Finding a injective mapping to prove the countability of $ \mathscr{P}_{\text{fin}}(\mathbb{N}) $. I have an assignment due Friday, and I have been struggling with one of the problems, which seemed to be easy enough at first glance. I have been working on it for a few days now and still cannot come up with any smart ideas.  I am asked to find an injective mapping  $$ \varphi_k : \mathscr{P}_k(\mathbb{N}) \to \mathbb{N}^k $$ where $ \mathscr{P}_k(\mathbb{N}) $ is the set of all subsets of the natural numbers with exactly k elements. From there I am supposed to proceed to prove that $ \mathscr{P}_{\text{fin}}(\mathbb{N}) $, the set of all finite subsets of the natural numbers, is countable.  I do not have a problem with the latter part, but I have tried to find such a mapping for days, but I seem to be missing some key concept.   I have tried all commutative operations, but always end up with two function values being the same for different sets. Vectors obviously don't work, at least not in a straight forward way, since  $$ \varphi_2(\{1, 2\}) = (1, 2) \neq (2, 1) = \varphi_2(\{2, 1\})$$ and $ \{1, 2\} = \{2, 1\}$. Can anyone get me on the right track?  Your help is most appreciated, thanks in advance!
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.).
Can words like "what" be the subject of a sentence? In a question like "Who hears a noise?",  is the subject of the sentence who?  I can think of a few tests for subjects like: "the subject is the phrase that inverts with the auxiliary to form a question". But this is a question and there is no auxiliary. We could say that's because it's a subject, but that would be just to presuppose that it's a subject in the first place.  I can think of tests like: "the form of the verb will agree with the number of the subject", but on the other hand who does not seem to have  fixed grammatical number, and it sometimes seems like who might actually be agreeing with noun phrases later in the sentence:   Who are the most prolific writers of our age? Who is the most prolific writer of our age?   There are also tests like "the subject is usually the first noun phrase in the sentence". But of course if we have a question like:   Who have you bitten?   ... you, not who seems to be the subject. The first noun phrase test is not very good.  I also know that some theories of grammar say that there is a gap in wh- questions that the wh- word is extracted from. So can this gap for instance be the real object of a question? If so, is [See community wiki post in the linked to question]  How can we show whether who is really the subject of my original example question? And how about other wh- words like what? Can they (also) function as subjects? I'm setting up an Nginx web server and and want to perform load balancing using the round robin method.  This is what I have found thus far:  upstream backend  {    server backend1.example.com;    server backend2.example.com;    server backend3.example.com; }   According to the page they says i need to add the above lines to my nginx.conf file and upload it to server to load balance.  I do not understand what it means by server backend3.example.com;.  How do I setup a backend instance? TTY doesn't wait for password There was a book (it actually had a second book as well) where a guy gets paid with a group of others to go to a planet that is enclosed in a metallic sphere.   There are many species in this planet. I don't recall if they actually met the races in the first or the second book but there an incident where an alien in the first book ate fruit from a tree that removed any signs of gender from it. How to get Beamer Math to look like Article Math In a LaTeX article, the math characters are more "curvy" than the text characters:  \documentclass[12pt]{article} \begin{document}  $x^2 + y^2 = z^2$ \end{document}     In a beamer presentation, the math characters do not have the same curves:  \documentclass{beamer} \begin{document} \begin{frame}  $x^2 + y^2 = z^2$ \end{frame} \end{document}     Question  How can I get the beamer math to look just like the article math?  I would prefer if the solution was a single line (that I could place at the top of my LaTeX file) as opposed to something that needs to appear next to every piece of math. Trying to get my head around the commutator subgroup. This is an excercise from Artin's Algebra:  Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian.  Here is what I've done:  Let $xC,yC \in G/C$ then $xyx^{-1}y^{-1}C = C$ since $xyx^{-1}y^{-1}$ is a commutator hence belongs to $C$. But then $xyC = yxC$ so $xC$ and $yC$ commute in $G/C$. This can be done for any elements, so $G/C$ is abelian.  This seems somewhat surprisingly short. Is that all there is to it?   Regards Mixing plural and singular list items with a single verb A friend wants to write,   There is no hardware to purchase, no additional software to install and no key fobs to worry about.   This is awkward because the verb "is" doesn't match up with the third item, which is plural and demands "There are" as the subject. However,    There is no hardware to purchase, no additional software to install and there are no key fobs to worry about.   sounds awkward as well.  Is there a good workaround for this problem where some list items are singular and some are plural, so they don't share a common verb? How do I make Bash my default shell on Ubuntu? Find when three numbers have the same remainder when divided by the same number If three numbers 112, 232, and 400 are each divided by the number D, each of their quotients will have the same remainder R. Find R where R>1  How should I approach this? How to apply for a UK standard visitor visa as a family? I am an Indian citizen residing in the United States. My wife and I are planning to travel to the UK for a week. We plan to apply for the UK standard visitor visa from the US. I was filling out the application and I have a few questions.   Travel History - The application asks for travel history to the United States, Canada, Australia and New Zealand. Considering that we're resident aliens in the United States, should we consider all the times we've traveled back to the US from India? Travel History - Should we also list out the times we've traveled to India from the US? Cost of the trip - Hypothetically if we plan to spend a total of ~4,000 USD on this trip, should I split up the cost and write down 2,000 USD on my application and 2,000 USD on my wife's application? Income - From what I've read on the forum, my wife must say that "someone else" i.e. me - is paying for her costs and mention my GWF number in the remarks on her application. Would that be sufficient considering that I'll be sending all income documents with my application? Savings - We have joint savings accounts. Should I mention the same amount on both our applications? Posting the documents - Is the up-to-date advice to send all documents in one packet? Or would sending them in separate packets at the same time be better? My thought was to annotate my wife's packet with my GWF number if we were to send them separately. Bank statements - We get e-statements. Would printing those out be enough? Elysia cron module does not execute cron job automatically I have installed the elysia cron module and setup the cron job. My module has a cron function test_cron() which inserts some data into a table.  This function lists in the elysia cron status page. I have setup the the default scheduled rule to */1 * * * * to run the cron after every minute.  When I click on the run infront of the test_cron() the cron runs and some data inserts into the database.  But this the manual method, I want to execute the cron after every minute therefore I defined the scheduled time for the test_cron(),  but it does not executed automatically after every minute.  What should I do? Is there any other configuration which I should do to run the cron automatically? Latex create a second keywordstyle using listing doesn't work Photoshop To CSS: How do you translate Fill to CSS I have upgraded 14.04 to 16.04 . Now I am trying to open the gedit from terminal by passing the command:    sudo gedit /some/where/file.txt   but it is throwing an error stating that:    (gedit:2090): Gtk-WARNING : Calling Inhibit failed: GDBus.Error:org.freedesktop.DBus.Error.ServiceUnknown:   The name org.gnome.SessionManager was not provided by any .service files**   However, it is opening the file. When I try to save any modification then it is showing this error after saving the file:    (gedit:2090): WARNING : Set document metadata failed: Setting attribute metadata::gedit-position not supported   I am not getting why this is happening in 16.04 only; it was not on 14.04. Sequence converges iff $\limsup = \liminf$ I want to prove that a sequence of real numbers $\{s_n\}$ converges to $s$ if and only if $\limsup_{n \to \infty} s_n = \liminf_{n \to \infty} s_n = s$.  Here are my definitions:  For any sequence of real numbers $\{s_n\}$, let $E$ be the set of all subsequential limits of $\{s_n\}$, including possibly $+\infty$ and/or $-\infty$ if any subsequence of $\{s_n\}$ diverges to infinity.  Then $\limsup_{n \to \infty} s_n = \sup E$, and $\liminf_{n \to \infty} s_n = \inf E$.  I know the theorem that a sequence converges to a point if and only if every one of its subsequences converge to that same point, so one direction of this proof is easy:  If $\{s_n\}$ converges to some point $s \in \mathbb{R}$,  then every subsequence of $\{s_n\}$ converges to $s$.  So the set $E$ of every subsequential limit of $\{s_n\}$ consists of the single point $s$, so $$\limsup_{n \to \infty} s_n = \sup \{s\} = s = \inf \{s\} = \liminf_{n \to \infty} s_n$$  But the other direction seems more tricky...  If $\limsup_{n \to \infty} s_n = \liminf_{n \to \infty} s_n = s$, then every convergent subsequence converges to the same point $s$.  Also, there can be no subsequences which diverge to infinity (otherwise $\limsup_{n \to \infty} s_n$ would be $+\infty$, or $\liminf_{n \to \infty} s_n$ would be $-\infty$).  But can't there be subsequences which diverge otherwise?  And wouldn't that throw off the convergence of $\{s_n\}$?    EDIT:  I'd also be willing to accept a solution which makes use of the "Pinching Theorem" (if $a_n \leq s_n \leq b_n$ for every $n \in \mathbb{N}$, and if $a_n \to s$ and $b_n \to s$, then $s_n \to s$). Selenium 2: Open link in new tab and close tabs Why does the symbol for the multiplication operation change shape? Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very curious. It seems like $\times$ is already quite sufficient as a descriptive symbol. Network reputation graph becomes nervous when clicking quickly, backs away slowly Find and replace text within multiple files I read in several places, including Wikipedia, that a tessellation of the plane by a single, convex, $n-$sided polygon is not possible for $n\geq7$. I was not able to locate a proof, or a paper that discusses this. Maybe it is too trivial, but I am not able to figure it out myself.   So I would like to ask you for help. Where can I find a proof for this?  Also, I am interested in the mathematical background behind plane tiling (especially periodic), If you can suggest papers on the topic it would really help.