I don't even know where to begin on this one.  Let $A$ be a countable set. In other words, $|A| = |\mathbb{N}|$. I'm trying to prove that there doesn't exist an $x$ such that $\mathcal{P}x = A$.   Maybe a proof by contradiction? Suppose there was a set $x$ whose power set was countable ...
Is it consistent with ZF for there to exist a set $S$ such that the power set $P(S)$ is countable? If so, what is the weakest form of the axiom of choice needed to prove that no such set exists?
I was trying to install minted package in TeXLive. I installed it and then run sudo tlmgr update --all. It was running slow so I stopped it using Ctrl+Z in an ubuntu terminal and then I run sudo tlmgr update --list. After that I installed texlive-latex-extra and python-pygments. Later on I run sudo tlmgr update --all again and let it finish.  After that I am trying to run a simple code using listings and greek but I get an error     Bad Character code (256)   A MWE that illustrated the problem is the following  \documentclass{book}  \usepackage[english,greek]{babel} \usepackage[iso-8859-7]{inputenc} % %\usepackage[utf8]{inputenc} \usepackage{kerkis} \usepackage{pifont} % %  \newcommand{\sw}{\selectlanguage{english}} \newcommand{\sq}{\selectlanguage{greek}} \newcommand{\eng}[1]{\latintext#1\greektext} \newcommand{\gre}[1]{\greektext#1\latintext}  \usepackage{xcolor} \usepackage{listings} \usepackage{textcomp}  \definecolor{listinggray}{gray}{0.9} \definecolor{lbcolor}{rgb}{0.9,0.9,0.9}  \lstdefinestyle{JavaStyle}{         backgroundcolor=\color{lbcolor},     tabsize=4, %   rulecolor=,     language=Java,      % choose the language of the code         basicstyle=\scriptsize,     % the size of the fonts that are used for the code         upquote=true,         aboveskip={1.5\baselineskip},         columns=fixed,         showstringspaces=false,         extendedchars=false,         breaklines=true,         prebreak = \raisebox{0ex}[0ex][0ex]{\ensuremath{\hookleftarrow}},         frame=single,         numbers=left,         showtabs=false,         showspaces=false,         showstringspaces=false,         identifierstyle=\ttfamily,         keywordstyle=\color[rgb]{0,0,1},         %commentstyle=\color[rgb]{0.026,0.112,0.095},         commentstyle=\itshape\color{green!40!black},         stringstyle=\itshape\color{red!90!black},         numberstyle=\itshape\color{yellow!50!black} }  \lstdefinestyle{CppStyle}{         backgroundcolor=\color{lbcolor},     tabsize=4, %   rulecolor=,     language=C++,       % choose the language of the code         basicstyle=\scriptsize,     % the size of the fonts that are used for the code         upquote=true,         aboveskip={1.5\baselineskip},         columns=fixed,         showstringspaces=false,         extendedchars=false,         breaklines=true,         prebreak = \raisebox{0ex}[0ex][0ex]{\ensuremath{\hookleftarrow}},         frame=single,         numbers=left,         showtabs=false,         showspaces=false,         showstringspaces=false,         identifierstyle=\ttfamily,         keywordstyle=\color[rgb]{0,0,1},         %commentstyle=\color[rgb]{0.026,0.112,0.095},         commentstyle=\itshape\color{green!40!black},         stringstyle=\itshape\color{red!90!black},         numberstyle=\itshape\color{yellow!50!black} } \begin{document}  This is a dummy text. kjhbki \end{document}   The .aux file says  \relax  \select@language{greek} \@writefile{toc}{\select@language{greek}} \@writefile{lof}{\select@language{greek}} \@writefile{lot}{\select@language{greek}}   while the .log file can be found   Any idea why is this happening and how to fix this? Note that if I remove the listings preamble, delete all produced files and build the document, everything works fine. Edited photos have reverted to originals on iphone Are features that allow −5 to attack to get +10 to damage mathematically sound? Some class features, monster traits, and feats allow you to take a −5 to your attack roll in order to gain a +10 to your damage roll. At first glance this looks really good, but when will it deal more damage than not using it on average?  For instance if you have a total attack bonus of +8 and you deal 1d12 damage. You have to roll a 7 to hit an AC of 15. If you take the −5 that makes it where you have to roll a 12. That's going to reduce your average damage by a lot. How to install Windows 7 after Ubuntu and dual boot? I've just started playing with Guice, and a use-case I can think of is that in a test I just want to override a single binding.  I think I'd like to use the rest of the production level bindings to ensure everything is setup correctly and to avoid duplication.  So imagine I have the following Module  public class ProductionModule implements Module {     public void configure(Binder binder) {         binder.bind(InterfaceA.class).to(ConcreteA.class);         binder.bind(InterfaceB.class).to(ConcreteB.class);         binder.bind(InterfaceC.class).to(ConcreteC.class);     } }   And in my test I only want to override InterfaceC, while keeping InterfaceA and InterfaceB in tact, so I'd want something like:  Module testModule = new Module() {     public void configure(Binder binder) {         binder.bind(InterfaceC.class).to(MockC.class);     } }; Guice.createInjector(new ProductionModule(), testModule);   I've also tried the following, with no luck:  Module testModule = new ProductionModule() {     public void configure(Binder binder) {         super.configure(binder);         binder.bind(InterfaceC.class).to(MockC.class);     } }; Guice.createInjector(testModule);   Does anyone know if it's possible to do what I want or am I completely barking up the wrong tree??  --- Follow up: It would seem I can achieve what I want if I make use of the @ImplementedBy tag on the interface and then just provide a binding in the test case, which works nicely when there is a 1-1 mapping between the interface and implementation.  Also, after discussing this with a colleague it would seem we'd head down the road of overriding an entire module and ensuring we have our modules defined correctly.  This seems like it might cause a problem though where a binding is misplaced in a module and needs to be moved, thus possibly breaking a load of tests as bindings may no longer be available to be overriden. Import Large Excel File Why didn't Escobar's hippos introduced in a single event die out due to inbreeding Today I read a  about how Pablo Escobar had once imported 4 hippos (1 male, 3 female) into his estate in Colombia for his private zoo. After his downfall, while other species were shipped out, hippos were considered too big to move and expected to not survive. However, to the surprise of all the hippos are thriving and are so numerous that there have been calls to cull them. From the report:  Numbers are projected to only get bigger. [Colombian biologist Nataly] Castelblanco and her peers say the population will reach over 1,400 specimens as early as 2034 without a cull - all of them descended from the original group of a male and three females. In the study, they envisaged an ideal scenario in which 30 animals need to be culled or castrated every year to stop that happening.  My understanding is that since there was only 1 male, the gene pool would be limited and lead to lot of inbreeding in the descendants. This would cause population to not explode because some individuals would be unfit to survive. Why has this not happened in case of hippos? Is it because there are 3 females (probably unrelated to each other) which keeps gene pool large enough? Or can mutations explain this phenomenon? Would results have been different if originally 2 females had been moved and only 1 retained? EDIT - I have just found that  is maybe less than 100 individuals which though big is not massive. EDIT2 - I have edited the question title to keep focus on the hippos although a general answer would be welcome. Is there a (basic) way to understand why a limit cannot be brought inside of a sum, in certain situations? I have seen that the method of "dominated convergence" implies that sometimes a limit cannot be brought inside of a sum, but is there a more basic way to understand this, preferably without using integrals?   A conceptual or simple answer would be helpful. Thank you! To my (British) ear, it never sounds correct to say you have a doubt about something.  I expect that doubt to be pluralised, even if there's only one specific thing that I'm doubtful of, in one specific way, so I would always say "I have doubts about something".  Answers to  say that using doubt as a synonym for question is "Indian English". That seems to be the case in  which starts with "I have a doubt on my construction", because the OP really is saying "I have a question" (which he goes on to actually ask). Okay, it should have been a question/doubt about something, not on, but I'm not concerned with that detail here.  But in  the OP says "...there is a doubt that can I use [some construction]". In that case it's not easy to see how the word doubt can be replaced with question.  I don't have a problem with, for example, , because there I interpret it as doubt=uncertainty (a state of mind, syntactically equivalent to words like happiness, agitation, sorrow, etc.). But you certainly can't "have a happiness about something". So my question is...     Is it acceptable today to use singular noun doubt in any contexts where it means something other than state of mental uncertainty? If so, what are those contexts? Why is the determinant the volume of a parallelepiped in any dimensions? For $n = 2$, I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculating the area by coordinates.  But how can one easily realize that it is true for any dimensions? It is stated as a problem in Spivak's Calculus and I can't wrap my head around it. Normally, LaTeX will only create references for the BibTeX entries cited in the text. Is there a way of extracting these entries into a different .bib file automatically? For example if I have a .bib file with two entries, and only one is cited in a particular text, I need a way of creating a new .bib file with just that reference. he became firm friend or friends? What are these three references from episode 9? Limiting floats to two decimal points I want a to be rounded to 13.95.  &gt;&gt;&gt; a 13.949999999999999 &gt;&gt;&gt; round(a, 2) 13.949999999999999   The  function does not work the way I expected. Is full wave rectifier better than half wave one? What's so special about standard deviation? Call to undefined function session_register() What are a few examples of noncyclic finite groups? I just want to make sure that every group $\mathbb{Z}_n$ for any $n$ is cyclic. Further, every group of prime order is cyclic because it is isomorphic to $\mathbb{Z}_p$. I think I have a handle on that, but what other finite groups are there that aren't cyclic? In particular, I'm looking for examples that aren't direct products of groups, just simply groups like $\mathbb{Z}_n$, not $\mathbb{Z}_a \times \mathbb{Z}_b$, etc. When do I use active vs passive perception? So, I'm trying to figure out how to run the scene from the beginning of Lost Mine of Phandelver (5e starter set) where the party is ambushed by goblins and I'm really confused as to how Passive Wisdom (Perception) works.  The included rulebook mentions on page 6 :     When you hide, there’s a chance someone will notice you even if they aren’t searching. To determine whether such a creature notices you, the DM compares your Dexterity (Stealth) check with that creature’s passive Wisdom (Perception) score, which equals 10 + the creature’s Wisdom modifier, as well as any other bonuses or penalties. If the creature has advantage, add 5. For disadvantage, subtract 5. For example, if a 1st-level character (with a proficiency bonus of +2) has a Wisdom of 15 (a +2 modifier) and proficiency in Perception, he or she has a passive Wisdom (Perception) of 14.   So the party is being ambushed by goblins and at that point the DM guide mentions on page 7 :     Check to see who, if anyone, is surprised. [...] Make a Dexterity (Stealth) check for the goblins, rolling once for all of them. Roll a d20, add the goblins' Stealth skill modifier (+6) to the roll and compare the result to the characters' Passive Wisdom (Perception) scores.   So to me this reads like I should roll 1d20+6 for the goblins and compare that to the static Passive Wisdom (Perception) score on the character sheet of a PC. The PCs don't get to roll anything as they're not actively looking for enemies, even though they are investigating the dead horses.  I would also think that if the characters were supposed to roll, the DM guide would clearly instruct so, given that this is the starter set.  However, watching  at around 8:01 I notice that the DM is explicitly asking his party for a perception roll. And the same thing happens on a couple of other videos where people are running the Lost Mine adventure. The DM always requests a roll.  So what gives? Shall I take it that the DMs ask for the roll because they consider the party to be actively seeking for hidden foes? Or am I just misreading the rulebook and the DM guide?