having trouble getting a good feel for what is allowed and what is not. Here's my work so far:  $u+v+u+v = 2(u+v) = 2u + 2v = u+u+v+v = u+(u+v)+v$  Here I just wanna claim that $u+(v+u)+v = u+(u+v)+v$   $\Rightarrow -u+u+(v+u)+v+(-v) = -u+u+(u+v)+v+(-v)$ : here im just adding -u to the right, and -v to the left. Question: is this "adding to both sides" really legit in this context? Why?   Quick help proof: $-v+v = (-1)v+(1)v = (-1+1)v = 0v = 0 = v-v$  And another: $ 0+v = v+(-v) + v = (1)v + (-1)v + v = (1-1)v + v = v = v+0$  We have $0 + (u+v) + 0 = 0+(v+u)+0 \Rightarrow u+v = v+u$  This feels ugly and not at all elegant, especially the great leap "add -u to both sides" feels completely out of place. Do I need more lemmas? Is there a more elegant way?   //not homework or anything, just for my own pleasure, feel free to provide theory, as it is more insightful than solutions. :) Thanks!   EDIT: corrected notation a little.
Is there any way to include an animated GIF directly? Is there any way to include an animated GIF directly in either PDFLaTeX or XeLaTeX? I realize the animate package can include animations in a PDF, but it doesn't support animated GIFs and you have to split them up manually into EPS or PNG files as far as I can tell. Can you clone a larger drive onto a smaller drive (used space fits on the smaller) I have a 500GB drive which is about 50GB full. I got a 80GB SSD, which has 74.5GB of available space. I want to clone the system drive onto the SSD.   DriveImage XML will not let me copy a larger drive to a smaller one. I'm running the Storm King's Thunder campaign, and we are at the portion where the PCs get an airship to travel around in, the airship is about 1000ft in the air. There are approximately 5-6 lvl 7 PCs, and they have 3 or 4 hard encounters a day. There are two wizards, a warlock, ranger, paladin, blood hunter, sorcerer, and fighter. For the most part, they have had fun with the different weapons and enemies they have faced; however, I find that one or two spellcasting PCs have been using spells such as mind sliver, polymorph, and hypnotic pattern to defeat more challenging encounters, such as one with a roc and wyvern. They make the creature make saving throws with hypnotic pattern and cause the enemy to fall to its death, and using mind sliver makes it even harder, especially on low wisdom creatures like rocs. Or with the polymorph spell, they cause the enemy to become a fish or chicken and throw it off the edge, and since the damage carries over, it kills the monster. I realize it was legal and made fights funnier, but it does get annoying when it's happening in almost every fight. Are there any suggestions on how to keep them on track without them making every encounter easy? I thought of giving the monsters immunity to these effects, but it just sounds like a bad excuse for me not being prepared. The validation tab in your script tool properties allows you to put parameters in categories. The :     Put parameters in different categories. Categories are expandable   window shades with a collection of parameters.     I am using the categories to organize my parameters, so I don't care about the expandable/collapsible feature;  it just adds a lot of time to the user.  This is all within a normal toolbox, not a script toolbox.  v10.1    Does anyone know how to have these categories expanded by default?  Here is my only validation code.  I am only utilizing categories.  The other parts are the default.  def initializeParameters(self):     """Refine the properties of a tool's parameters.  This method is     called when the tool is opened."""     self.params[0].category = "General"     self.params[1].category = "General"     self.params[2].category = "Inudaciones"     self.params[3].category = "Cienegas"     self.params[4].category = "DrenajesPrincipal"     self.params[5].category = "DrenajesSensillos"     self.params[6].category = "Inudaciones"     self.params[7].category = "Cienegas"     self.params[8].category = "DrenajesPrincipal"     self.params[9].category = "DrenajesSensillos"     return Prove that any two nontrivial subgroups of $\mathbb{Q}$ have nontrivial intersection I need to prove that any two nontrivial subgroups of $\mathbb{Q}$ have a nontrivial intersection as part of a larger proof that $\mathbb{Q}$ cannot be represented as a nontrivial direct product.    (Yes, I realize that there are other ways to prove that $\mathbb{Q}$ cannot be represented as a nontrivial direct product, but I am not interested in those, and will not accept answers showing me how to do that).  Based on the density of the rationals, I guess this idea kind of makes sense, but I am doing this proof for an abstract algebra class, and I don't think real analysis proofs would be accepted.   Everywhere I've looked online has said this is "trivial" or "easy to see" without bothering to actually prove it for those of us mere mortals who don't see it as such. As for myself, I have literally no idea where to even get started proving this. Could somebody help me figure this out (without giving me cutesy examples that aren't really all that related with the expectation that I should be able to figure it out from looking at those - that's not really how my brain works)?  Thank you. SaveAsActivity is not allowed with whatId that is not supported as a task whatId Systems of equations with a single number Class is not key value coding-compliant Prove that a group is infinite and nonabelian. I'm trying to solve a question of Hungerford's Algebra:     Show that the group defined by generators $a,b$ and relations   $a^2=b^3=e$ is infinite and nonabelian.   I guess a good approach would be to find an infinite and nonabelian group with two generators satisfying the same conditions and let Van Dyck do all the work. I've been thinking about the group of permutations $S_{\mathbb N}$, some groups of matrices, but I couldn't get it. Could anyone give me a hint? Thanks in advance! Why don't files from Xbox 360 show up when USB drive is plugged into computer? I'm using a Lexar JumpDrive S73 32GB USB flash drive. I've configured it on my Xbox 360 and got a message saying it's compatible and meets the standards, but when I plug in it in my computer no files show up, even in Horizon.   After the driver was installed, both the USB mass storage and Lexar JumpDrive USB device were ready to use (installed properly), but it said no driver found for Xbox 360.    I tried looking for some sort of driver (which sounds silly) but I couldn't find anything.  I'm attempting to mod Minecraft on my Xbox and I don't want to buy another USB. Audits bug in the filtered review queue Just recently the . I've got  this morning which seems to be out of scope for my filtered list. I have filtered my list to only review VBA and Excel-VBA tags while the audit is tagged with perl http pipes which have nothing in common with VBA. I see a problem with the way the system currently works  Since I have a filtered list of tags, once I see a differently tagged question I already know it's an audit. Come on, that's too easy!  Can this please be fixed? update I have ran into  which I decided to edit - the question was tagged VBA and I thought it had nothing to do with VBA so I selected Edit which was the correct action. Now, once the first audit was clearly one to close - nothing to do with VBA and really easy to know that it actually was an audit - the second one was a very good audit. I am just thinking now if this is a proper bug or just a matter of malfunction in the system. I think the first one may have been made an audit for the wrong reason. Let's turn this into a discussion as I can't clearly work the system out. how to remove frost free outdoor faucet drain valve assembly I've tried unscrewing the part outlined in red below. No dice. Do I just need to remove the whole rod (above the red outline) and replace it?) If so, can anyone point me to instructions for them.  I can't see in there. Worried that I'll pull it out and not be able to put it back in.  (The deeper I go in the plumbing the more things I find that there is no modern fitting-part :)  My new assembly  replaces the parts outlined in red. the new assembly has a female thread. I am HOPING that the thicker part of the pipe (just above the black "collar" stopper)  is screwed onto the pipe above.  However, I cannot unscrew it. There is nothing to get a hold of. It's perfectly round. I've tried dw40, and let it sit for 20 minutes. No love.  I'll try wacking it with a board in the direction along which it moves in when it screws (perpendicular to the threads per [this answer][2]) I have read a lot about the SB-800/SB-900 but I am on a low budget (around ~120$). Is there an external flash that I can have to start learning to use it? I do not need TTL compatibility so it can be an older model. I have a Nikon D5000. Almost every time I turn my XBox on and start up Fallout 4, I am unable to use the "Continue" option because the most recent save is corrupt.  I instead use the "Load" option. Here, I can see that the last save I manually performed is perfectly fine, and load it with no problem. However, there is an additional, corrupt save with the same time stamp immediately after it.  This doesn't happen quite every time, but often enough that I'm surprised when there's no corrupt save waiting for me to delete it.  I've tried a few things to solve this, including exiting to the game's main menu before turning the XBox off, going out to the guide before turning the XBox off, turning it off immediately after saving, and waiting a while after saving.  Why do I get all these corrupt saves? Do others have this problem? How do I fix it? Is Python/ArcPy capable of delimiting strings with commas into different fields, and running a calculation based on those separate columns Using ArcGIS ArcMap 10.6: I have a text field in an attribute table that has comma separated values (example: 12,24, 64, 87,29). For this example, I will name this field VALUES. Please note that VALUES includes data collected from field work so the spacing between commas will be varied. My current workflow is as followed:  Export the ID and VALUES fields in the attribute table into a .csv file using Table to Table in ArcMap 10.6 Open the .csv in Excel and run the 'text to columns' function under the Data ribbon Separate the VALUES field into multiple columns based on commas Run the calculation: Copy Column I values and ID columns into a new .csv file Save and close the new .csv file and join the shapefile to the new .csv file based on ID Export the newly joined shapefile into a new file and delete the duplicate ID field in the new shapefile  I am wondering if there is a way to automate this Python/ArcPy so I can create a custom script tool to do this without leaving the ArcGIS environment. Is there a way to separate values in a text field based on commas into different columns, run the above calculation, and have the values from that calculation entered in a new field named VALUES_CALCULATED? I've been trying to solve the following problem:  Suppose that $f$ and $f&#39;$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\infty}f&#39;(x)$ exist. Show that $\displaystyle\lim_{x\to\infty}f&#39;(x) = 0$.  I'm not entirely sure what to do. Since there's not a lot of information given, I guess there isn't very much one can do. I tried using the definition of the derivative and showing that it went to $0$ as $x$ went to $\infty$ but that didn't really work out. Now I'm thinking I should assume $\displaystyle\lim_{x\to\infty}f&#39;(x) = L \neq 0$ and try to get a contradiction, but I'm not sure where the contradiction would come from.   Could somebody point me in the right direction (e.g. a certain theorem or property I have to use?) Thanks Can $\frac{\sqrt{3}}{\sin20^{\circ}}-\frac{1}{\cos20^{\circ}}$ have two values? Hello I am taking a Discrete Mathematics course and having some trouble with this question about sets:  Under what conditions is $A- B = B- A$  Diagram:   Diagram:   Maybe I'm understanding incorrectly, but how can A-B and B-A be equal if they contain elements that are not in each other? Which Content Management System (CMS)/Wiki should I use? How do I fix overscan on my HDMI HDTV? I've connected a HDTV with HDMI on my Radeon 4670, using the open source ati driver. However there is "overscan" which cuts off a bit of each edge of the screen, about the size of the gnome-panel. How can I fix this so it displays the full resolution?