I'm completely lost, and I haven't been able to get anything yet. I looked at some that were similar to this problem, and still got pretty lost.
How to define a bijection between $(0,1)$ and $(0,1]$?   Or any other open and closed intervals?   If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it): I make a function $f : (-1, 2)\rightarrow (-5, 4)$ of the form $f(x)=mx+b$ by \begin{align*} -5 = f(-1) &amp;= m(-1)+b \\ 4 = f(2) &amp;= m(2) + b \end{align*} Solving for $m$ and $b$ I find $m=3\text{ and }b=-2$ so then $f(x)=3x-2.$  Then I show that $f$ is a bijection by showing that it is injective and surjective.
Commutator of two normal subgroups with trivial intersections So I tried looking everywhere but I'm really lost with this proof. I need to to show that:  Let $H$, $K$ be normal subgroups of a group $G$, satisfying $H\cap K = \{e\}$. Let $h\in H$, $k \in K$. (a) Prove that the commutator $w = hkh^{-1}k^{−1}$ satisfies $w \in H$ and $w \in K$. Then conclude that $w = e$ (b) Prove that $hk = kh$, for all $h \in H$, $k \in K$. How do you run update-grub? JavaScript hard refresh of current page No longer earning rep for editing posts Easily measure elapsed time Using getopts to process long and short command line options Real-life examples of common distributions Book about a town that is transported by an explosion into a freezing desert It starts with a town that was hit by a mysterious explosion which transports that town into a desert. The weather is cold and people burn coal but they soon run out and send out an expedition to look for signs of life - they navigate the desert for a couple of days and come across a glass dome covering a city. Apparently they have been transported far in the future.  And that's where I stopped reading and want to resume now. Don't remember the title or the author and it's at least 20 years old. Cannot add custom launcher to Ubuntu Dock ('Add to Favorites' option does not show up) I have a custom .desktop file: [Desktop Entry] Type=Application Terminal=false Icon[en_US]=/home/alex/bin/idea-IC-173.4127.27/bin/idea.png Name[en_US]=IntelliJ Exec=/home/alex/bin/idea-IC-173.4127.27/bin/idea.sh Name=IntelliJ Icon=/home/alex/bin/idea-IC-173.4127.27/bin/idea.png  Output of ls -lha .local/share/applications/intellij.desktop: -rwx--x--x 1 alex alex 248 Dez 30 10:54 .local/share/applications/intellij.desktop  Answers posted to a previous question () don't work for me. I can find this launcher in Activities and I can double click on that file and IntelliJ will start. Nevertheless, I cannot add this launcher as a favorite to the dock. The option does not show up. Temporarily deleted answer missing from asker's inbox I  and . This answer does not appear in my inbox. I never saw a notification, and as of now the latest notification from CS is  {"SiteUrl":"cs.stackexchange.com","FaviconUrl":"https://cs.stackexchange.com/favicon.ico","CreationDate":"17 hours ago","Url":"https://cs.stackexchange.com/questions/641/language-of-the-graph-of-an-affine-function/667#667","Title":"Language of the graph of an affine function","Summary":"Hint:\n\n$53 \\cdot 100000010000000000 + 4 = 5300000530000000004$.\n\nSketch:\n\n\n  ! Assume $a$ has $n$...","Type":"answers","Count":3,"IsNew":false}   This answer was posted, deleted and undeleted while I wasn't online. I suppose the deletion removed the entry from my inbox. Then the undeletion should have recreated an entry but didn't. Java Generics: List, List, List Show that the set of functions $\mathbb{N}\to\{0,1\}$ is not countable Recall that a countable set $S$ implies that there exists a bijection $\mathbb{N}\to S$. Now, I consider (0,1). I want to prove by contradiction that $(0,1)$ is not countable.  First, I assume the contrary that there exists a bijection $f$, and I can find an element in $S$, but not in the range of $f$. But I can't find such element. How can you construct such $f$? Output of which command used for input to cd I would like to take the output of a which command, and cd to the parent directory. For example, say I have the following:  which someprogram   With output:  /home/me/somedirectory/someprogram   And I would like to cd to the directory that someprogram lives in:  cd /home/me/somedirectory   I'd like to accomplish this in one line. What is the most elegant, tricky, short way to do this? Contour integral for $x^3/(e^x-1)$? What contour and integrand do we use to evaluate $$ \int_0^\infty \frac{x^3}{e^x-1} dx $$  Or is this going to need some other technique? This post is in reference to the following blog post:   To start off, a quick introduction of myself; my name is Jascha Drel, Product Manager for Advertising at Stack Overflow. In this role, I am part of ’s organization.   Our team is continuously making improvements to the ads experience on our sites. One of the changes we’re excited about sharing is the new Report this Ad feature now available on some of our sites. We want to continue to give our communities more native control over their ad experience. We needed a quantifiable way to receive feedback and this feature will allow that.  We enforce strict  on the types of advertisers we work with but cannot fully control what is shown due to how advertising delivery works. However, being able to hear from our users about problematic ads will be a huge benefit as we can address those problems faster and with more accuracy. The Report this Ad feature will allow us to quickly see and investigate ads reported by you which matters to us because we want to hear your thoughts.  Furthermore, this new feature ensures that the members of our communities have a built-in way to report ads they consider problematic when we test any new ad formats/solutions.  We are launching this feature on 5 sites initially (, , , , and ) to get a good understanding of what we can expect in terms of numbers of reports whenever launching this to the entire network. Prior to launching it network-wide, we’ll share general results we’ve seen here.  What happens once an ad is reported?  All reports will be consolidated into a dashboard that the team will look at several times per week to spot irregularities in terms of the total number of reports, report categories, and on an advertising campaign basis. We will not be replying to individual reports, but we will reach out to our advertisers, to share feedback or notify them if the ad being reported does not fit our guidelines. We may also evaluate our policies in response to large numbers of reports not covered by them. Since we don’t have a baseline for the number of ads that are going to be reported using this new feature, we will consider reevaluating the process if the current process doesn’t scale.    We’d like to share an update on the progress of this feature:   We've launched this feature on 5 SE sites (Server Fault, Database Administrators, Arqade, Game Development, and Unix &amp; Linux) initially for logged in users only, and for anonymous users afterward. The report this ad button was shown several million times (of which roughly 65% paid ads and 35% house ads). We showed several dozens of different ad campaigns. So far we have received 0 reports. We've tested the feature several ways, found some minor bugs (which have been/are being fixed), however, in the vast majority of the scenarios the feature works well (we’re happy to be ).   We’re currently graduating this feature and are releasing it network-wide this week. Feel free to comment below if you have any questions/comments. I think it would make sense to split up the flag "This is not an answer" into three flag choices:   This should be a comment This should be an edit This should be a new question   If we had that, it would make it easier (and therefore more efficient) for mods to figure out what to do with answers flagged thusly. What is a good book to learn number theory? What would be a good book to learn basic number theory? If possible a book which also has a collection of practice problems? Thanks. MBP 13' (mid-2010; 7,1) 2.4 Core 2 Duo P8600 supporting 16 GB RAM under Win7? I’ve got a MBP 13′ (mid-2010) 2.4 Core 2 Duo (full specs below) that I’m now operating solely under Win7 64Bit. It formerly run as well under OS X 10.6.8, relatively to which the firmware seems up to date (MBP71.0039.B0B).  Would my system recognize 16 GB RAM if operated under Win7 64 Bit? Or would it require the latest firmware (MBP71.0039.B0E) of OS-X 10.7.5 (which was never installed on my MBP) anyway?   As far as I have read elsewhere, using 16 GB RAM under OS X seems to definitely require this later version of OS X (including the respective firmware update); so I'd like to know whether this is an issue of OS X or (as I suspect) the firmware.    Modellname: MacBook Pro Modell-Identifizierung: MacBookPro7,1 Prozessortyp: Intel Core 2 Duo Prozessorgeschwindigkeit: 2,4 GHz Anzahl der Prozessoren: 1 Gesamtzahl der Kerne: 2 L2-Cache: 3 MB Speicher: 4 GB Busgeschwindigkeit: 1,07 GHz Boot-ROM-Version: MBP71.0039.B0B SMC-Version (System): 1.62f6 Simplest Light triggered relay I am building a large array of Light Switched relays, and due to the size of the array i want to keep the part count down.  Each 'Pixel module' has a 12V SPDT, bulb and an activate-on-dark circuit. There is a strong spotlight on the device and the viewers shadow is obscuring the light in their shadow so the swing is large.  I have seen the obvious LDR-and-Pot circuits with a junction out to a NPN Transistor, and the more stable comparator alternative  I was looking at "solar cell" types of Photodiodes (PiN Type), and I can't see why I cannot just lay one across a PNP transistor (Collector to Base?) or something so it applies a Base voltage when lit up.  Can someone briefly explain to me why I am wrong?  I figure it would be all over the net if it was that easy, and i cant find any examples!  Is this possibly the cheapest way to make a light switched 12V relay driver?  Note: I prefer solutions using discrete circuitry, as I have a thing for human readability, visual components etc :P I have a front end module that generates an (ECG) signal that varies from +/-2.5 V. I want to shift this signal to 0 - 5V. What is the best way to do this? Would a summing amplifier like the below circuit good enough? With R1 = R2 and V1 = 2.5V, V2 = my signal, V3 = V4 = GND