I need to find a bijection between the power set $P(\mathbb{N})$ and the set of functions $J=\{f|f:\mathbb{N}\to \{0,1\}\}$   I considered the map $g:P\to J$ as $g(A)=f $ where $f$  is a function who is mapping $A$ to {0} and $\mathbb{N}\setminus A$ to ${1}$, then i think this will be a required bijection!
I recently got the book "selected problems in real analysis", and I'm stuck solving the very first problem  $(u_n)$ is a binary sequence iff it only contains $0$ and $1$ in the sequence  Let $A$ be the set of all binary sequences  I have to prove that $A$ and ${2}^{\mathbb{N}}$ have the same cardinality, that is to say there exists a 1-1 function from one set to another  I've thought about maybe considering integers as base-2 numbers  Thanks for your help
I've noticed that, if I add \n to a pattern for substituting using sed, it does not match. Example:  $ cat &gt; alpha.txt This is a test Please do not be alarmed  $ sed -i'.original' 's/a test\nPlease do not/not a test\nBe/' alpha.txt  $ diff alpha.txt{,.original}  $ # No differences printed out   How can I get this to work? Fight 707 arrives at/in Washington D.C. in ten minutes Fight 707 arrives at Washington D.C. in ten minutes. Fight 707 arrives in Washington D.C. in ten minutes.     Do we have to use 'at' or 'in' in this case? Washington D.C. is a big city, so we have to use 'in' normally. However, because the plane leaves the city soon, the city can be a particular point on a map. So, can we use 'at' in this case? What are the areas where graph theory can be applied?  Cause I wonder what applications this have on the real world.  Does it solve certain problems and stuff?  Areas such as communication networks and coding in IT, genomics, computation, and scheduling in operations research Upon answering the telephone, the person calling asks if Joan is available. If Joan is the person who answered the phone, should she say "This is her" or "This is she"? I'm a fan of extension methods in C#, but haven't had any success adding an extension method to a static class, such as Console.  For example, if I want to add an extension to Console, called 'WriteBlueLine', so that I can go:  Console.WriteBlueLine("This text is blue");   I tried this by adding a local, public static method, with Console as a 'this' parameter... but no dice!      public static class Helpers {     public static void WriteBlueLine(this Console c, string text)     {         Console.ForegroundColor = ConsoleColor.Blue;         Console.WriteLine(text);         Console.ResetColor();     } }   This didn't add a 'WriteBlueLine' method to Console... am I doing it wrong? Or asking for the impossible? How to find the 'sizeof' (a pointer pointing to an array)? First off, here is some code:  int main()  {     int days[] = {1,2,3,4,5};     int *ptr = days;     printf("%u\n", sizeof(days));     printf("%u\n", sizeof(ptr));      return 0; }   Is there a way to find out the size of the array that ptr is pointing to (instead of just giving its size, which is four bytes on a 32-bit system)? Integrate $\int \frac{25 - x^2}{x^4} \operatorname d\!x$ I think we use trig substitution for this. But I'm not sure where to even begin with this   $$\int \frac{25 - x^2}{x^4} \operatorname d\!x$$ I have the following code:  if ($_POST['submit'] == "Next") {     foreach($_POST['info'] as $key =&gt; $value) {         echo $value;     } }   How do I get the foreach function to start from the 2nd key in the array? Get different and common items in two arrays with LINQ Do I need a transit visa for waiting 23 hours in Dubai airport? I am an Indian citizen living in Ukraine. I will be traveling to India but I need to wait in Dubai airport 23 hours for my next flight to India. I will not be leaving Dubai airport.  So in this case do I need a transit visa? How to allow non-superusers to mount any filesystem? Is it possible to allow some particular users (e.g. members of a group) to mount any filesystem without superuser privileges on Linux?  Another question might have been "in what ways a user can harm a system by mounting filesystems?" Atheros AR9485 wifi disconnects randomly display only 3 foreach result per row Which characters played the parts of the lineup discussed by Rusty and Daniel? In planning the heist, Rusty names a number of roles that are going to be needed to pull off the job.      Off the top of my head, I'd say you're looking at a Boesky, a Jim Brown, a Miss Daisy, two Jethros, and a Leon Spinks. Not to mention the biggest Ella Fitzgerald ever.   Which characters in the film lined up with these named roles? Index entries duplicated for captions Chapter number starts at 2 not 1 How can I start my chapter from "Chapter 2" instead of "Chapter 1".  I am preparing LaTeX document for each chapter. when I put \chapter{Introduction}, it produces "Chapter 1" Introduction. However I want it to be "chapter 2" Introduction instead. Why is time evolution unitary? Is the reason why the time evolution operator is unitary based on purely physical arguments, i.e. that the physical processes that an isolated system undergoes shouldn't depend on any particular instant in time (homogeneity of time); thus two experimenters who conduct the same experiment starting from the same initial state, but at different times, should have the same probability amplitude for that state?! Or is there some mathematical argument as well?  Also, is the reason why the time evolution operator is linear implied by the superposition principle (as an arbitrary state can be expressed as a linear combination of basis states the operator should act linearly as otherwise the state as a whole would evolve differently to the superposition of states that it was initially represented by)?! Hibernating Ubuntu on EC2 instance? I've got an application I'd like to move to EC2. It will likely spend more than half the day completely dormant, so I'm trying to come up with a good solution for starting and stopping it as needed. It takes a few minutes to start up from nothing, so it would be nice if i could hibernate the OS for quicker resumes. I've seen a couple forum discussions on the topic of hibernation within EC2, but never anything conclusive. Has anyone found a working solution to this, or at least some resources that could help me out? How do I build a house for my NPCs? I read on the wiki that people will move in under the following conditions:   You do something to please them An empty house is available with a light source, table, and chair (also with walls).   I have a house which is about 7x8, then a roof space above that; about half the area. I could put furniture up there too, but will anyone move in? I am writing a client-side Swing application (graphical font designer) on Java 5. Recently, I am running into java.lang.OutOfMemoryError: Java heap space error because I am not being conservative on memory usage. The user can open unlimited number of files, and the program keeps the opened objects in the memory. After a quick research I found  and others saying on Windows machine the JVM defaults max heap size as 64MB.  Given this situation, how should I deal with this constraint?  I could increase the max heap size using command line option to java, but that would require figuring out available RAM and writing some launching program or script. Besides, increasing to some finite max does not ultimately get rid of the issue.  I could rewrite some of my code to persist objects to file system frequently (using database is the same thing) to free up the memory. It could work, but it's probably a lot work too.  If you could point me to details of above ideas or some alternatives like automatic virtual memory, extending heap size dynamically, that will be great.