Let $f: A\rightarrow A$. Prove that if $(f ◦ g)$ is surjective, then $f$ is surjective. Let $f: A\rightarrow A$. Prove that if $(f ◦ g)$ is surjective, then $f$ is surjective.  We know that $(f ◦ g)$ is surjective, so that means that $\forall x\in A, \exists a$ such that $f(g(a)) = x$  Let $y = g(a) \in A$  $\implies f(y) = f(g(a)) = x$  $\implies \forall x\in A, \exists y$ such that $f(y) = x$  $\implies f$ is surjective  Is this proof correct?
If $f \circ g$ is surjective, then f is surjective Let $f: X \rightarrow Y$ and $g: Y \rightarrow X$  If $f \circ g$ is surjective, then f is surjective, too.  I think that is true.   Question: How can I proove that? I have so far:   $\forall y \in Y: \exists x \in X : f(x)=y$  $\forall x \in Y : \exists y \in Y : f \circ g(y)= x$
Adapt BibTeX style to give all initials I'm using a modification of the plainyr BibTeX style. Although my bib-file contains all initials for some authors (e.g. B.D. McKay), I end up with just the first initial in my document (e.g. B. McKay). I thought about editing the BibTeX style myself, but after seeing the code, I decided against it. Maybe there are some people who are more knowledgeable in this area of expertise.  I want my final document to include the initials for all the names, and not just those of the first name.  This is the relevant code of the BibTeX style, or at least, I think it is the relevant code.  FUNCTION {format.names} { 's :=   #1 'nameptr :=   s num.names$ 'numnames :=   numnames 'namesleft :=     { namesleft #0 &gt; }     { s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't :=       nameptr #1 &gt;         { namesleft #1 &gt;             { ", " * t * }             { numnames #2 &gt;                 { "," * }                 'skip$               if$               t "others" =                 { " et~al." * }                 { " and " * t * }               if$             }           if$         }         't       if$       nameptr #1 + 'nameptr :=       namesleft #1 - 'namesleft :=     }   while$ } Performance of Non Clustered Indexes on Heaps vs Clustered Indexes compares the performance for individual select/insert/delete/update and range select statements on a table organized as a clustered index vs that on a table organized as a heap with a non clustered index on the same key columns as the CI table.  Generally the clustered index option performed better in the tests as there is only one structure to maintain and because there is no need for bookmark lookups.  One potentially interesting case not covered by the paper would have been a comparison between a non clustered index on a heap vs a non clustered index on a clustered index. In that instance I would have expected the heap might even perform better as once at the NCI leaf level SQL Server has a RID to follow directly rather than needing to traverse the clustered index.   Is anyone aware of similar formal testing that has been carried out in this area and if so what were the results? When should a verb be followed by a gerund instead of an infinitive? Some verbs are followed by ing, e.g. I enjoy swimming. We can't say I enjoy to swim. Likewise, some verbs are followed by to, e.g. I decided to make a plan.   Which particular verbs are followed by ing and to? Can you please provide a list for that?  Moreover, which verbs can be followed by both without having the meaning changed? Difference between $(this) and event.target? Trying to locate a file, how can I search the entire hard drive for a file? Should a math paper use non-alphabetical ordering of authors' names for unequal contributions? Using alphabetical ordering for authors' names is a long established tradition in mathematics. In the situations of extremely unequal contributions, is it reasonable to break from this tradition?  Has there been any examples?  If it is common, what is the threshold for breaking from alphabetical ordering? Invest money and get 120% back - a site says I want to create a circuit that can detect audio frequencies and flash RGB LEDs different colors depending on the detected frequency. The frequency range will be from 20 Hz to 12 kHz, after looking at the chart below.   (source: )    So my question is: what circuit design should I choose to make a good frequency detector? I suppose I would need    Preamp for the microphone (is it necessary for the ADC?) Band pass filters (active or passive?) A microcontroller with an onboard ADC that has a sampling rate of at least 24 kHz.   Before I get too excited, are there any circuits out there that already do this? How do I force Excel (and other Microsoft Office products) to stop opening files in the same application? Whenever I "double click" on an Excel file and another Excel file is open, the newly opened file automatically opens in the same application window as the previously opened Excel file. This isn't limited to just Excel, as I've seen Word do this as well. This poses a problem when wanting to compare documents side by side.    The current solution I have for this is to actually open another Excel or Word instance, and then open the file from within that application window itself. Is there a way to force Office to open a new instance of the application when double clicking on the file icons?  I'm currently using Office 2007 and Windows XP, but I've seen this on Office 2010 and Windows Vista and 7. I'm looking for an overall solution if possible. How would you count occurrences of a string (actually a char) within a string? How do I remove an OpenID or login? How do I remove login credentials (emails or OpenIDs) from my account?     For more information, see "" in the . align vs equation I always use align in my documents, and avoid equation. Is there anything wrong with that? My reasoning behind this: align > equation, so why not use it? Factorial of $i$ With $i^2=-1$, what is $i!$ ?  I'm not sure if it is $i$ or if it is an infinite product $i(i-1)(i-2)(i-3)(i-4)(i-5)\dots$  It would make sense either way.  Is it not defined? Prove that a continuous function on a closed interval attains a maximum Can the Room of Requirement provide anything if needed? For example:   James Potter wanted to hide from Voldemort, it would have served as a perfect hiding place. Harry Potter wanted all the Hocruxes, he should have gone straight to the room of requirement where he would have found all the Hocruxes along with the Sword. All the muggle borns of the wizarding community could have taken refuge in the room of requirement. Can someone meet a dead person in the room of Requirement if needed? Would it have been possible for Harry to go in past in order to see Dumbledore for the explanation of many things? Can the room duplicate something? For instance, Harry needed the Elder wand; with Voldemort having it, could he still get it? Someone wishes to be knowledgeable than Dumbledore, would the seeker find his desired wish by finding books and things?   Who created the room of requirement (surely not Salazar)? Was it beyond Dumbledore's magic to create such a place? Herstein Topics in Algebra, $\pi \mid a$ or $(\pi,a) = 1$ How to make bash put prompt on a new line after cat command? What I get:  host:~ user$ cat example.txt some texthost:~ stas$   What I want to get:  host:~ user$ cat example.txt some text host:~ stas$   Is there a way I can make cat behave like this?  I'm using bash on Mac OS X. How to permanently turn off online results I don't like getting online results when searching in my dashboard, so I went to System Settings &gt; Security &amp; Privacy &gt; Search, and I turned the setting off:    But when I restart my machine, the setting is always reset to:    So how can I permanently disable it without it turning on again after a restart?  Information Update:  I have had a look at the post that this has been marked as a possible duplicate of, and I don't think that my question really is a duplicate of it, as the other question is asking about how to set it globally, and I am not, I am only wanting to set it for one user, so the fact that it is not global is not the problem, the problem is that it is not sticking for that one user, and just resetting to its default state after a restart. In fact, I don't want it to be set globally, and the answers from the other question only really explain how to do that.    OS Information:  Description:    Ubuntu 15.04 Release:    15.04   Package Information:  unity:   Installed: 7.3.2+15.04.20150420-0ubuntu1   Candidate: 7.3.2+15.04.20150420-0ubuntu1   Version table:  *** 7.3.2+15.04.20150420-0ubuntu1 0         500 http://gb.archive.ubuntu.com/ubuntu/ vivid/main amd64 Packages         100 /var/lib/dpkg/status Why can't I set a default output audio device in Ubuntu 19.10? How can I do a full shutdown of windows 10 without logging in? I have a computer with both Windows 10 and Ubuntu 17.04 installed on it. I can't log in to Windows, and think it's a problem with the registry. Luckily, there is a way to restore the registry from backup from Ubuntu, but you have to be able to modify files on the Windows partition. Right now, I can't write to the NTFS partition because Windows didn't do a full shutdown.  How can I force windows 10 to do a full shutdown without being able to log in?