Easy proof the set of finite Set in countable is countable Suppose I know a result that the set of finite sets in $\mathbb{N}$ is countable.   Is there a very quick way to show that the set of finite sets in any $X$ countable is countable?  Idea...two sets have same cardinality if there is a bijection, we biject set of finite sets, bijection preserves cardinality. End of proof.
Neatest proof that set of finite subsets is countable? I am looking for a beautiful way of showing the following basic result in elementary set theory:     If $A$ is a countable set then the set of finite subsets of $A$ is   countable.   I proved it as follows but my proof is somewhat fugly so I was wondering if there is a neat way of showing it:  Let $|A| \le \aleph_0$. If $A$ is finite then $P(A)$ is finite and hence countable. If $|A| = \aleph_0$ then there is a bijection $A \to \omega$ so that we may assume that we are talking about finite subsets of $\omega$ from now on. Define a map $\varphi: [A]^{&lt;\aleph_0} \to (0,1) \cap \mathbb Q$ as $B \mapsto \sum_{n \in \omega} \frac{\chi_B (n)}{2^n}$. Then $\varphi$ is injective hence the claim follows. (The proof of which is what is the core-fugly part of the proof and I omit it.).
Obtaining remote host name in bash script Sorry my knowledge of Linux bash commands is pretty basic, I've been searching for a while but I'm not 100% sure what I need to search for.  I was wondering if there's a way to grab the current logged in users remote host name in a Linux bash script? I have a script in which I need to log each time a user runs it. I'm obtaining the date like so:  cdat=`/bin/date +%a' '%d' '%h' '%Y', '%H':'%M`;   I now need to add the users remote host (not the username they logged in with). I'm not 100% sure I'm using the right terminology here either, just to clarify; by 'remote host name' I mean the same output that prints on the screen on most servers I've logged into over ssh, for example:  Last login: Mon Jul 22 16:35:09 2013 from win7-i7-stuart.my.domain.com   I'm looking for the win7-i7-stuart.my.domain.com bit. L293, L298 and SN754410 H-bridge drivers on low voltage power supply This question is about the following three integrated H-bridge drivers: -    or L293D (D = protection diodes added)  (protection diodes included)  (no protection diodes)   Time after time the same question keeps coming up - someone is using one of these devices (on a low voltage, usually around 6V or less) and they are just not performing adequately. The reasons are listed further below but my question is this: -  What H-bridge drivers are preferred when controlling a low-voltage motor?   Information  The L293 and the SN754410 are nearly identical and crucially, if you try and control a 1 amp load, you are faced with dismal performance: -    The tables tell you (typical conditions) that the upper transistor drops (loses) about 1.4 volts when driving a 1A load and, the lower transistor drops (loses) about 1.2 volts when driving a 1A load. The upshot is that if you have a 6V, 1A motor and 6V battery, don't expect to see more than 3.4 volts across the motor: -  \$V_{OUT} = 6V - (1.4V + 1.2V) = 3.4V\$  Worst case scenario is you might only see 2.4 volts across it.  What about the L298? It's got a nice big heat-sink whereas the L293 and SN754410 are regular-looking chips. Here's what the volt drop (losses) look like: -    It's the same story - for a 1A load, you can expect to lose up to 3.2 volts and, what you thought might be 6V across your motor, is at best 4.2 volts and at worst only 2.8 volts.  Clearly none of the devices listed are suitable for low voltage applications where the motor might be expected to draw in excess of 0.5 amps. I think it would be cool if a question would eventually open up to moderators to accept answers on the user's behalf, once a time period has passed. The asking user would get the rep score for the accept.  I know I've seen questions successfully answered with lots of upvotes and no accepted answer that sit there for months unaccepted. This would give superusers a little more functionality and would tie up a lot of loose ends.  What do you think? Since I upgraded to the latest version of Ubuntu, I cannot install the Java OpenJDK package needed to compile Android custom ROMs. When I run sudo apt-get install openjdk-7-jdk, the following error appears:  Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  openjdk-7-jdk :  Depends: openjdk-7-jre (= 7u51-2.4.6-1ubuntu4) E: Unable to correct  problems, you have held broken packages.   How do I fix this problem? Every Finite Group is Isomorphic to a Subgroup of $A_n$ What do I need in a camera to take non-blurry photos of a fast-moving child? Can there be multiple paths for which Fermat's least time principle can be obeyed? Can there be more than one path for which Fermat's least time principle is obeyed? How do I set OS X to boot by default? I set Windows 7 to my Boot Camp. Windows 7 starts now by default in the normal boot. However, I want that OSX boots by default and Windows 7 only as I press ALT and then select it.  How can you set OSX to boot by default? We have $n$ charged and $n$ uncharged batteries and a radio which needs two charged batteries to work. I have a mathematical transformation that takes 16 parameters (grouped into 3+8+5) and would like to make a latex command for it, so that I can easily change the notation for it if the need arises.  As far as I know, both \def and \newcommand take a maximum of 9 arguments, is there any (recommended) way to extend this? HTML text input allow only numeric input I have a file with a number of lines in a file filename.  I want to count how many lines start with character 'a', with 'b' and so on in one go.  What command i should execute.? A word for "every two days" Reclassifying raster layer with certain colors for value ranges in QGIS? I'm using QGIS. I want a raster layer with 6 different colors each color representing a certain value range (e.g. -1 to 0 red, 0 to 0.2 orange).  In the layer style menu I can only approximate colors. Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective?  I tried the following: $$f:\mathbb{R}\rightarrow \mathbb{R^{+}} $$ $$f(x)=x^{2}$$  and  $$g:\mathbb{R^{+}}\rightarrow \mathbb{R}$$ $$g(x)=\sqrt{x}$$  $f$ is not injective, and $g$ is not surjective, but $f\circ g$ is bijective  Any other examples? Is voltage the speed of electrons? PHPExcel Sumif and Skip Understanding imaginary exponents How to get a vintage/dreamy look like this photo?   It has some yellow color cast and seems like extra blur added in post process? Person falls into radiotelescope and his mind gets transported to the far future