I want to find functions  $f: A \to B$ and $g: B \to A$ such that $g(f) = i_A$ but $f(g) \neq i_B$.   Is this possible? An exercise for one of my classes is to actually prove that if $g(f) = i_A$, then $f(g) = i_B$.   I've found functions where $f(g(x)) = x$ for most $x$, but failed for an exceptional choice of $x$. Would that be a counter example?   Are there examples where $g(f(x)) = x$ for all $x \in A$ and $f(g(x)) \neq x$ for all $x \in B$?
Two functions $f:A\rightarrow B$ and $g:B\rightarrow A$. Can someone give me an example where $g\circ f=id_A$ but $f\circ g\neq id_B$?
prevent "action" if top of page I have a manual list of stuff that are sometimes separated by an hrule(or my own custom ruler) for visual purposes.  Sometimes the hrule ends at the top or bottom of a page which makes it useless.  Is there a way to prevent it from being added if it is at the top or bottom of a page?    MWE of problem:  \documentclass{book} \usepackage{tikz} \begin{document} \newcommand\rulesep{%   \par % make sure we end a paragraph   \vskip5pt % space above the rule   \leaders\vrule width \textwidth\vskip0.4pt % rule thickness is 0.4pt   \nointerlineskip % disable interline glue here   \vskip5pt % space below the rule } \def\dotikzcircle#1{\begin{tikzpicture}\draw(0,0)circle(#1);\end{tikzpicture}} \rulesep \noindent This is a test\noindent \rulesep \noindent End of test \rulesep \dotikzcircle3 \rulesep \begin{center}\dotikzcircle3\end{center} \rulesep \begin{center}\dotikzcircle3\end{center} \rulesep \dotikzcircle1 \rulesep \dotikzcircle2 \rulesep \dotikzcircle3 \rulesep \dotikzcircle4 \rulesep \dotikzcircle5 \rulesep  \end{document} How to enable fast user switching when Login Options are disabled? I have IIS7 on my computer, and I can access it through other computers in my house on the same network through my computer's IP address - 192.168.1.10x  Instead of typing in http : //192.168.1.10x/ I want to be able to go to http : //somename/  I know you can change the hosts file to add it but I'd have to do that on each computer and I'd still have to type in the IP to access it from my iPod/Wii... etc. Ligatures in figures and pdftex PreparedStatement IN clause alternatives? What are the best workarounds for using a SQL IN clause with instances of java.sql.PreparedStatement, which is not supported for multiple values due to SQL injection attack security issues: One ? placeholder represents one value, rather than a list of values.  Consider the following SQL statement:  SELECT my_column FROM my_table where search_column IN (?)   Using preparedStatement.setString( 1, "'A', 'B', 'C'" ); is essentially a non-working attempt at a workaround of the reasons for using ? in the first place.    What workarounds are available? Prove that $a^{2^n}=1 \mod 2^{n+2}$ I would like to prove that  $$a^{2^n}\equiv 1 \pmod {2^{n+2}}$$  I tried induction but could not get it.  Thank you very much! Showing that the set of fixed points is a smooth manifold Let $M$ be a smooth manifold and Let $G$ be a finite group acting on $M$ by diffeomorphisms. Show that the set of fixed point $F=\{m \in M: g.m=m \}$ is a smooth manifold. I am unable to deal with the set of fixed points. It doesn't seem to be an open set. It would have been trivial if it was open. Any hints? Indian passport threading coming off My Indian passport threading is coming off i.e. the cover of the passport is coming off, there are only 2 threads remaining.   Will it be a problem at the airport when going abroad ? Also anyone who has experience of getting a replacement passport fairly quick ? Because I need to get a passport in like 15 days. Update query cancelled by user Why can't you use the keyword 'this' in a static method in .Net? I was looking at coreutils and found this as one of the files included as part of coreutils:  /usr/bin/[. What is [ and what does it do? It is an executable. I just don't know what it does or how to use it. $ file /usr/bin/[   /usr/bin/[: ELF 32-bit LSB executable, Intel 80386, version 1 (SYSV), dynamically linked (uses shared  libs), for GNU/Linux 2.6.15, stripped    When I try to run it, I think it is defaulting to the bash built in line expansion. Instead of actually running the file. $ &quot;/usr/bin/[&quot; /usr/bin/[: missing ‘]’ $ /usr/bin/\[ /usr/bin/[: missing ‘]’ Labeling all sides of the matrix with kbordermatrix I have a matrix with labels on the left and top. Now I want to have a label on all sides of the matrix. How can I change my code?  \documentclass[10 pt, a4paper, leqno, oneside] {report} \usepackage{kbordermatrix} \begin{document}  \renewcommand{\kbldelim}{(}% Left delimiter \renewcommand{\kbrdelim}{)}% Right delimiter \[   \tilde{\textbf{m}} = \kbordermatrix{     \mbox{} &amp; c_1 &amp; c_2 &amp; c_3 &amp; c_4 &amp; c_5 \\     % &amp; c_1 &amp; c_2 &amp; c_3 &amp; c_4 &amp; c_5 \\     r_1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 \\     r_2 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 \\     r_3 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 1 \\     r_4 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 1 \\     r_5 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1    } \]  \end{document} Problem with glossary I am copying the very first piece of code from  in order to create a glossary. While I get no problem in the compilation I do not get the "glossary part" that appears in the example. I literally copy paste.  Also, I want to create a document that only consists of a glossary of various terms. That is to put everything in alphabetical order (with the possibility to include references as well). How could I possibly do this? I tried some itemize options with alphabetical order but does not look that good. Convolution operator has norm 1 with respect to $L^2$ norm Let $C_c(\mathbb{R})$ be the following:  $$C_c = \{ f \in C(\mathbb{R}) \mid \exists \text{ } T &gt; 0 \text{ s.t. } f(t) = 0  \text{ for } |t| \geq T\}$$  Let $T_n \in L(C_c(\mathbb{R}))$ be a linear operator such that:  $$T_n u = \delta_n *u, \forall u \in C_c(\mathbb{R}),$$  where  $$ \delta_n(t)= \begin{cases}        n^2(t+1/n) &amp; -1/n \leq t \leq 0 \\       -n^2(t-1/n) &amp; 0 &lt; t \leq 1/n \\       0 &amp; \text{elsewhere}     \end{cases} $$  I have to prove that with respect to the 2-norm, the operator $T_n$ has $\|T_n\| = 1$ and I have succeeded in proving that $\|T_nu\|_2 \leq \|u\|_2, \forall u \in C_c(\mathbb{R}).$  Now, I remained  to prove that $\exists \text{ } u \in C_c(\mathbb{R}) \text{ s.t. } \|T_nu\|_2 = \|u\|_2$, but I really don't get how $u$ should be shaped in order to satisfy it. Is there a way that I can configure Visual Studio 2008 to understand CamelCase?  Specifically, I'd like to be able to get Ctrl + right or left cursor to take me to a subsection of a variable or type name. i.e., if my cursor was at the start of this line: LongNamedExampleClass longNamed = new LongNamedExampleClass();  and I hit Ctrl + right, I'd like to jump to &quot;Named&quot; in the class name, rather than the start of the variable name.  If I hit it again, I'd like to go to &quot;Example&quot;.  I hope that makes sense. I used this navigational feature in Eclipse quite a lot, and find it difficult to unlearn something so ingrained. Surface integral of a scalar over a unit cube. Evaluate the following integral $$\iint_S (x+y+z) \, dS$$  where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$  Honestly, I don't know what to do. All I know is that you have to evaluate the integral for each of the 6 surfaces of the cube. The normals for them I can do and the equations of the planes, but other than that I couldn't do much.  Can someone break it down for me, please?  PS: No divergence theorem yet for me. Is it correct to use that/which in the relative clause when it is referring to the place as an object pronoun? How do I tackle this combinatorics problem about married couples around a table? How can I know when my computer running Windows 7 was last restarted?   I prefer a solution that doesn't involve searching the event log, but something like wmic or maybe cmd commands. Hello [Comma?] John,