Let $X$ be a set, $P(X)$ the power set of $X$, and $2^X$ the set of all function from $X$ to $\{0, 1\}$.  Now define $J: P(X) \rightarrow 2^X$ by $J(A) = \chi_A$, where $\chi_A$ is the characteristic function.  Show that $J$ is a bijection from $P(X)$ to $2^X$.  I can demonstrate that $J$ is a bijection from $P(X)$ to $\{\chi_A | A \subset X\}$.  But how do I know for sure that $\{\chi_A | A \subset X\} = 2^X$?  Or, is there a function in $2^X$ that does not take the form of $\chi_A$?  I am not seeing how the answer is no, but I can't think of any counter examples either.
How can I formalize that the function $$f:\mathcal{P}(A)\to(A\to \{0,1\})$$ is bijective? It seems clear to me but I don't know how to write a proof. Thanks in advance.
What will happen to my reputation when the Community Wiki status of a post is reversed? Sample size for logistic regression? I want to make a logistic model from my survey data. It is a small survey of four residential colonies in which only 154 respondents were interviewed. My dependent variable is "satisfactory transition to work". I found that, of the 154 respondents, 73 said that they have satisfactorily transitioned to work, while the rest did not. So the dependent variable is binary in nature and I decided to use logistic regression. I have seven independent variables (three continuous and four nominal). One guideline suggest that there should be 10 cases for each predictor / independent variable (Agresti, 2007). Based on this guideline I feel that it is OK to run logistic regression.    Am I right? If not please let me know how to decide the number of independent variables? Is there a difference between Bayesian and Classical sufficiency? The title pretty much says it all. 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How do I access a specific element in a vector of vectors clojure What happens when a creature enchanted with Soul Link would deal lethal damage to a player? Player A has 3 life left. Player B attacks with a  that has been enchanted by Player A's . Does player B win, or does nothing happen?  On the one hand, "whenever" makes it sound like the trigger (dealing damage) and the effect (healing damage) happen simultaneously.  On the other hand, combat damage doesn't go on the stack, while most other things do. So, possibly the combat damage is dealt, state-based actions are checked, and now it's too late for the Soul Link effect to heal player A.  If anyone knows which way is correct, my thanks! When I read about , I see that they should appear as solution to prevent  and  but when I actually try to write a code which would use Lightning promises it still doesn't look like synchronous code, it still resembles both pyramid of doom and callback hell.  main: function(component, helper) {     helper.promise1(component, helper).then(         $A.getCallback(function(record) {             return helper.promise2(component, helper, record).then(                 $A.getCallback(function(){                     console.log("Success");                     helper.showToastMessage(component, "Great Success!!!!")                 }),                  $A.getCallback(function(errorMessage){                     console.log("Rejected by second promise2");                     helper.showToastMessage(component, "Failure " + errorMessage);                 })              )         }),          $A.getCallback(function(errors){             console.log("Rejected by first promise1");             let message = errors &amp;&amp; Array.isArray(errors) &amp;&amp; errors.length &gt; 0 ? errors[0].message :'Unknown error';             helper.showToastMessage(component, "Failure " + message);         })      ) }   Is that possible to refactor this code to look more like synchronous code and without callback hell or pyramid of doom? 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So the computer here is configured in English and i have to switch it to French. Here is the content of my ~./dt :  C                     errorlog              sessionlogs Desktop               errorlog.old          sessions Trash                 errorlog.older        simulateur.dtwmrc appconfig             help                  startlog appmanager            icons                 startlog.old asr.dtwmrc            outils.dtwmrc         startlog.older bin                   palettes              tmp bookmarks             sdtfonts              types dtwmrc                sdtpcaction.conf      wsmenu en_US.ISO8859-15      sdtpcaction.conf.bak   Thanks Is it acceptable to take photos of slides at a conferences for private use? Is it accepted/appropriate to take pictures of presentation slides at conferences for private use?   I am not talking about putting those pictures out in the public afterwards, I know this is illegal without consent of the presenter.    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For example man gcc brings:  NAME        gcc - GNU project C and C++ compiler  SYNOPSIS        gcc [-c|-S|-E] [-std=standard]            [-g] [-pg] [-Olevel]            [-Wwarn...] [-Wpedantic]            [-Idir...] [-Ldir...]            [-Dmacro[=defn]...] [-Umacro]            [-foption...] [-mmachine-option...]            [-o outfile] [@file] infile...         Only the most useful options are listed here; see below for the        remainder.  g++ accepts mostly the same options as gcc. .... many text   But what if I need only some part of it? What is the best way to jump to the section where -Idir is described, for example? Let $g_n= {2^2}^n +1 $. Prove $g_0 · g_1 · · · g_{n−1} = g_{n} − 2$ Let $g_n= {2^2}^n +1 $. Prove $g_0 · g_1 · · · g_{n−1} = g_{n} − 2$.    I'm not sure how to start this proof, if it should be done algebraically or if I should try to use a proof by induction. Greyed-Out vs Invisible Can there be a notification for those people that participated in the question in the past (either who asked or answered it) once a bounty is started on it? Why does this PNG image on twitter look different when clicked? This morning on twitter there was a tweet with the text "click on this picture".  When picture is clicked, it's opened in wider viewer, but looks completely different.  I tried to analyze this picture in ACDSee, and I also tried uploading on Facebook, but it's happening only when embedded on twitter. I don't have much knowledge about graphics so please explain this to me. Why does this PNG image change only when clicked on Twitter?  Screenshots (left = before clicking; right = after clicking):     Here's the original . I just installed Ubuntu 13.10, and I can't configure my wi-fi Epson TX430W printer.  Using the Settings-&gt;Printers interface, I can detect the printer. When I click add, I briefly see a "Searching for drivers" dialog. 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