Prove that the set $S$ of rational points in the plane $\mathbb R^2$ is denumerable.
Show that $\mathbb{Q}\times \mathbb{Q}$ is denumerable
Usernames vs. Real Names I want to create a social network mainly geared towards user-to-user interactions such as commenting and discussion boards. From my observations, I've found that users are more likely to engage in constructive conversations when using their full names as opposed to creating flame wars when hiding behind a username. While the latter still has the possibility to exist using one's real name, it's not as evident considering that the user is bearing his or her identity—either through the site's main registration or through Facebook Connect.  Don't get me wrong I've seen many communities manage to still offer insightful user contributions using usernames. All in all, what are the pros and cons of the two? Facebook changed how their API serves profile pictures that are requested with a non-application scope UID. Now because before Graph API 2 all applications shared the original UID, if users like me has joined Stack Exchange websites before  2, rolling out their profile pictures are not shown properly. Facebook suggests to use an access token to query those profile pictures, either an application access token or a user access token. A user access token is not a reliable option at all, because it would only work for users who joined website with Facebook and are currently logged in. Application access tokens, by the way, can be used by creating sort of a proxy that would retrieve a profile picture from Facebook and display it. Yet still it's not perfect. I think the best solution is to communicate with Facebook and ask them to offer an opportunity to convert old Graph API 1 UID's to application scope user IDs. Recent movie involving dimensions and time travel/displacement It's a (relatively) recent movie about a kid whose scientist father is missing or presumed dead.  The kid experiences strange situations (e.g., books from a case fall off).  In the end, it was the father the one that was throwing those books to draw the attention of the kid.  Does anyone know the name of this film? This question came up after reading , in which it is re-iterated (as it has been many times on this site), that investing in the stock market, especially an index fund that tracks the top 100 / 500 companies in a country, basically always has great returns in the long run.  So my question is: Why isn't everybody rich?  If it is so easy (basically invest as heavily as you can in your youth to be rich into middle age and older), why doesn't everyone do this? There would be no need for pensions or any other savings or financial security that way, why even bother investing in assets like property that you have to manage when you can just have an index fund work for you?  I'm guessing that something is being missed here. What keeps mass from turning into energy? I understand the energy and mass can change back and forth according to Einstein. It is fluid; it can go from one to the other. So, what keeps mass from just turning into energy? Is there some force holding a subatomic particle together? What keeps mass in it's state? I hope this is not a silly question but I am clueless. Thanks I've had a couple of answers today where the OP indicated via a comment that I solved their problem, but the question wasn't upvoted or accepted.  In both cases the person probably wasn't able to upvote since they were new, but they could have accepted my answer. I felt a little guilty asking them to accept the answer, but since it solved their problem I felt that it would help others who stumbled on the question trying to solve their own problem to know that my solution really was the answer. Is asking the OP to accept your answer when they've said that it solved their problem a reasonable thing to do or does that feel like nagging for reputation? Select Command not working while run shell scripts on remote machine What should we do about Stack Overflow-specific questions on Meta SE? Energy-momentum tensor transformation I have a long line of code that I want to break up among multiple lines. What do I use and what is the syntax?  For example, adding a bunch of strings,  e = 'a' + 'b' + 'c' + 'd'   and have it in two lines like this:  e = 'a' + 'b' +     'c' + 'd' Why are real symmetric matrices diagonalizable? I wanted a simple environment for making lists of equations, so I made one that includes an equation and a split environment. The code won't compile and the error seems unrelated. The most pressing thing I noticed was the &quot;You can't use `\end' in internal vertical mode.&quot; The problem is definitely the split environment. Here is a MWE. \documentclass{article} \usepackage{amsmath}  \newenvironment{env}     {\begin{equation}\begin{split}}     {\end{split}\end{equation}}  \begin{document} \begin{env}     x \end{env} \end{document}  If I remove the split environment from env, everything compiles just fine. Additionally, If the following example compiles just fine. What is the problem then? \documentclass{article} \usepackage{amsmath}  \begin{document} \begin{equation} \begin{split}     x \end{split} \end{equation} \end{document}  Here is the LaTeX error upon running the MWE. math.tex|11 error| LaTeX Error: \begin{split} on input line 9 ended by \end{env}. math.tex|12 error| LaTeX Error: \begin{equation} on input line 9 ended by \end{document}. math.tex|12 error| Missing $ inserted. math.tex|12 error| Missing } inserted. math.tex|12 error| Missing } inserted. math.tex|12 error| Missing } inserted. math.tex|12 error| Missing \cr inserted. math.tex|12 error| Missing { inserted. math.tex|12 error| Missing { inserted. math.tex|12 error| Extra }, or forgotten $. math.tex|12 error| Missing $ inserted. math.tex|12 error| Missing } inserted. math.tex|12 error| You can't use `\end' in internal vertical mode. math.tex|12 error| LaTeX Error: \begin{equation} on input line 9 ended by \end{document}. math.tex|12 error| Missing } inserted. math.tex|12 error| Missing $ inserted. math.tex|12 error| Display math should end with $$. math.tex|| Emergency stop. math.tex|| ==&gt; Fatal error occurred, no output PDF file produced! A word for "every two days" In an electromagnetic theory class, my professor introduced the concept of "action at a distance in physics". He said that:     If two charges are at some very large distance, and if any one of the charge moves, then the force associated with the charges changes instantaneously.   But according to Einstein, no information can travel faster than the speed of light.   So photons (the information carriers in electromagnetic force) cannot instantaneously deliver information.   So that we associate a field with the two charges and if any charge moves, there is a deformation in that field and this deformation travels with the speed of light and conveys the information.   If the field deformation information cannot travel more than the speed of light, how does the force instantaneously change at very large distances? I couldn't find anything about Ubuntu acting as a  receiver or sender.   Can it work at all? Are there hardware prerequisites? Is WiFi a requirement or can it work over LAN or another kind of network connection? WiFi direct seems to be a necessary requirement, is it a sufficient one? (i.e. if a system supports WiFi direct does that mean it supports Miracast?) Are there differences in support between receiving/sending? How is the latency? (compared to the competition, i.e. VNC, commercial Miracast devices, etc.) How do I actually use it, if it's difficult?   Specifically, I plan to use it together with an Android phone (4.x Jelly Bean). What happened after the Ring was dropped into Mount Doom? In the Peter Jackson movie, once the One Ring is thrown into the volcano in Mount Doom, why does the mountain and tower collapse, and the ground disappears everywhere where the orcs are standing but not where the humans are? Where are all the orcs running to, what do they do next? Show that if $f(z)$ is an entire function such that it takes real values on the lines $Im(z)=0$ and $Im(z)=a$ (for some $a \in \mathbb{R}_{\neq 0})$, then $f$ is periodic with period $2 i a$.    So here is my thought process: First of all by reflection principle $f(\bar{z})=\overline{f(z)}$ so if our function $f$ is given in the strip $0 \leq Im(z) \leq a$, then its values on $-a \leq Im(z) \leq a$ follow by reflection principle. Then by imposing $f(z)=f(z+2 ia)$ we can intuitively see that this will define an entire function by looking at how the strips $na \leq Im(z) \leq 2na$ seamlessly stack on top of each other. Is this logical? And how would one prove this analytically? What is the most efficient way to add rows and columns to a matrix? Ask question post button remains disabled I've noticed this twice in the last ~2 weeks or so. The &quot;post your question&quot; button can remain disabled for some reason even when your question is shown as &quot;ready to publish&quot;. I.e., you've gotten to the &quot;review&quot; page after the initial edit. If I remember correctly, one case was on Stack Overflow, and the other was on . I'm not sure what exact scenario leads up to this. I haven't paid attention to the button until the last step when I'm ready to click it. The workaround has been to reload the page. (I copied my content for safe-keeping first.) This works, and the button then shows up as active. Afterwards, however, on both occasions, it forced me through the &quot;are you a human&quot; verification.  Update - this question itself did not exhibit this, nor did it do the .  Update - I'm using Chrome on Windows and am not sure what version it was when this happened.  It has  installed, but there aren't any scripts for any Stack Exchange sites.  It has  installed.  The virus scanner on the PC I probably was using was This is an exam question I was practising. I have the general understanding of Linear programming, but how would you go about finding the Decision Variables, Objective function and Constraints for this?  My Opinions: Decision variables: $M_1$- # of Model $1$ to be produced $M_2$ and $M_3$(similar definition as $M_1$)  Objective function: I'd Multiple the Cost to make in house by cost to buy from subtractor  to get Minimise: $3900M_1+7500M_2+27300M_3$  Constraints: $3M_1+1.5M_2+2M_3\ge0$  $1.25M_1+2.5M_2+0.75M_3\ge8500$  $M_1,M_2,M_3\ge 5500$  No idea where number order would go? Do I use it in the Constraints and multiply by the Hours?  Now, am I on the right track?