I am unsure as to where to start with this problem. The way I read it is that $f^{-1}(f(A)) ⊃ A$ means that $A$ is a subset of the preimage of the image of $A$. But I am unsure.
More homework help.  Given the function $f:A \to B$.  Let $C$ be a subset of $A$ and let $D$ be a subset of $B$.  Prove that:  $C$ is a subset of $f^{-1}[f(C)]$  So I have to show that every element of $C$ is in the set $f^{-1}[f(C)]$  I know that $f(C)$ is the image of $C$ in $B$ and that $f^{-1}[f(C)]$ is the pre-image of $f(C)$ into $A$.  Where I'm stuck is how to use all of this information to show/prove that $C$ is indeed a subset.    Do I start with an arbitrary element (hey, let's call it $x$) of $C$? and then show that $f^{-1}[f(x)]$ is $x$?  I could use a little direction here...  Thanks.
prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit Is logic subjective? If logic is constructed from axioms, and axioms are depended on observation which in term could be subjective, does this means that logic could be limited to our observation, and not really absolute and fundamental? Fade in each element - one after another I am trying to find a way to load a JSON page to display my content, which I currently have. But I am trying to fade in each element one after another? Is anyone familiar with a way to do that?  Fade in each element with a slight delay?  Here is an example of my code, I am using the jquery framework.  CODE: Active vs passive voice with inanimate object as the subject I have asked the  in ELL Stackexchange forum. There all the answers and comments I received was really helpful, but still I was looking for something more. So I have decided to ask this same question here also.   I have seen some sentences like this -         The film was announced to be released on this coming Friday.   The film was announced to release on this coming Friday.      Both the sentences are used, and I know they are interchangeable and does bear the same meaning. But I have some doubt about the second sentence - why is it "to release", when the film cant release itself.   Some other sentences of similar structure that arises my confusion -         The product sells well.   The book reads well.   The glass broke.      The sentences are understandable, but in each of the sentences the subjects can't perform any job, as they all are inanimate object. The verbs are active, but they acts like they are used as passive.  Now my question is how to decide which verb to use this way and which not? Lily Potter chose to die instead of sacrificing her son Harry to Voldemort. In doing so, protective enchantments were created that protected Harry and made him less vulnerable to Voldemort and Voldemort's attempts to kill Harry.   Yet, I don't understand how Lily's enchantments, when Voldemort stole Harry's blood in the graveyard scene in Goblet of Fire (chapter 33, The Death Eaters), were helpful to Voldemort at all.     ‘He took my blood,’ said Harry.   ‘Precisely!’ said Dumbledore. ‘He took your blood and rebuilt his living body with it! Your blood in his veins, Harry, Lily’s protection inside both of you! He tethered you to life while he lives!’   Deathly Hallows - page 568 - Bloomsbury - chapter 35, Kings Cross   Yet I wonder about the following canon information:    Lily's enchantment was contingent upon Petunia Dursley giving Harry a home, which Petunia     grudgingly did&#185;. It was the blood bond between Harry and Petunia, plus the fact that Petunia gave Harry a home, where he could return to once per year, that sealed Lily's   protective enchantments and made them strong. Voldemort was never granted a home at the Dursleys, and in fact never stepped foot into the Dursleys' home at any point in canon. Never having been given a home at the Dursleys, how, then, did Lily, Petunia, and Harry's blood bond exist within Voldemort? Or did it? If it did, would it be because ? Lily's enchantments broke the moment Harry turned 17 and could not protect Harry any more&#178;. According to the 's page on Voldemort, Voldemort was 69 years old when he stole Harry's blood. If Lily's enchantment broke for Harry at age 17, why would the enchantments work for Voldemort when he was 69? Dumbledore tells Harry in Order of the Phoenix that Voldemort would not possess Harry because the love that fills Harry -- Lily's love -- is a force that Voldemort detests so greatly that he cannot bear to come into contact with it&#179;. Yet the love Lily infused into Harry runs in his veins; Voldemort took Harry's blood, the blood that is imbued with Lily's love. How can he bear to be in contact with the love given to Harry in his own veins?   Given this canon information, why did Lily's enchantments work for Voldemort?  &#185;Half-Blood Prince - chapter 3, Will and Won't and Order of the Phoenix - chapter 37, The Lost Prophecy &#178;Half-Blood Prince - chapter 3, Will and Won't &#179;Order of the Phoenix - chapter 37, The Lost Prophecy Proving $(A\le B)\vee (B\le A)$ for sets $A$ and $B$ Switch command output language from native language to english Can I live in the UK for a long time through repeat visits? I had recently entered the UK without a visa to be with my partner, from 21 July 2016 until 16 December 2016. I hope to return on the 9th of January. We don't meet the income requirement for a Spouse visa. I recently applied to university, and got in, and was intending to return to the UK on a student visa, but my university did not tell me about maintenance fees and my loan wasn't large enough to cover it all, so my visa would have been denied.  Now, I am hoping to attend university in the next semester, and therefore return to the US to apply for that visa. Do you think that if I try to return to the UK without a visa, I will be turned away?  I won't be working or studying during this time, however, I'm afraid that they'll consider it living in the UK illegally; I don't have a criminal record or anything like that. Using Ubuntu 11.10, how do I format an external or USB drive? I don't find an option to 'format' when I right click. In the new version, the drives don't come up on the desktop, but instead come on the Launcher. I just bought a SATA SSD, and wish to install Windows 7 directly on it, but when Windows asks me where do I want to install it, it doesn't recognize any drives.   I have disconnected all my other HDDs My BIOS recognizes the SSD. I went back to my original installation of Windows and my SSD is recognized Align multiline equation with expression after equal sign I am trying to align a multiline equation using array within the equation but it turns out it doesn't do very well. The code I am using is as follows:  \begin{equation}    \begin{array}{lcl} \mathcal{L}^{-1}\left\{f(d)\right\} &amp;= \mathcal{L}^{-1}\left\{f_1(\delta).f_2(\delta)\right\} \\ &amp; = exp(mt) \star \left\{\frac{l}{2\sqrt{\pi t^3}} exp(-l^2/{4t})\right\} \\ &amp; = F_1 * F_2 \end{array} \end{equation}   Can anyone suggest better way to deal with it ? pgfplots -> contour gnuplot with lualatex instead of pdflatex Problems with booting the system I have yesterday updated packages and today the system cannot boot properly. After seeing the Kubuntu logo there is a black screen which shows repeatedly.     Stopped User Manager for UID 121.      Removed slice User Slice of gdm.      Created slice User Slice of gdm.       Starting User Manager for UID 121...      Started Session c3 of user gdm.       Started User Manager for UID 121.   Please see the screenshot below What does "period" mean when someone says "Sth... Period"? I often hear US citizens say something like this:     &lt;Some phrases>. Period.   What does period mean when it appears in a sentence by itself in this type of situation? $\epsilon$ undefined using Minion Pro with pdftex I'm using MiKTeX 2.9 and have recently installed the Minion Pro fonts for use with pdftex. I followed the instructions available on  to install them.  Everything works as expected, but in place of the Greek letter epsilon I get a box with a cross (see below), and the following warning:    I found the issue was reported , but I still can't fix it. xelatex works fine, but I'd like to to stick to pdftex so I can use microtype.  \documentclass{article}  \usepackage[T1]{fontenc} \usepackage{MinionPro}  \begin{document}  Text here. $\epsilon$  \end{document}       UPDATE: a set of instructions for the installation of the up-to-date FontPro scripts with MiKTeX can be found . Dynamically set value of a file input I am going to change my machine: from a laptop with Ubuntu to another laptop where I am going to install Ubuntu...  I am quite happy with the current system, what I am afraid of is having to rebuild everything from 0.  So what do people often do to clone an Ubuntu system? That may include the data, packages, shortcut settings, etc... Python __str__ and lists I have a doubt on something. I have a theorem that says that if $R$ is a ring, then $R/I$ is a domain iff $I$ is prime. Since $X^2-1$ is note prime, $\mathbb Z[X]/(X^2-1)$ is not a domain. But, $X^2-1=(X-1)(X+1)$, and since $X-1$ and $X+1$ are prime in $\mathbb Z[X]$, there are coprime, and thus, by the chinese remainder $$\mathbb Z[X]/(X^2-1)\cong \mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1),$$ since $\mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1)$ is a domain, then so is $\mathbb Z[X/(X^2-1)$. What is the mistakes here ? show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$