Let $\varphi: G\rightarrow G'$ be a homomorphism. Show that for each $a \in G$, $| \varphi (a) | \le | a |$.
Order of a homomorphism of groups
Are there any cheap, fast lenses capable of highschool sports photography? I recently took on my first paying client, and she is also into photgraphy. She usually does sports for her highschool yearbook. They just have an entry level Canon DSLR with a kit zoom on it that goes up to 200mm, but it's not very fast (f/4 or 5.6 or so). She told me that she has a tough time - especially at basketball games - getting shots with any kind of reasonable lighting, and has to do a lot of work in post just to get them reasonable.   mentions the Cannon EF 300mm f/2.8L IS USM II for professional sports photographers. At $6k USD I suspect the yearbook won't be springing for that lens anytime soon.  So I was curious, are there any reasonable quality, fast lenses for Canon cameras that are lower priced (certainly no more than say, $1k)? I'd love to be able to point them to a couple of options. I assume the largest print they'd be making would be a full spread in the yearbook - maybe two 17"x11" (two 8.5" side by side) What are exchanges like "How are you," "I'm fine," and "See you later" called? Answer turned to CW, votes pre-CW era doesn't count in tag specific stat My understanding is that if an answer becomes CW (one way or another), original author gets to keep the votes pre-CW.  Does this also keep the pre-CW vote count for the tag-specific stat? It doesn't look like it, because I checked recently and suddenly my number dropped. It looks like the tag-specific stat just counts votes for non-CW answers, and doesn't "discriminate" between the pre-CW and post-CW era. If it's eventually CW, then it simply contributes 0 votes.  I'm fine with the fact that my answer became CW (in this case, through the question having 30+ answers), but I would like to keep the votes I earned before it did for the tag stat.  Otherwise I think the system is "broken". Someone can go posting 30+ answers to questions where Jon Skeet answered, and all of Jon Skeet's answers would become CW, and he'd then be 0 at all tags. What does the word "it" mean? Even if determinism isn't true for everything that happens—even if   some things just happen without being determined by causes that were   there in advance—it would still be very significant if everything we   did were determined before we did it. However free you might feel when   choosing      [Thomas Nagel - What does it all mean? p.52]   What does the word "it" mean? Buying shares when the price goes down 2% and selling shares when it goes up 2% Installing 2 versions of Ubuntu on same hard drive? Get top results for each group (in Oracle) Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges. How do I configure swappiness? It's my understanding that "easily" is the regular adverb for "easy", e.g. "He makes friends easily".  But in some collocations one still uses "easy", e.g. "Take it easy", "Rest easy" or "Slow down easy". Is there any rule when "easy" would be used over "easily", or does one just have to know these collocations? When having a number of files opened in Texmaker, at the top of editor pane, there is a drop down panel with the names of open documents in it. I prefer to have tabs of open documents at the top of editor pane, similar to Texstudio and Winedt. Is there a way that I can change this setting? vs. . Which to use? Using INSERT INTO...EXEC AT linked server into temp table fails with Msg 7391 How to make putty to not break my session after some time? Each time putty is closing the session after some time if it is idle.  There is no time parameter on putty, so how can I keep my putty ssh session always Alive? build folder usage issues Manually Installed new Python to /usr/bin, how to get Repo version back? Yes, I'm a moron, and I tried to manually install a higher point release of python 2.7.16 to /usr/bin/  (i did ./configure --prefix=/usr) on my Ubuntu 16.04 machine. It seems to work fine EXCEPT for packages that depended on python, like apt-listchanges, which fails with messages like ImportError: No module named apt_pkg  there's no make uninstall available, I've tried manually removing files, and still when I try apt --reinstall install python, Nothing gets overwritten and the python executable still gives the new 2.7.16 version...   Any idea how to get the official repository version back?? How to disable captive portal detection? How to remove exclamation mark on Wi-Fi or signal icon in Lollipop? How can you explain that this double negation is not a double negation? Is there a rule in English about this kind of sentence?  PS / Do I have to mention Pink Floyd Copyright ? :-)  Edit : Since there are a lot of Pink Floyd related explanation, I'll bring a Freddy Mercury one : "I don't have time for no monkey business", which I also understand as "I don't have time for monkey business". Am I right ? How can I detect the encoding/codepage of a text file In our application, we receive text files (.txt, .csv, etc.) from diverse sources. When reading, these files sometimes contain garbage, because the files where created in a different/unknown codepage.  Is there a way to (automatically) detect the codepage of a text file?   The detectEncodingFromByteOrderMarks, on the StreamReader constructor, works for UTF8  and other unicode marked files, but I'm looking for a way to detect code pages, like ibm850, windows1252.     Thanks for your answers, this is what I've done.  The files we receive are from end-users, they do not have a clue about codepages. The receivers are also end-users, by now this is what they know about codepages: Codepages exist, and are annoying.  Solution:    Open the received file in Notepad, look at a garbled piece of text. If somebody is called François or something, with your human intelligence you can guess this. I've created a small app that the user can use to open the file with, and enter a text that user knows it will appear in the file, when the correct codepage is used.  Loop through all codepages, and display the ones that give a solution with the user provided text.  If more as one codepage pops up, ask the user to specify more text. Cardinality of equivalence relations in $\mathbb{N}$ I came across this long proof on this site:    But I would like to know whether my direction can work.  Say we want to find the cardinality of all equivalence relations in $\mathbb{N}$. Since it is a subset of all relations in $\mathbb{N}$, I conclude it has a cardinality smaller or equal to $\aleph$. Now, define an injective function from $P(\mathbb{N})$ to the set of equivalence relations by matching each subset of $\mathbb{N}$ with the identity relation (which is an equivalence relation in $\mathbb{N}$.  Therefore the cardinality of all equivalence relations in $\mathbb{N}$ is greater or equal to $\aleph$ and using CSB we get the desired result.  Seems legit?