I am pretty stuck on this problem.From which direction should I start?
I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it?
Method to create a live USB disk (with persistence) which actually works? I've tried so many combinations - all I want is a fast Ubuntu Live USB disk with persistence that works. What I've tried so far:   Using unetbootin on my Mac, choosing some "Space used to preserve files across reboots (Ubuntu only)".   Result: Doesn't boot at all.  Using unetbootin on my Mac, choosing no such space.  Result: Boots fine and very fast (doesn't even show the annoying dialog "Try ubuntu/install ubuntu" at startup!), but has no persistence. (So I have to choose my keyboard layout, install programs I need, etc. every time)  Using Startup Disk Creator from my Ubuntu 11.04 installation.  Result: I cannot choose "reserved extra space" at all. It's greyed out. I know I know, there was bug , and many people say that the workaround from  works for them. But it doesn't work with Ubuntu 11.04 anymore (see also ).  Using Startup Disk Creator from my Kubuntu 10.10 installation, gnome version (usb-creator-gtk).   Result: I can select the "reserved extra space" option, but installation doesn't work (fails when installing the bootloader).  Using Startup Disk Creator from my Kubuntu 10.10 installation, kde version (usb-creator-kde).   Result: Actually works: Reserved space can be selected, installation works. But after booting, I get the screen "Try ubuntu/install ubuntu". Clicking on "Try ubuntu" takes me to the desktop - but only after waiting 5 minutes (every time!). It also doesn't seem to keep my keyboard settings.    I know, that instead of creating a startup disk, I could install a full ubuntu system on the USB stick. But as far as I know, this isn't very good, because /tmp and similar directories will then also be on the USB disk, which is a) slow, and b) not very good for the USB disk due to frequent overwriting.  I'm frustrated. So has anybody managed to create a working Ubuntu 11.04 Live USB disk with persistence? Who paid for Harry's Nimbus 2000? The Nimbus 2000 was the fastest broom at the time of the Harry's first year at Hogwarts.  Professor McGonagall wanted to win the Quidditch Cup so she spoke with Dumbledore and got an exception from the rule that no first-years could have a broom for Harry.  My question is who paid for the broom?  It couldn't have been a cheap broom.  They wouldn't have taken gold out of Harry's Gringotts account without permission.  It wouldn't be right for the school to pay for it and I doubt McGonagall paid for it out of her own pocket.  So who paid for the broom? Yesterday, I was browsing the mobile Stack Overflow site. I noticed one thing: the textareas where you write your comments/answers are very buggy. I'm using a Samsung Galaxy S2, with the standard Android browser (Android version 4.1.2). There are a number of things that aren't working when typing your comment/answer.   Automatic spacing with Swype is disabled (however this might be because of Swype, not the site). Text navigation buttons (step left, step right, select text, copy text etc.) doesn't work with the on-screen keyboard, and selecting text by holding your finger over text is extremely difficult. When you paste something it always seems to be pasted at the top of your text, no matter where you try to paste it. Replying to another comment ("@Example User") does not show suggested people to reply to, like the full site does.   Is there anything I can do to solve these issues/bugs, or is it in the hands of Stack Overflow itself, and if so, should it be looked into?  Edit: I was using the mobile version of the site when noticing these bugs. What is a StackOverflowError? What is a StackOverflowError, what causes it, and how should I deal with them? Where to place log4j.xml What is the agreed upon definition of a "positive definite matrix"? In a topological space, why the intersection only has to be finite? I was wondering if there is a reason why in the definition of a topologic space, the reunion of any collection of open sets is also an open set but the intersection has to be finite to be a set.  Is there because of the definition of an open set itself ? I have always seen it as a definition, but now I want to know why.  Thanks for your help. If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative. 4 fast cores or 6 slower ones? Canonical Post #2: What is the perfect, and how should I use it? Where is Jon going? Export each image from a collection in Google Earth Engine Polynomial ring $R[X] $ has zero Jacobson radical, for $R$ a domain Is a transit visa required to pass through Bangkok Airport? Verifying that the ideal $(x^3-y^2)$ is prime How to prove that the ideal $I=(x^3-y^2)$ in $k[x,y]$ is prime?    I have constructed a map from $k[x,y]$ to $k[t]$, which maps $x$ to $t^2$, and $y$ to $t^3$. Then, I want to show that the kernel is exactly $I$. But the difficulty is that how to prove the kernel is contained in $I$.  Could you help me? Thanks a lot! Why is current a scalar quantity? What's the difference between the 'ref' and 'out' keywords? I'm creating a function where I need to pass an object so that it can be modified by the function. What is the difference between:  public void myFunction(ref MyClass someClass)   and  public void myFunction(out MyClass someClass)   Which should I use and why? A matrix and its transpose have the same set of eigenvalues/other version: $A$ and $A^T$ have the same spectrum I'm wondering something. Whenever i right click on minecraft and click open with another application, i try to find Java JDK. I have installed it for minecraft itself, but i just cant find where java is on the list of other applications. Any help? Will reformatting my hard drive fix bad sectors on it?