I recently  a fact that surprised me:  The power set of the natural numbers is equinumerous with $\mathbb{R}$.  In other words, $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}|$.  I don't intuitively see why this is true. Can anyone explain why?  Edit: duplicate has been found, please vote to close.
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?
Why aren't we sterilizing N95 masks? Can someone use DC generator along with electric motor to make electricity along with mechanical energy? I know this might sound stupid and maybe stupid totally, but I need some clarification....  I came through electric motor principle.  and then I came across the following link on DC generators    Now my question is, can't we combine them both to provide DC current with little DC used in motor side? By "combine," I mean using a DC generator coil only at the other side of the coil of the motor in the motor itself. The condition is that the 2 coils are connected via some apparatus that causes the 2nd coil to move along with the 1st coil.   In this way we can use both the mechanical energy produced by the 1st coil and the electrical energy produced by the 2nd coil, right?   This is a great concept if it works, right? Please throw some light on this. I have two given names and one surname. Is it a problem if I omit the surname from my ticket? I have two names as my given name in the passport and I typically just use this as my first and last name. However, my passport has a surname too which I almost never use. Due to this, I mistakenly provided my given name as first and last name.   As per the passport,  Surname : AAAA Given name(s) : BBBB CCCC   In the ticket, First name : BBBB Last name : CCCC  The credit card I've used to make this booking has my name as BBBB CCCC.  Will this create a problem at the travel time?  Update : the airline says that they can't update the name in the ticket. They are saying that since they have made a remark of the passport details boarding will not be a problem but I might face problems at immigration. Do they check name details in itenerary to exactly match passport at the time of immigration? Is that true in your experience?  Update : Travelled without any problem, neither at boarding gate, immigration nor after arrival at border police control. I'm in trouble with this homework.  Find the sum of the series $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$, where the terms are the inverse of the positive integers whose only prime factors are 2 and 3.  Hint: write the series as a product of two geometric series.  Ok, I found some patterns in these numbers, but I can't find the two series. Blank lines in an equation and some other similar function Why blank lines between equation makes error? there are a lot of commands in latex and almost for all of them these blank lines are not important,  but why in this case is important and is there any similar function? Renting a car in Ireland & Great Britain: 1 rental or 2? My wife, two kids and I will be traveling in Ireland and the UK Nov. 1st through 21st. We're flying into and out of Dublin. Our plan is to do a clockwise tour: Fly into Dublin -> Belfast -> Scotland -> England/London -> Back to Dublin to fly home.  We've been planning to rent a car when we land in Dublin (doing just things outside Dublin at the start of the trip) and have a car all the way until London, then turn in the car and take a train/ferry to get back to Dublin before flying home.  However, reading through the car rental section of Rick Steve's 2012 Ireland book is causing us to question if this is the right plan.  He says (p. 480):     If your trip covers both Ireland and Great Britain, you're better off   with two separate car rentals, rather than paying for your car to ride   the ferry between the two islands. On an all-Ireland trip, you can   drive your rental car from the Republic of Ireland into Northern   Ireland, but be aware of drop-off charges ($75-150) if you return it   in the North.   So it seems like he recommends against doing one car rental for both islands, but he also recommends against renting in the Republic of Ireland and returning it in Northern Ireland (our other option for our current itinerary).  So we're not sure which is best here.  A few questions about this:   If we do one car rental all the way from Dublin to London, will there be a similar $75-$150 fee? Is there an easy place to return a car on the Northern Ireland side of one of the Northern Ireland -> Scotland ferries, and an easy place on the Scotland side to rent a car?  We'd like to minimize the number of buses/whatever we need to take between car rentals since we'll be lugging around car seats for both of our children (on top of carrying them and our luggage). Will one way be significantly cheaper than the other?  Besides the basic cost of taking the car on the ferry and the drop-off charges, our credit card will cover insurance in Scotland/England but not in Ireland, so we could potentially get a better rental rate by renting a second vehicle once we're on Great Britain. Although, I know you often get a better rate when keeping a car rental for longer, so that may negate any benefit from the credit card covering the insurance. I have this problem: I'm used to compile and run .c files with CodeBlocks. If I do that from my Desktop (ext4 partition) it works without problems, but doing this from my secondary hard drive (NTFS partition) makes CodeBlocks to say permission denied while running a .c file.  I have read it's because of the partition type is it correct? But I don't know how to solve it. I tried to edit fstab (sudo gedit /etc/fstab) adding this line:   /dev/sda /media/Dati ntfs user,exec 0 1   but after rebooting Ubuntu I had a warning on boot about a problem mounting a drive, that's because I set KDE to auto mount my Dati drive on boot.   So I should auto mount my NTFS drive with execute permission? How can I do that? I have 3 old computers lying around right now running a P2 at 800 MHz(?), Intel Mobile 1.6 GHz, AMD Athlon XP 2000+ at 1.66 GHz, and (might not use this) P4 at 2.7 GHz, all with 512 MB Ram, and am considering clustering them together for fun/knowledge. They would be running an undecided version of linux, preferably ubuntu based.   The issue is what I want to use it for: general computing and occasional video encoding. By general computing I mean day to day tasks. However I'm not sure if every program started by a single X session is going to exist on the same machine, defeating the purpose of such a system. Will programs be split up or exist on one machine?  Second, assuming this is running 100baseT ethernet (not sure if the PCI slot itself could handle Gigabit), would the speed of having a program exist over the network be an issue? It seems that the constant asking of various things in RAM would be quite slow.  And before you say "buy another computer!", that's not the point of this question. I'm asking would it be usable, not necessarily practical. And yes I know, this is going to be extreamly power consuming. Take as an example the well-known . In the original paper, the regression model was used to model the amount people were willing to pay for improved air quality.  But the 'people' whose behaviour was being modelled was never specified. Presumably, it was not limited to 'residents of Greater Boston'. Only implausibly would it be 'humans anywhere on the planet'. Nor, 'all US residents'.   So what inference are we invited by the authors to draw about the scope they intend for their theory? Perhaps 'all residents of large US metro areas'. But - what about other First World metro areas? The first two sections of the Boston paper scrupulously avoid any reference which might suggest that they had a particular 'population' to which their 'sample' was to apply.  There is also the dimension of time - the data came from the 70s, whereas the theory is has no time limitations spelt out.  Plus - going back to first principles - how could the Boston data be treated as a sample of a population including, say, the Chicago metro area when no data from the Chicago area was considered?   Besides, lots of the covariates in the Boston regression weren't samples at all, but totals - so, for example, the proportion of blacks was supplied from census stats.  My question: how can we conceive the Boston data as supporting the geographically unlimited propositions in the paper? Is there a mathematical way of showing equivalence or approximation of the Boston data analysis to other cities? Or is it just assumed as a matter of common sense - such that the paper's authors would be amazed to think anyone could have any doubts on the matter? I'm currently doing a couple of exercises on logarithmic expressions, and I was a bit confused when presented with the following: $5^{\log_5 17}$.  The answer is $17$, which is the argument of the logarithm in the exponent, but I don't understand the reason why.  A previous question in the exercise was $12^{\log_{12} 144}$, which is pretty straight forward since $\log_{12} 144=2$ and $12^2=144$ but since $\log_5 17$ is an irrational number, I couldn't calculate it the same way.  So my question is, why is $x^{\log_x n}=n$?  [Update]  I've now managed to reason this out and realized how simple it is.  Basically, if $y=\log_x n$, then $y$ is the number that  $x$ must be raised to to become $n$; therefore, if I raise $x$ to that power $y$, I will naturally get $n$.  I guess my problem was that I tried to work out the question mechanically instead of intelligently and that's why I failed, because once I realized that I can't calculate $\log_5 17$, I was stumped.  Now that I reasoned it out, as explained in the previous paragraph, I understand it. Why does Math.Floor(Double) return a value of type Double? Why don't planets have Circular orbits? How do I start Nautilus as root? I've got a problem with Nautilus in Ubuntu 12.04 LTS.  When I run the command gksu nautilus /, the password ask box appears and I type my super-user password, then the password box disappears and Nautilus is not opened.  When I try to open Nautilus as a normal user in with the command nautilus, the folder is opened.  When I try in terminal   $ sudo nautilus   I got the following error  $ sudo nautilus /  ** (nautilus:8523): WARNING **: Command line `dbus-launch --autolaunch=2c8ce9b7da2257c2609b749700000007 --binary-syntax --close-stderr' exited with non-zero exit status 1: Autolaunch error: X11 initialization failed.\n Could not parse arguments: Cannot open display:   How can I fix this error? Is the cube root of a prime number rational? The question is: if $P$ is prime, is $P^{1/3}$ rational?   I have been able to prove that if $P$ is prime then the square root of $P$ isn't rational (by contradiction) how would I go about the cube root? I'm using Bash on both client and server. When running a command over SSH:   ssh &lt;host&gt; 'declare' gives a list of shell variables. ssh &lt;host&gt; 'mount' gives a list of mountpoints.   However, declare is a Bash builtin, while mount is an external command. How does SSH know which to run if there is a shell builtin and an external command with the same name on the server? Is it possible to have an object ignore a certain light source but still be affected by others? I want to create an effect on a character where their skin and clothing are effected by two different light sources. The clothing should only reflect a bright bounce light, but have the skin should not be affected by this light source.    In the example image I have two objects a cube, a sphere and three emission  lights . I want the two white lights to hit both objects, and I need the purple light to hit and reflect off of the sphere, but ignore the cube entirely (no reflections).  Someone suggested to composite the layers together, it works... but this is very impractical and cumbersome because it forces the background to be transparent and it will be a pain to try and layer together a more complex scene, especially if its animated which is what I want this effect for.  I know you can have objects ignore shadows and all kinds of other things so im hoping someone knows how to get an object to ignore light with a few settings. Can't install php5 on Ubuntu 16.04 so I've recently installed my first Ubuntu, to be precise, a daily build of 16.04, on my new laptop since the hardware was to new for 15.10. Now I'm in the process of setting it up and I want to install php5 and some extensions.  When I execute sudo apt-get install libapache2-mod-php5 I get the error message that there is no installation package available for php5 and libapache2.  The error message is in german, so I doubt that this would help, but anyways here it is. :-) E: For package »libapache2-mod-php5« existiert kein Installationskandidat.  Do I need to add a apt-get repository? Or what am I doing wrong? Implicit newline at the end of each paragraph? In certain styles of text I define \parindent 0pt in the preamble and write the text in this manner:  paragraph 1\\  paragraph 2\\   and so forth. I wonder if there is anything I can include in the preamble for instance to modify how the end of paragraphs are interpreted so I don't have to include the \\ at the end of each? I did the following proof and I was wondering if its valid. It feels wrong because I didn't actually test the case when purportedly $n$ is not prime, but please feel free to correct me.  Assume there exists $n$ such that $\phi(n) = 14$. Assume $n$ is prime. Then $\phi(n) = n-1$. Then here, n-1 = 14, so n = 15.  We know since Euler's totient function is multiplicative that $\phi(xy)$ = $\phi(x)\phi(y)$, so $\phi(15) = \phi(3)\phi(5)$, but alas $\phi(3)\phi(5) = 2\cdot4 = 8 \ne 14$. If $n$ is not prime a similar argument follows since we know then that $n$ must be composed of prime numbers by the prime factorization theorem. How to convert decimal to hexadecimal in JavaScript How do you convert decimal values to their hexadecimal equivalent in JavaScript?