Suppose $|X|=2^{2^{\aleph_0}}$, $A \subset X$ and $|A|=2^{\aleph_0}$. Is it possible to prove that $|X \setminus A|=2^{2^{\aleph_0}}$ without using the axiom of choice and its consequences?
I'm currently reading a little deeper into the Axiom of Choice, and I'm pleasantly surprised to find it makes the arithmetic of infinite cardinals seem easy. With AC follows the Absorption Law of Cardinal Arithmetic, which states that for $\kappa$ and $\lambda$ cardinal numbers, the larger infinite and the smaller nonzero, then $\kappa+\lambda=\kappa\cdot\lambda=\max(\kappa,\lambda)$.  I was playing around with the equation $k+\aleph_0=\mathfrak{c}$ for some cardinal $k$. From the above, it follows that $\mathfrak{c}=k+\aleph_0=\max(k,\aleph_0)$, which implies $k=\mathfrak{c}$.  I'm curious, can we still show $k=\mathfrak{c}$ without the Axiom of Choice? Is it maybe possible to bound $\mathfrak{c}-\aleph_0$ above and below by $\mathfrak{c}$? But then I'm not quite sure such algebraic manipulations even mean anything, or work like that here. Certainly normal arithmetic does not! Thanks.
How do I move child windows without moving or minimizing parent in GNOME 3? Producing PDF/X compliant document with XeLaTeX Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$? I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:     Find a function $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)) = -x$ for all $x \in \mathbb{R}$.   I am also wondering, can we find $f$ so that is continuous?  I was thinking of letting $f$ be a periodic function, and adding half the period to x each time. I had no success with this, and am now thinking that such a function does not exist.  Source: Some questions are getting closed as duplicates - good, the system is working. Unfortunately, some are getting closed as duplicates of other questions which are already deleted. This leaves all 1-9,999 reputation users left scratching their heads when they try to view the duplicate. Even worse, imagine being the author of the question and having your question closed as a duplicate when the original question isn't accessible.  I think something is wrong. Today I undeleted a question (left it closed) since it was referenced in the close-reason for another question. Comments indicated that people were clearly irritated that they could not access the duplicate     [Comment] Ok, who thought it was a good idea to close this as duplicate of a deleted question, which most people cannot even see? For the record, I'm reopening - clearly there's enough interest in this question. – Kobi 14 hours ago   ---   [Comment] The PosibleDuplicate link is broken. Voting to reopen. – Rahul G 7 hours ago   ---   [Flag] Closed as duplicate of a deleted question, fail. Thanks for moderating. – {username removed} 14 hours ago   My undelete-action was short-lived, as a few users returned to delete the post again.  It certainly cannot be desirable to reference posts that the vast majority of users cannot even see, especially when the author of the redundant question cannot see (at the time of writing this, the author has a reputation of 18).  I'm not sure what the solution is to this, but I'd like to hear some ideas.  Questions  Question:  Duplicate of: ML estimate of exponential distribution (with censored data) In Survival Analysis, you assume the survival time of a r.v. $X_i$ to be exponentially distributed. Considering now that I have $x_1,\dots,x_n$ "outcomes" of i.i.d r.v.'s $X_i$. Only some proportion of these outcomes are in fact "fully realized", i.e. the remaining observations are still "alive".   If I wanted to perform a ML estimate for the rate parameter $\lambda$ of the distribution, how can I utilize the non-realized observations in a coherent/appropriate manner? I believe they still contain useful information for the estimation.   Could someone guide me to literature on this topic? I am sure it exists. I am however having trouble finding good keywords/search terms for the topic. I am trying to add jQuery to my page (it is not being added by default). The file is adding to the page but after drupal.js causing the error:  SCRIPT5009: 'jQuery' is undefined  drupal.js, line 5 character 1    The code that adds drupal.js to the page in common.inc  ... 'misc/drupal.js' =&gt; array(           'data' =&gt; 'misc/drupal.js',           'type' =&gt; 'file',           'scope' =&gt; 'header',           'group' =&gt; JS_LIBRARY,           'every_page' =&gt; TRUE,           'weight' =&gt; -1,           'preprocess' =&gt; TRUE,           'cache' =&gt; TRUE,           'defer' =&gt; FALSE,         ), ...   In my module hook_init I have:    drupal_add_js('misc/jquery.js', array(     'group' =&gt; JS_LIBRARY,     'weight' =&gt; -20,     'cache' =&gt; TRUE,   ));   I would think with a lower weight it would load before drupal.js?  The header of the page contains the following scripts:  &lt;script type="text/javascript" src="http://mysite.com/misc/drupal.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/misc/jquery.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/misc/jquery.once.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/sites/default/files/languages/nl_c9aC92_wtuULa79-hTFcGPM53oqk1yKwciOa_Q3yO28.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/sites/all/modules/lang_dropdown/msdropdown/jquery.dd.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/sites/all/modules/lang_dropdown/lang_dropdown.js?mqdxrf"&gt;&lt;/script&gt; &lt;script type="text/javascript" src="http://mysite.com/sites/all/modules/devel/devel_krumo_path.js?mqdxrf"&gt;&lt;/script&gt; Unable to install any updates through update manager/apt-get upgrade I have been running Ubuntu 12.04 for the past few weeks. I recently tried to download and install some files through the Update Manager. It located a few files, downloaded them, but then failed to install them. It gave the error:  installArchives() failed: (Reading database ...  (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error   I tried using apt-get upgrade but a similar error was produced:  Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be upgraded:   icedtea-6-jre-cacao icedtea-6-jre-jamvm openjdk-6-jre openjdk-6-jre-headless   openjdk-6-jre-lib 5 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/34.6 MB of archives. After this operation, 1,024 B of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error E: Sub-process /usr/bin/dpkg returned an error code (2)   Any help would be greatly appreciated. Thanks in advance.  Edit: Here's some extra information that may be useful. My original problem was that the Ubuntu Software Centre would not open. I found that the Update Manager and Synaptic Package Manager also failed to open. They all gave an error along the lines of:   "installArchives() failed: dpkg: error: reading package info file '/var/lib/dpkg/available': Input/output errorError in function:"   After much googling I seemed to fix that problem as the package managers would open again. When I then tried to download and install any updates from the Update Manager I got the error at the beginning of this post. Assume I wield a Whip in one hand and a dagger in the other.  If a creature moves from 5ft to 10ft it has technically moved out of my reach (dagger), but at the same time it is still inside my reach (whip). When does it provoke opportunity attacks from me?  I am fully aware that this would not enable me to make off-hand attacks since the whip is not a light weapon, just to cut that issue out of the question. What does *~ mean? At the end of a makefile I saw   rm -f *~ *.class   I understand the *.class, but what's *~? Last digits, numbers Can anyone please help me?  1) Find the last digit of $7^{12345}$   2) Find the last 2 digits of $3^{3^{2014}}$.  Attempt: 1)  By just setting the powers of $7$ we have $7^1 = 7$, $7^2=49$, $7^3=343$, $7^4 = 2401$, $7^5 = 16807$, $7^6 = 117649$, $\dots$  After the power of $4$, the last digits will repeat. Then by noticing the pattern the digits will end in $7,9,3$ and $1$.  Then we can divide the exponent $(12345)$ by $4$ since this is the cycle that makes it repeat. Then $12345 : 4$ has remainder $1$. So $7^1 = 7$ is the unit digit to $7^{12345}$.   So the last digit is $7$.   I know how to do it like this, the problem does not state how to find the last digit, but I know it has something do do with Euler's theorem.  for part $2$) I don't know how to start. Can anyone please help me?   Thank you for the help. Finding a 2x2 Matrix raised to the power of 1000 A generalized version of the second Borel-Cantelli lemma says     Theorem 5.3.2. Second Borel-Cantelli lemma, II. Let $\mathcal F_n, n \ge 0$ be a filtration   with $F_0 = \{\emptyset, \Omega\}$ and $A_n , n \ge 1$ a sequence of events with $A_n ∈ \mathcal F_n$ . Then   $$ \{A_n \,i.o.\} = \left\{\sum_{n \ge 1} P (A_n |\mathcal F_{n−1}) =\infty \right\}. $$   An exercise of this lemma is     Exercise 5.3.6. Show    $\sum_{n \ge 2} \mathbb P (A_n | \cap_{m=1}^{n−1} A_m^c ) = \infty$ implies $P (∩_{m \ge 1} A_m^c ) = 0$.   I think we can actually prove that $\cap_{m \ge 1} A_m^c = \emptyset$. My proof is like this: Let $B_n = \cup_{m = 1}^n A_n$. Let $\mathcal F_n = \sigma(A_1,\dots,A_n)$, which forms a filtration. It is easy to verify that $$ \mathbb P (A_n | B_{n-1}^c ) = \mathbb P (A_n | \mathcal F_{n-1})(\omega) $$ when $\omega \in B_{n-1}^c$.  Assume there exist $\omega \in \cap_{m \ge 1} A_m^c$. It follows from the previous lemma that $$ \omega \in \left\{\sum_{n \ge 1} P (A_n |\mathcal F_{n−1}) =\infty \right\} = \{A_n \,i.o.\}. $$ In other words, $\omega \in \cap_{m \ge 1} \cup_{n \ge m} A_n$, which contradicts $\omega \in \cap_{m \ge 1} A_m^c$.  Is this the right proof? Let $n \in ℕ$. Show that the ring $ℤ/nℤ$ is a field if and only if $n$   is prime.   Let $n$ prime. I need to show that if $\bar{a} \neq 0$ then $∃\bar b: \bar{a} \cdot \bar{b} = \bar{1}$. Any hints for this ?  Suppose $ℤ/nℤ$ is a field. Therefore: for every $\bar{a} \neq 0$ $∃\bar b: \bar{a}\cdot \bar{b}=1$. How can I show that $n$ must be prime ? What is /etc/mtab in Linux? What is /etc/mtab in Linux?   Why is it needed and advantages of having it? This morning, I read Wikipedia's informal definition of a limit:     Informally, a function f assigns an output $f(x)$ to every input $x$. The   function has a limit $L$ at an input $p$ if $f(x)$ is "close" to $L$ whenever   $x$ is "close" to $p$. In other words, $f(x)$ becomes closer and closer to $L$   as $x$ moves closer and closer to $p$.   To me that sounds like something that might be better described as a 'target'.  If I take a simple function, say one that only multiplies the input by $2$; and if my limit is $10$ at an input $5$: then I've described something that seems to match the elements contained in Wikipedia's definition. I don't believe that that's right. To me it looks like an elementary-algebra problem ($2p = 10$). To make it more calculusy, I could graph the function's output when I use inputs other than $p$, but that really wouldn't give me anything but an illustration of the fact that one's answer moves farther from the right answer as it becomes more wrong (go figure).   So limits are important; what I've just described is trivial. I do not understand them. I know calculus is often used for solving real-world challenges, and that limits are an important element of calculus, so I assume there must be some simple real-world examples of what it is that limits describe.   What is a simple example of a limit in the real world?  Thank you  -Hal. Data Dump of Attachments I am trying to do an export of approx 30,000 attachments in our Salesforce Org,  Would anyone have been through a similar exercise and can advise best practice or the best way / tool to do this?  Kind Regards  matt I had recently faced a problem:     Solve the Diophantine Equation $x^2 - y! = 2001$.   Solving it was quite easy. You show how $\forall y \ge 6$, $9|y!$ and since $3$ divides the RHS, it must divide the LHS and if $3|x^2 \implies 9|x^2$ and so the LHS is divisible by $9$ and the RHS is not. Contradiction. Hence, the only solution is $(45, 4)$.  That made me wonder, how we can solve the Diophantine Equation $x^2 - y! = 2016$. We cannot apply the same logic here. $2016$ is a multiple of $9$ and it is clear that $3|x$ and $9 \nmid x$. How do I proceed from here? java.lang.NoClassDefFoundError: org/apache/xpath/XPathAPI app engine datastore auto-clears every time project runs Skyrim crashes as soon as I boot up into the menu. How do I fix this? Whenever I boot up Skyrim, it crashes as soon as it reaches the main menu with no dialog or anything. I thought this was related to my using dual monitors somehow, but I disconnected the other monitor and the game still crashed. I tried running the game in different resolutions, both full-screen and windowed, at various quality settings; no dice. My computer does meet the minimum requirements to play Skyrim.  How do I fix this problem?