Why do people say "explain to me", not "explain me"?
Can I rephrase the sentence "Can you explain this word to me?" into "Can you explain me this word."
How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number? This is my first post on this forum :)  I've been trying for a while to solve this problem about symmetric difference of two sets:     Given two sets $A$ and $B$, we call symmetric difference of $A$ and $B$ the following:      $$A\mathrel{\triangle}B = (A\cup B)-(A\cap B)$$      Show that if $A\mathrel{\triangle} B = \emptyset$, then $(A\subset B) \vee (B\subset A)$.   I don’t know where to start; thank you very much for the help :) How can I prove that the additive group of rationals is not isomorphic to a direct product of two nontrivial groups? I am working through Paolo Aluffi's new GSM text on my own (self-study). On page 63, he asks the reader to     Prove that $\Bbb{Q}$ is not the direct product of two nontrivial groups.   For some context, this is an exercise following a section entitled "The category Grp".  I am assuming that he means "is not isomorphic to the direct product of two nontrivial groups", and I can see two possible ways to proceed with this proof, but have been unsuccessful with either approach.    Approach 1:  Show that the additive group of rationals has a property that is preserved by isomorphism that the direct product of two nontrivial groups does not have or vice-versa.  This seems challenging unless I can significantly narrow down the properties that a direct product of two nontrivial groups that was isomorphic to $\Bbb{Q}$ would necessarily have.  Approach 2:  Considering the section in which this question occurs, show that if $G$ and $H$ are nontrivial groups and $\Bbb{Q} \cong G \times H$, then there are homomorphisms $\varphi_{G}:\Bbb{Q} \rightarrow G$ and $\varphi_{H}:\Bbb{Q} \rightarrow H$ which do not factor or do not factor uniquely through the product $G \times H$.  This would be a contradiction, as $G \times H$ is a final object in the category Grp.  I would greatly appreciate suggestions on how to proceed further with either of these approaches or with alternate approaches. How many combinations can I make? let $n \gt 1$ be an integer, and consider $n$ people; $P_1, P_2,..., P_n$ let $A_n$ be the number of ways these $n$ people can be divided into groups, such that each group have either one or two people  determine $A_1, A_2, A_3$  So I have  $A_1 = 1$ way $A_2 = 4$ way $A_3 = 12$ way  But I am not sure if this is right... Can anyone confirm?  edit:  My way of getting this is  $A_1 = 1 $ because {P1} only has 1 way to sort  $A_2 = 4$ because {P1,P2} = {P1,P2},{P2,P1},{P1P2},{P2P1}  etc. How do I get attention for one of my own questions without a good answer? How to reference sections, equations, and figures in a separate document (e.g., supplemental information)? I have a main document (manuscript) and a supplemental document.   I wonder if there is a way that I can "link" the two (make them recognize each other) so that section, equation, and figure references across documents are recognized? Fullscreen Vista's Command Prompt, or Replacement Opening the command prompt ("cmd") on Vista into full screen causes the application to take up 100% vertical space, but only about 50% horizontal. Is there not a way I can make this fullscreen? If it's simply not possible, what is a good replacement? What's /System/Volumes/Data? New mac user here (Linux background). I'm running macOS Catalina (Beta5). I've noticed there are 5 volumes in my main disk container. One is mounted at /, that I understand, it's the system, another is /System/Volumes/Data, which I tough it would be the Linux /home, but it's not, so what it it?  Also what are the other two unmounted volumes? One might be the recovery one, but the other? Thanks! Domain of ProbabilityDistributions I've been asked the following question:  What five odd integers from the set  $\{1, 3, 5, 7, 9, 11, 13, 15\}$ that when summed together equals to $30$? Note that any integer can be used more than once.   If my limited knowledge of maths is correct, there should be no answer, as no odd number of odd integers summed together can give an even number. Novel about a guy who is possessed by the divine essence and the world ends? I read this novel back in the mid-80's but I'm pretty sure it was written in the 70's.  Strange things start to happen around a homeless man - like the laws of chance are changed.  The investigators discover that they can figure out what is going to happen next by interpreting the dreams/vision of the man.  One such dream is of a large body of water that is split into two pieces. Turns out that means that "C" (sea) - the speed of light has been halved.  The homeless man had become the embodiment of the creator essence. The creator was hiding, because it was tired of trying to keep the universe working and from the fight with is adversary - chaos.  At the end of the book, the creator allows the world to be destroyed and recreates it - except that planets are cubes and move in squares.  Yes - it was rather bizarre. I'm reading Baby Rudin and exercise 28 of chapter 2 reads "Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable." I don't understand why the set must be closed in order for this to be true.  My proof is as follows, and I don't make any reference to the closedness of the set:  Let $E$ be a set in a separable metric space.  Let $B$ be a countable base of that metric space.  Let $P$ be the set of condensation points of $E$.  I claim that $P$ is perfect and $E-P$ is at most countable.  If $p$ is a limit point of $P$, then every neighborhood of $p$ includes at least one point $q$ of P.  For a neighborhood $N_{p}$ of $p$, choose a neighborhood $N_{q} \subset N_{p}$ of $q$.  Since $q$ is a condensation point of $E$, uncountably many points of $E$ are in $N_{q}$, and thus in $N_{p}$.  So $p\in P$ and $P$ is closed.  If $p \in P$, then $p$ is a condensation point of $E$, so every neighborhood $N_{p}$ of $p$ must include uncountably many points of $E$.  If $p$ is not a limit point of $P$, then there exists a neighborhood $N_{p}$ of $p$ such that $N_p$ includes no condensation points of $E$ except for $p$.  This means that every point $q\in N_{p}-\{p\}$ must have a neighborhood $N_{q}$ that does not have uncountably many points of $E$.  For each $b_{k}\in B \cap \mathscr{P}(N_{p})$, choose such a neighborhood $N_{q_{k}} \supset b_{k}$ of some point $q\in N_{p}-\{p\}$.  Now $\{N_{q_{k}}\}$ covers $N_{p}$, and $\bigcup_{k=1}^{\infty}N_{q_{k}}\supset N_{p}$ has at most countably many points of $E$, for it is a union of countably many sets that have at most countably many points of $E$.  This contradicts the assumption that $p\in P$.  So $P$ is perfect.  Let $S=E-P$. There are no condensation points in $S$.  So there exists an open cover $\{N_s\}$, where each $N_s$ is a neighborhood of $s \in S$ with at most countably many elements of $S$. For each $b_{k}\in B \cap \mathscr{P}(S)$, choose such a neighborhood $N_{s_{k}} \supset b_{k}$ of some point $s\in S$.  Then we have a countable subcover $\{N_{p_{k}}\}$ of $S$. As $N_{p_{k}} \cap S$ is at most countable for all $k$, $S$ must be at most countable.  So what am I doing wrong?  Or is Rudin's assumption of closedness extraneous? Additional Drivers for Ubuntu 17.10 The Additional Drivers settings interface is no longer available in Ubuntu 17.10  How can I install proprietary drivers in Ubuntu 17.10?  I need drivers for Wifi &amp; Bluetooth (BCM43142) and Intel Graphics.  The earlier Additional Drivers program showed a package named jockey but it no longer seem to exist in repositories. Food in the Star Wars Universe? Where is the denoiser? In my experimentation, I've used only BJTs as switches (for turning on and off things like LEDs and such) for my MCU outputs. I've been repeatedly told, however, that N-channel enhancement-mode MOSFETs are a better choice for switches (see  and , for examples), but I'm not sure I understand why. I do know that a MOSFET wastes no current on the gate, where a BJT's base does, but this is not an issue for me, as I'm not running on batteries. A MOSFET also requires no resistor in series with the gate, but generally DOES require a pull-down resistor so the gate doesn't float when the MCU is rebooted (right?). No reduction in parts count, then. There doesn't seem to be a great surplus of logic-level MOSFETs that can switch the current that cheap BJTs can (~600-800mA for a 2N2222, for example), and the ones that do exist (TN0702, for example) are hard to find and significantly more expensive. When is a MOSFET more appropriate than a BJT? Why am I continually being told that I should be using MOSFETs? Standard packages for compact SMT crystals There is an old and bog standard form factor for SMT crystals - HC49/US. The problem is that HC49/US takes a lot of real estate on the board.  It's 12mm long.  Another problem with crystals and oscillators in general is that they often become long lead time items.  [This is my experience.]  There are lots of small form factor crystals from various manufacturers.  But, at a glance, it seems that there is no standardization of form factor.  It’s nice when packages are standardized.  It alleviates the long lead time problem.  Crystals from manufacturer B may be used, if crystals from manufacturer A are out of stock.  Are there standard form factors for compact SMT crystals?  Update  Armed with the insights from Gribo and Gustavo, I did a more focused survey.  Found that 12MHz in 3.2x2.5mm package are produced by: Abracon, Citizen, Epson, Fox, ECS, TCS, NDK.  To be included in this list, the part had to be in stock at DigiKey.  It turns out that 3.2x2.5mm package is common across many manufacturers. Basic Web Development IDE/Editor like Dreamweaver? I've used Windows for many years, using Dreamweaver as my editor as it had a file tree which is based on a project (via Local, FTP or SFTP).  I need a web development IDE with the following functionality:   syntax highlighting for PHP, HTML, CSS, JavaScript  file tree - Local, FTP, SFTP, Project-Based (e.g. via the methods of Local/FTP/SFTP) in other words, like Dreamweaver can open multiple files (tabbed)   Additionally, I wouldn't mind features like debugging, class listings, etc, as long as I can hide them away. I'm trying to identify a science fiction series on UK TV during the 70s, but I can only recall the following:   A party of adults and children. There was a woman with a telepathic bond to her cat: she could look into its eyes to see what it had seen. Time travel was involved. I think it may have been inadvertent or uncontrolled time travel. On one occasion the party encountered an Elizabethan seaman who demanded to know if Elizabeth was still queen. They party replied "err... yes" because they didn't want to have to explain time travel. Is there an easy way to attach source in Eclipse? I'm a big fan of the way Visual Studio will give you the comment documentation / parameter names when completing code that you have written and ALSO code that you are referencing (various libraries/assemblies).  Is there an easy way to get inline javadoc/parameter names in Eclipse when doing code complete or hovering over methods? Via plugin? Via some setting? It's extremely annoying to use a lot of libraries (as happens often in Java) and then have to go to the website or local javadoc location to lookup information when you have it in the source jars right there!