Prove that $A \cup B = A \cap B$ if and only if $A=B$. Prove that $A \cup B = A \cap B$ if and only if $A=B$.  My method was:  We must prove two implications, so we will proceed by proving the first implication. We will do this by proving the contrapositive: If $A \neq B$, then $A \cup B \neq A\cap B$. So, we assume that $A \neq B$. Let $k \in A$ and $k \notin B$. Then $k$ is in $A \cup B$ and not $A \cap B$, and $A \cup B \neq A \cap B$. Therefore this is true by definition.   Since $A = B$, for every $k \in A$, $k$ is also in $B$. Hence $A \cup B = A=B$ and $A \cap B = A=B$. Therefore, $A \cup B = A \cap B$.  I know I'm close in the first part of the proof, but I am not sure about the second part when I try to prove the other direction.
If the union of two sets is contained in the intersection, then one is contained in the other ($[A\cup B\subseteq A\cap B]\implies A \subseteq B$) Let $A$ and $B$ be sets. Let $A\cup B\subseteq A\cap B.$ Prove that $A\subseteq  B.$   My understanding of this question is that all the elements of set $A$ that intersects with set $B$ exists in the union of sets $A$ and $B$ and because in order for the elements to intersect, the elements must exist in both sets $A$ and $B$. Therefore set $A$ is a subset of $B$.  My question is how do I write the proof down so that it is an acceptable answer? Also does this mean set $B$ is a subset of $A$?
How can I prevent Chrome from offsetting newly opened Window? When I open a new Window in Chrome on Windows 7 (by pressing CTRL+N), the new window does not exactly overlap the existing window.   Instead it is offset to the right slightly.   So, immediately after opening a new window I have to manually move it to the left so that it uses the full screen real estate.   Is there a way to prevent this offset? I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.  $$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$  I want to check whether that would be: 0,3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and so on.  Meaning that it would include 0, 3, 5, 8. Then from 9 and on, every Natural Number. But how would I explain it as a set? or prove that these are the numbers in the set? Set Text Color for textView Android How to get hardware specs of Android device? How can I find information on the hardware of my Android device like processor architecture, RAM, etc? Why can't I find the Karin Kuruma? Differences between no child vs no children a) They have no child. b) They have no children.  Are both a) and b) correct? Though b) seems more common. I'm trying to merge two directory trees that have many common elements but also each have elements that are only present in one of the two trees. The main issue I am having is that when mv encounters two sub-directories with the same relative paths it either keeps the source (with -f) or the destination (with -n) but I can't make it take the union of both sub-directories. I could of course use rsync with --remove-source-files but this will actually copy the data and then delete the old files as opposed to a true move. The two directory trees contain several hundred GB of data and are both on same partition so I would love to do a true move if possible for the sake of time.   Right now I have find * -type file -exec mv -n {} /destination/{} but this only moves  files from the source to the destination when the target directory already exists. I can preface this command with something like mv -n * /destination/ but this only brings over the top level directories. Is there any way to do this in a single line? I could  always write a script to check if the directory exists before copying the file but this seems like such a basic task, it seems like there should be an easier way. How do hashes really ensure uniqueness? This might seem an impractical and unnecessary conversation, but I feel it's something I need to clarify. Especially, as I just got my first developer job in a blockchain startup.  So hashes are said to generate the same thing for any information it sees, and it's only a one-way interaction.   As far as I understand, hashes are just long alphanumeric strings. If one computes hashes across all documents, keys, information, files, etc, over and over again- it's simply a matter of time until the same combination comes up again for different information (applying it across all possible contexts)? This might be an impractical test given size and possibility, yes, but is it this that makes hashes powerful, that it's practically impossible to recreate a hash that for any foreseeable endeavor it's fine, or is there some element that I'm missing that reduces even that extremely low probability to zero? Is there any design constraint that prevents a very powerful computer to back-calculate the original data from a hash? Or is it simply that the design is so complex that it's simply a futile exercise to dream of such a large computer required for this task? What's stopping a hacker or malicious middle man to hack open the software program or library that creates this "hash" and then use that library to create hashes of his/her own or mislabel some target company's hash with his/her own pointing to their version of the file? Especially since many applications, languages, and developers use hashing independently. Whichever is most weakly secured, we can use that to take on the rest?   Any resources are also appreciated. I need Mac OS X 10.6.8 (Snow Leopard) in order to work with some legacy software and I want to buy new Mac Mini computer.  Is it possible to install this OS on a new Mac Mini or it's not compatible with it?  If so, what is the latest model of Mac Mini that I can use in this scenario? Is there an unsubdivide function in Blender? Difference between SQL batch, statement and RPC? What is the difference between SQL batch, T-SQL statement and Remote Procedure Call? How can I tell if part of the T-SQL code is a batch or statement? Going into a directory linked by a link When I cd a link, my current path is prefixed with the link's path, rather than the path of the directory the link links to.  E.g.  ~/dirlinks/maths$ ls -l logic lrwxrwxrwx 1 tim tim 71 Jul 27 10:24 logic -&gt; /windows-d/academic discipline/study objects/areas/formal systems/logic  ~/dirlinks/maths$ cd logic ~/dirlinks/maths/logic$ pwd /home/tim/dirlinks/maths/logic  ~/dirlinks/maths/logic$ cd .. ~/dirlinks/maths$   I would like to have my current path changed to the path of the linked dir, so that I can work with the parent dirs of the linked dir as well. Besides ls the link to find out the linked dir, and then cd into it, what are some simpler ways to accomplish that? For example, after cd into a link, how do you change your current path to the path of the linked dir? Is $K[a,b]=K(a,b)$ for algebraic $a, b$? When I'm looking down the front page on , I often wish I could tell at a glance what happened to the question to bump it there. Sometimes it is obvious from looking at the number of answers, other times you have to follow a link to find out if it has been edited or if an answer has been added or updated.  I'm guessing this is just as applicable on other SE sites - this is the kind of thing I'm suggesting:    A couple of suggestions from the comments that might be good:   To reduce clutter the word 'ago' could be removed at the same time. The new text could be visible only to established users - ie users who are likely to be reviewing/editing and would benefit from seeing at a glance what kind of change has occurred. Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$ I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because otherwise $e$ wouldn't be that special, but is there any proof of it? Is it possible to create an iron door which responds to levers on both sides Is the external micro SD card path the same for all devices? I am using a Samsung Tab 2.  There is a path: /mnt/external_sdcard/myfile. Here, I can save, read, or write my file to the external micro SD card.  I'm just wondering if this path is the same for all devices, or if it may be different. Which Pokémon can speak with humans? Inspired by Detective Pikachu, I was wondering which Pokémon can "speak" with humans.  I would like to include  Pokémon like Meowth in the anime that speak English/Japanese with their mouths, Pokémon like Mewtwo in the anime/movies that speak telepathically, or other Pokémon we've seen like Lugia/Entei that seem to speak without moving their mouths but still out loud for all to hear (presumably).  The format doesn't matter (manga, anime, video games). And while this is inspired by Detective Pikachu, it's not about it, as the movie is not out yet      And it seems like the human character has the special ability, not the Pikachu What is DOCTYPE and why do I want to use it? What are the different DOCTYPEs I can use? What is the difference between standards and quirks mode, and what are some quirks I may run into with differently set DOCTYPEs?   Lastly, what is the proper DOCTYPE that I should be using? I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof:  Proof. Let $\mathfrak s$ denote the scalar matrices. Clearly $\mathfrak s\subset\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$. Suppose that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is generated by more than one element, so that $X\in\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$, but $X\notin\mathfrak s$. We can change basis such that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is upper-triangular. (The change of basis leaves the scalar matrices invariant.) Then $X$ is upper-triangular.  I want to conclude that there exists a $Y\in\mathfrak{gl}_n\mathbb F$ s.t. $[X,Y]$ is not upper-triangular. How can I see that $Y$ always exists, except by waving hands?