My question reads:   If A is finite and B is infinite, then B-A is infinite.
Prove: If $A$ is a finite set and $B$ is an infinite set, then $B \setminus A$ is an infinite set.   I've tried proving the contrapositive, that if $B \setminus A$ is a finite set then $A$ is infinite or $B$ is finite, but I don't think this is the way to go.
Referring to unidimensional motion, it is obvious that it doesn't always make sense to write the speed as a function of position. Seems to me that this is a necessary condition to derive formulas like:  $$v^2=v_0 ^2 +2\int_{x_0}^{x}a\cdot dx$$  In fact, in the first step of the demonstration (the one I saw, but I think that this step is crucial) it's required to write $a=dv/dt=(dv/dx)(dx/dt)$, that doesn't make sense if $v$ isn't a function of $x$.   When can one rigorously write $v=v(x)$? Dual booting Windows and Ubuntu My laptop has 500gb hard disk in which for 250gb I installed Windows xp on C drive(primary partition). I want to install Ubuntu 12.04 in the remaing hard drive with dual boot option. When I am going to install Ubuntu it is showing something like  /boot /boot/   Can anybody help me in this regard that where and how to install Ubuntu? Why is $(2, 1+\sqrt{-5})$ not principal in $\mathbb{Z}[\sqrt{-5}]$?   Say $(2,1+\sqrt{-5})=(\alpha)$, then since $2\in(2,1+\sqrt{-5})$ we have $2\in (\alpha)$, so $\alpha\mid2$ in $\mathbb Z[\sqrt{-5}]$. Writing $2=\alpha\beta$ in $\mathbb Z[\sqrt{-5}]$ and taking norms, $4=N(\alpha)N(\beta)$ in $\mathbb Z$, so $N(\alpha)\mid4$ in $\mathbb Z$. Similarly, since $\sqrt{-5}\in(\alpha)$ we get $N(\alpha)\mid5$, thus $N(\alpha)$ is a common divisor of $4$ and $5$, therefore $N(\alpha)=(1)$, so $\alpha$ is a unit. But that means $1\in (2,1+\sqrt{-5})=(\alpha)$, so it must be the whole ring, but it cannot to reach a contradiction, so how can I find an element, which is not in $(2, 1+\sqrt{-5})$.  Or is there a simpler method ? (Maybe not UFD would imply not PID) Very large log files, what should I do? ( deals with a similar issue, but it talks about a rotated log file.)  Today I got a system message regarding very low /var space.   As usual I executed the commands in the line of sudo apt-get clean which improved the scenario only slightly. Then I deleted the rotated log files which again provided very little improvement.  Upon examination I find that some log files in the /var/log has grown up to be very huge ones. To be specific, ls -lSh /var/log gives,   total 28G -rw-r----- 1 syslog            adm      14G Aug 23 21:56 kern.log -rw-r----- 1 syslog            adm      14G Aug 23 21:56 syslog -rw-rw-r-- 1 root              utmp    390K Aug 23 21:47 wtmp -rw-r--r-- 1 root              root    287K Aug 23 21:42 dpkg.log -rw-rw-r-- 1 root              utmp    287K Aug 23 20:43 lastlog    As we can see, the first two are the offending ones. I am mildly surprised why such large files have not been rotated.  So, what should I do? Simply delete these files and then reboot? Or go for some more prudent steps?  I am using Ubuntu 14.04.  UPDATE 1  To begin with, the system is only several months old. I had to install the system from scratch couple of months back after a hard disk crash.  Now, as advised in ,  I first checked the offending log files using tail, no surprise there. Then, for deeper inspection, I executed this script from the .  for log in /var/log/{syslog,kern.log}; do    echo "${log} :"   sed -e 's/\[[^]]\+\]//' -e 's/.*[0-9]\{2\}:[0-9]\{2\}:[0-9]\{2\}//' ${log} \   | sort | uniq -c | sort -hr | head -10 done   The process took several hours.  The output was in the line of,   /var/log/syslog : 71209229  Rafid-Hamiz-Dell kernel:  sda3: rw=1, want=7638104968240336200, limit=1681522688 53929977  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device 17280298  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device    1639  Rafid-Hamiz-Dell kernel:  EXT4-fs warning (device sda3): ext4_end_bio:317: I/O error -5 writing to inode 6819258 (offset 0 size 4096 starting block 54763121030042024)        &lt;snipped&gt;  /var/log/kern.log.1 : 71210257  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device 71209212  Rafid-Hamiz-Dell kernel:  sda3: rw=1, want=7638104968240336200, limit=1681522688    1639  Rafid-Hamiz-Dell kernel:  EXT4-fs warning (device sda3): ext4_end_bio:317: I/O error -5 writing to inode 6819258 (offset 0 size 4096 starting block 954763121030042024)    (/dev/sda3 is  my home directory. As we can find,   lsblk /dev/sda NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 931.5G  0 disk  ├─sda1   8:1    0 122.1G  0 part / ├─sda2   8:2    0   7.6G  0 part [SWAP] └─sda3   8:3    0 801.8G  0 part /home    Why a process will want to write beyond the limit is actually outside the scope of my comprehension. Perhaps I will want to ask a different question in this forum if this continues even after a system update.)  Then, from  (you may want to check  for a deeper understanding), I executed,   sudo su - &gt; kern.log &gt; syslog   Now, these files have zero sizes. The system is running fine before and after a reboot.   I will watch these files (along with others) in the next few days and report back should they behave out-of-line.  As a final note, both  the offending files (kern.log and syslog), are set to be rotated, as inspection of the files (grep helped) inside  /etc/logrotate.d/ shows.  UPDATE 2  The log files are actually rotated. Looks like the large sizes were attained on a single day. How do you use the Dungeon Defender's Portal Gun? The topic pretty much sums up my question. I have not yet figured out how the gun operates aside from the fact that it creates Portal portals. The weapon description indicates that it deals 250+ base damage, but how is that damage applied to the monsters? Canonical Question  I want to do &lt;some really cool thing&gt; with a lightning:datatable, but I can't seem to figure out how to do it. I checked the , but I did not see a way to implement &lt;this really cool feature I need&gt;.  Perhaps I want to provide tooltips, or a pick list, or alter the behavior of the menus, or display images, or show a custom component I created, or define a new row-level click behavior, or alter the keyboard event handlers, etc.  Can I somehow do this? I am struggling with this question from Halmos's text, please ignore the imperative language.  "Suppose that $L, M$  and $N$ are subspaces of a vector space. Show that the equation    $$L \cap (M + N) = (L \cap M) + (L \cap N)$$   is not necessarily true.  Since each of these subspaces has origin in them, clearly there intersections could not be empty. I wasn't able to formulate an example where this result did n't hold. Any help would be highly appreciated. Placing the object of an infinitive before it instead of after it Which badges do not cause a notification for experienced users? How can faces be rotated around their individual normals? In edit mode with these settings:   Face selection Pivot point: individual origins Transformation orientation: Normal   The faces don't rotate around the normals, is there another required setting that I overlooked? As mentioned on :     For something that can be done so   easily by accident, there should be   the ability to undo mistakes.   I know that I have clicked several times on a comment up-vote without actually wanting to.  I would suggest being able to cancel the up-vote (maybe only for a couple of minutes or even less than a minute, no problem, as long as I can undo a miss-click). Under what conditions does loot despawn? Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$ Proof that Epicycloids are Algebraic Curves? Epicycloids are most commonly described by the parametric equations,   $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$  $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right).$  Where $R$ is the radius of the fixed circle and $a$ is the radius of the rolling circle.   With $R = ka$ we also have,   $x(t) = a[(k + 1)\cos(t) - \cos((k + 1)t)],$  $y(t) = a[(k + 1)\sin(t) - \sin((k + 1)t)].$  Several books discussing epicycloids mention that if the ratio of the radii of the circles $\left( \frac{R}{a} = k \right)$ is rational, then they are algebraic curves. However, I’ve only been able to find the Cartesian equations for the cardioid, nephroid and ranunculoid. With the cardioid being,   $(ax + x^2 + y^2)^2 = a^2(x^2 + y^2).$   The nephroid,   $(-4a^2 + x^2 + y^2)^3 = 108a^4y^2.$  And the ranunculoid,   $-52521875a^{12} – 1286250a^{10} (x^2 + y^2) – 32025a^8 (x^2 + y^2)^2 + 93312a^7 (x^5 – 10x^3y^2 + 5xy^4) – 812a^6 (x^2 + y^2)^3 – 21a^4 (x^2 + y^2)^4 – 42a^2(x^2 + y^2)^5 + (x^2 + y^2)^6 = 0.$  Clearly this doesn’t cover all epicycloids where $\frac{R}{a}$ is rational.   What is the proof that shows that epicycloids, where the ratio of the radii are rational, are algebraic curves? Radius of convergence of a sum of power series I have  two series  $\displaystyle\sum_{n=1}^{\infty} a_n x^{n}$   $\displaystyle\sum_{n=1}^{\infty} b_n x^{n}$  with radius of convergence $2$ and $3$ respectively.  How can I find the radius of convergence for the following?     $\displaystyle\sum_{n=1}^{\infty} (a_n +b_n)x^{n}$   Can anyone help me? When writing answers on here, it's often necessary to create a new script and chmod +x it to make it executable. I won't go into whether this is a good thing or not, but explaining to people that they need to fire in three separate command (and teaching them to use nano!) is a bit of a waffle.  Is there anything easier? How does an attacker's reach interact with the Mounted Combatant feat's ability to force an attack against the mount to target the rider instead? How to redirect output of an entire shell script within the script itself? Is it possible to redirect all of the output of a Bourne shell script to somewhere, but with shell commands inside the script itself? Redirecting the output of a single command is easy, but I want something more like this: #!/bin/sh if [ ! -t 0 ]; then     # redirect all of my output to a file here fi  # rest of script...  Meaning: if the script is run non-interactively (for example, cron), save off the output of everything to a file.  If run interactively from a shell, let the output go to stdout as usual. I want to do this for a script normally run by the FreeBSD periodic utility.  It's part of the daily run, which I don't normally care to see every day in email, so I don't have it sent.  However, if something inside this one particular script fails, that's important to me and I'd like to be able to capture and email the output of this one part of the daily jobs. Update: Joshua's answer is spot-on, but I also wanted to save and restore stdout and stderr around the entire script, which is done like this: # save stdout and stderr to file  # descriptors 3 and 4,  # then redirect them to &quot;foo&quot; exec 3&gt;&amp;1 4&gt;&amp;2 &gt;foo 2&gt;&amp;1  # ...  # restore stdout and stderr exec 1&gt;&amp;3 2&gt;&amp;4 How can I know if $2^{2^{2^{2^{2}}}}+1=?$ is prime? I've seen a couple questions around here like , which mentions:     Unfortunately that same browser won't allow me to test HTTP PUT, DELETE, and to a certain degree even HTTP POST.   I've also heard that browsers support only GET and POST, from some other sources like:        However, a few quick tests in Firefox show that sending PUT and DELETE requests works as expected -- the XMLHttpRequest completes successfully, and the request shows up in the server logs with the right method. Is there some aspect to this I'm missing, such as cross-browser compatibility or non-obvious limitations?