Prove that $\mathcal{P}(A) \sim \{0,1\}^A$   We might can find a bijection function, however I am trying to prove it in other way. I want to F, G 1-1 functions and prove that those functions are inverse each other, once I did it, we know from a another proved claim that function is inverse if and only if function is bijection.  I define $F:\mathcal{P}(A) \to {0,1}^A$ by the following rule, for all $X\in \mathcal{P}(A)$, we have function $F(X): A \to \{0,1\}$ and for all $a\in A$ we define function $F(X)(a)=0$ its easy to prove that this function is 1-1.  Now I have to define $G:\{0,1\}^A \to \mathcal{P}(A)$ by the rule: for all $f\in \{0,1\}$, $G(f)=?$  My question is how can we define $G(f)$ to be 1-1 function? Moreover, any suggestions on this strategy will be highly appreciated.
I got this question in homework:     Let $\{0,1\}^A$ the set of all functions from A (not necessarily a finite set)       to $\{0,1\}$. Find a correspondence (function) between $\{0,1\}^A$ and       $\mathcal P(A)$ (The power set of $A$).       Prove that this correspondence is one-to-one and onto.   I don't know where to start, so I need a hint. What does it mean to find a correspondence? I'm not really supposed to define a function, right?  I guess once I have the correspondence defined somehow, the proof will be easier.  Any ideas? Thanks!
In Windows I used command-line clipboard copy-and-paste utilities...      pclip.exe and gclip.exe     These were UnixUtils ports for Windows (but they only handled plain text). There were a couple of other native Windows utilities which could write/extract any format.    I've looked for something similar in Synaptic Package Manager, but I can't find anything.    Is there something there, that I've missed? ... or maybe this is available in Bash scripting?    The type of utility I'd like will be able to read/write via std-in/std-out or file-in/file-out, and handle Unicode, Rich Text Format, picture, etc. clipboard formats...    NB: I'm not after a clipboard manager. How to Implement Page Object and Page Factory Pattern in Selenium Webdriver? I am a beginner in learning selenium Webdriver. I am not able to understand how page object and factory models works and how it can be beneficial?    Kindly help me explain it with detailed explanation. Increase iPhone call history How can see call history in iPhone one month back, i also tried with tried party software but that is not perfect , is the any solution from apple? How can I check the actual size used in an NTFS directory with many hardlinks? On a Win7 NTFS volume, I'm using cwrsync which supports --link-dest correctly to create "snapshot" type backups. So I have:  z:\backups\2010-11-28\cygdrive\c\Users\... z:\backups\2010-12-02\cygdrive\c\Users\...   The content of 2010-12-02 is mostly hardlinks back to files in the 2010-11-28 directory, but there are a few new or changed files only in 2010-12-02. On linux, the 'du' utility will tell me the actual size taken by each incremental snapshot. On Windows, explorer and du under cygwin are both fooled by hardlinks and shows 2010-12-02 taking up a little more space than 2010-11-28.   Is there a Windows utility that will show the correct space acutally used? Tweak showlabels/showkeys - wrap the label Can showlabels wrap the label in the margin? A very similar question is , but the solution of @DavidCarlisle does not solve the more general case. Consider the following MWE:  \documentclass{article} \begin{document} \begin{figure}[ht]   \centering   \rule{1cm}{1cm}   \caption{default}   \label{fig:a very very very long and detailed label} \end{figure} \end{document}   When adding the package showkeys the output is:    Similar problem occurs with the showlabels package (whether the hack of @DavidCarlisle is used or not):   So, is there a way to fix it? If a charge travels in a circle it must accelerate, thereby producing EM.  However, a wire in a circular loop is analogous to many charges moving in a circle. So, why don't circular currents produce EM? (I have not found any evidence that circular currents can produce EM).  A similar question is: . However, my question asks about circular wires and direct current. Thanks. I am trying to evaluate the following sum. Here $x\in\mathbb{R^{+}}$. $$f(x)=\sum_{n=1}^{\infty}\frac{(n-1)!}{\prod_{r=1}^{n}(x+r)}$$ At present I do not know how to proceed. Any hints are appreciated. Thanks. What 70s (60s?) TV show / TV movie had a generation ship whose inhabitants did not know they were on a spaceship? I have the vaguest memory of a TV show (maybe a BBC import?) in which the protagonists were moving through the sections of a massive generation ship whose inhabitants had forgotten that they were on a spaceship... I know there was an RPG with the same plot (which, after all, is a pretty great blank slate) but I definitely feel I saw this on a static-filled 13" B&amp;W TV back in the day... If $A,B,C$ are the angle of a triangle, then show that $\sin A+\sin B-\cos C\le \dfrac32$ I tried substituting $C=180^\circ-(A+B)$ and got stuck. I also tried using the formula $\sin A+\sin B=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}$ without any progress. Please help! Why are we limited to six questions per day? How do I disable this:    after taking screenshot? Does everyone see the same Pokémon? I was wondering if everyone sees the same Pokémon on the map. If I catch all the Pokémon around me will others be unable to get any? A little look through an etymology dictionary shows that the root is Latin gigas with adjective form gigant.  So in its derivation to English, why did the second "g" get retained in gigantic but was dropped from giant?  I have an inkling that the word giant may have travelled through French on its way to English, where it may have been pronounced "gee-yant" or "jee-yant" with a change to the second "g" that was then dropped when taken into English.  However, the "g" is retained as a hard "g" in gigantic. Let $\{a_n\}$ be an unbounded, strictly increasing sequence of positive real numbers and let $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$. Which of the following statements is/are correct? (CSIR NET December 2014)         For all $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;1-\bf{\frac{a_m}{a_n}}$   There exist $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;\frac{1}{2}$      My attempt:  Analysing question in light of $a_k=k\implies x_k=\frac{k+1-k}{k+1}=\frac1{k+1}$, I feel both are correct, I doubt if  I can generalise the result in light of this one example.   Any way to prove that this true for general case?    Note: Please do not confuse  with my question. The first two options are different, I've marked the difference in bold. How can I correct this code to get an answer? Is there BitTorrent software that runs in a terminal? I want to deploy it on my VPS server to download resources quickly.  This could save me a lot of time since some resources are in a foreign country, induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$ I encountered the following induction proof on a practice exam for calculus:  $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$  I have to prove this statement with induction.  Can anyone please help me with this proof? Favorite questions in the Android app I haven't been able to find my favorite questions in the official Stack Exchange Android app.  Looking at my accounts page gives nothing on them. Cascading Dropdown I am new to SharePoint 2013. What I need to figure out is based on the following situation. I have several school districts with multiple schools within the school district. I would like to create a cascading dropdown that 1st lists the districts and then will automatically populate a list of schools within a particular district when the district is selected. Can someone tell me if this is possible and if so what steps do I take to achieve it?    Thank you! I got multiple entry schengen visa issued by Greece. Can I directly travel from India - Qatar - Romania or through schengen state