(i) $Y^\emptyset$ has exactly one element, namely $\emptyset$, whether $Y$ is empty or not, and (ii) if $X$ is not empty, then $\emptyset^X$ is empty.  How do you prove these statements to be true?
Let $ D = 2^\mathbb{N} $, i.e., D is the set of all sets of natural numbers.   What's the meaning of this definition? Intuitively, I would suggest that $ D = \{1,2,4,...\} $ but the explanation "set of all sets" leads me to the guess that this is wrong.
default permissions for /var/www I was trying to open a file and write to it with PHP at /var/wwwfolder but it wasn't working so I did  sudo chmod 777 /var/www   now I want to set the permissions for /var/www to the default. what are the default permissions for /var/www? Weight Painting Won't Get Rid of Weird Protrusions Confidence interval around binomial estimate of 0 or 1 What is the best technique to calculate a confidence interval of a binomial experiment, if your estimate is that $p=0$ (or similarly $p=1$) and sample size is relatively small, for example $n=25$? Compute Taylor error of $10^{-5}$ for $\cos(\pi/20)$ As, the above link shows, other countries have similar murder rates but they are not in the radar of international travelers.  For example, as seen in the list, Puerto Rico has similar statistics. But, personally, I never heard of Puerto Rico to be a dangerous place to travel.  Why is it considered so dangerous to travel in South Africa? Is that due to the incompetent law enforcement agencies? Undefined Control Sequence. 1.60 \@writefi nmake fatal error U1034: syntax error : separator missing For each positive even integer $n$, set   $$P_n = \displaystyle {{n} \choose {\frac{n}{2}}}\frac{1}{2^n}.$$  Show that $\displaystyle \lim_{n \to \infty} P_n$ exists and determine its value.   Here's what I have so far: each $P_n$ can be thought of as the probability of tossing a fair coin $n$ times, and obtaining exactly $\frac{n}{2}$ heads. My expectation is that $P_n \to 1$, since in any large trial, one would expect to record just about as many heads as tails. But I'm not sure how to mathematically justify this hunch.  Hints or solutions are greatly appreciated. What is a good focal length for street photography? With a long lens you don´t disturb people but you are left "out of the scene", while with a short focal lens you are "inside the scene" but you may disturb people. What is a better alternative for street photography? Upgrading from Yosemite to El Capitan, but not Sierra We have an older iMac (Model A1311, Mid 2011). It has been running Yosemite. We have run into some of the WIFI issues associated with Yosemite.   Is it possible to upgrade to El Capitan but not Sierra?   El Cap has been out longer. Its my wife's computer, its older, and I really don't want to be debugging Sierra until a few  more "dot" releases come out. What's the difference between 'allow' and 'allow for'? Is there a point of beating Vault of Glass raid on Hard after finishing on Normal? I want to append some data to a layer in my PostGIS DB using a shapefile, but when I use the Import layer/file tool in QGIS, it doesn't allow me to choose the append option.  Why is this and how do I get it to allow me to append? I understand we cannot state with arbitrary precision the position and momentum of a micro-particle as we superpose infinite waves to create a wave packet at the exact position of the particle and hence cannot say which wave is corresponding to its momentum. This is theoretically said. Also some people say that measurement of one disturbs the other and some say it's reality.  Firstly, the theoretical explanation: why should I consider that a wave out of many superposed will give us the momentum or position? We might use another single wave which is not sine in nature and might get the results. Or why won't a single sine wave suffice? This all considers the micro-particle as wave because of de Broglie.  Secondly, the practical thing seems legit.  Thirdly, the electron must have some kinetic energy and position. It is who we are measuring and can't get it exact so why blame the electron as in theoretical concept. How we quantify the uncertainty, it must depend on the measuring device and various other factors.  I might need to edit this post based on your answer so that I can explain more clearly my thinking.  Edit: From here is what gets the measurement problem:        Text: Imagine that you're blind and over time you've developed a technique for determining how far away an object is by throwing a medicine ball at it. If you throw your medicine ball at a nearby stool, the ball will return quickly, and you'll know that it's close. If you throw the ball at something across the street from you, it'll take longer to return, and you'll know that the object is far away.   The problem i­s that when you throw a ball -- especially a heavy one like a medicine ball -- at something like a stool, the ball will knock the stool across the room and may even have enough momentum to bounce back. You can say where the stool was, but not where it is now. What's more, you could calculate the velocity of the stool after you hit it with the ball, but you have no idea what its velocity was before you hit it.   This is the problem revealed by Heisenberg's Uncertainty Principle. To know the velocity of a quark we must measure it, and to measure it, we are forced to affect it. The same goes for observing an object's position. Uncertainty about an object's position and velocity makes it difficult for a physicist to determine much about the object.   Edit: After seeing this answer one of my doubts is very well cleared, the other answers are also written and helpful:        Text: There is a definine velocity and momentum, we just don't know it.   Nope. There is no definite velocity--this was the older interpretation. The particle has all (possible) velocities at once;it is in a wavefunction, a superposition of all of these states. This can actually be verified by stuff like the double-slit experiment with one photon--we cannot explain single-photon-fringes unless we accept the fact that the photon is in "both slits at once".So, it's not a knowledge limit. The particle really has no definite position/whatever. How does MySQL process ORDER BY and LIMIT in a query? Command "find -name" must be enclosed in quotes or it doesn't work. Why is that? Can Windows Virtual PC run a 64-bit OS? What do you call a person who keeps talking about art; studies art, criticizes art, and thinks he'd be amazing with it. But he doesn't practice it How to disable pylint 'Undefined variable' error for a specific variable in a file? Showing that localization is an exact functor