Why is cardinality the of real numbers $2^{\aleph_0}$? While I know that real numbers can be constructed using sets of natural numbers, that solely does not mean that the real numbers has cardinality $2^{\aleph_0}$. So, what makes the real numbers cardinality $2^{\aleph_0}$?  I can't find why it is like this in my textbook...
What is the simplest way of proving (to a non-mathematician) that the power set of the set of natural numbers has the same cardinality as the set of the real numbers, i.e. how to construct a bijection from $\mathcal{P}(\mathbb{N})$ to $\mathbb{R}$?
How do I embed a gather environment within a custom environment? I've tried defining the following custom environment, but I get errors:     \begin{gather} ended by \end{solution}   etc...   Here's a MWE showing my problem:  \documentclass{article}  \usepackage[fleqn]{mathtools}  \newenvironment{solution}{%  &lt;-Solution   \par\addvspace{1em}   \itshape{Solution:}\par\begin{gather}}   {\end{gather}\par\normalfont\hfill\qed\par\addvspace{\medskipamount}}%  \begin{document}   \begin{gather}     testing   \end{gather}    %% This does not work:   \begin{solution}     testing   \end{solution} \end{document} Send authors an inbox message if their question is closed I think it would really help if you get notified that a question of yours was closed.  Especially for the multi-site power user, usually (at least for me) a question is asked and then left alone for while. I rely on the network-wide inbox to tell me when someone has commented or answered, but if I don't receive any messages then I tend to assume that the question wasn't answered or even noticed. It's only when I check the question a day or two later to see what I can add do I find out it was closed.  This would be especially helpful for duplicates, where the answers you need are already there. A message to the effect of, "Closed as duplicate of question" is just like notifying me that a new answer has been added!  Unfortunately, question closures can remain unnoticed by their askers until months have passed. By then, the questions may either be too old to migrate to the correct site, or may , depending on the specific closing reason. We should notify users of these question closures so that they can improve them and learn from their mistakes.  Could closed questions get added to the inbox? What would happen if two wizards made each other a Horcrux? Would they be invincible? How would you kill one of them? I was trying to install Lightworks 11.5.1 (amd64) on Ubuntu 14.04(amd64) but there's a notice: Dependency is not satisfiable: libtiff4. Is there anyway to solve this dependency issue?  Thanks in advance. How to use unicode characters in Windows command line? ASP.NET "special" tags What is the official name for the "special" ASP.NET tags like this:  &lt;%# %&gt; &lt;%= %&gt; &lt;%@ %&gt; &lt;%$ %&gt;   I can't seem to figure out the conceptual or well known name for these, so I'm having trouble searching for more info.  As a bonus, can anyone give me a quick rundown of all of the possible "special tags" and what each one of them does (or point me to a resource)? Finding the maximum likelihood estimator Find the MLE of the unknown parameter $\theta$ when $X_1,X_2,...,X_n$ is a sample from the distribution whose density function is: $$f_X(x) = \frac12e^{-|x-\theta|}, -\infty&lt;x&lt;\infty$$ What I did so far: $$\text{maximize }(\frac12)^ne^{-|x_1-\theta|-|x_2-\theta|-...-|x_n-\theta|} \\ \text{thus we want to minimize }|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|$$ Since $|x_i-\theta|$ is positive for all $i$, we want to solve for $\theta$ so: $$|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|=0$$ What do I do from here? What's the proper way to typeset a differential operator? I can't seem to find any consensus on the right way to typeset a differential operator, whether it is:   in a standalone context:  as part of a derivative:  as part of an integral:    In all of these cases, I have seen them sometimes italicized like variables (as in the second and third examples above) and sometimes not italicized like operator names (as in the first example above). There is also additional variation in how much spacing goes between the "d" and whatever it is acting on.  Seeing as this is an incredibly common symbol in many branches of mathematics, I am curious about the lack of standardization -- I've seen all possible combinations used in all possible contexts by many respected authors and publishers.  Are there any rules of thumb that I can follow? I personally tend to prefer the non-italicized version with very little space next to it, so I add:  \renewcommand{\d}[1]{\ensuremath{\operatorname{d}\!{#1}}}   to my default preamble, and use that everywhere, but I'd like to know if I'm breaking any hard-and-fast rules here. I am a newbie in linux. I wanted to install ubuntu to my desktop. I currently have two partitions, Windows 8 in installed on one, and i have some important files in another partition. While installing ubuntu, if I chose "Replace Windows 8 with Ubuntu", do I lose the other partition too?   And are there other ways to leave the other partition as it is, and install ubuntu in the partition  where windows was installed. no space left in '/' but a lot space left in '/home' i install ubuntu and followed a guide to put 20gb in '/' and the rest (150gb) in '/home'. Now i just found out that a lot of the things installed in ubuntu usually goes to '/' instead of '/home'. I have a lot more things to install.  So what is the point of '/home' ? How do I fix this ? Should I transfer 150gb from '/home' to / ?  Filesystem      Size  Used Avail Use% Mounted on udev            7.8G     0  7.8G   0% /dev tmpfs           1.6G  9.5M  1.6G   1% /run /dev/sda6        19G   16G  1.8G  91% / tmpfs           7.8G  676K  7.8G   1% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           7.8G     0  7.8G   0% /sys/fs/cgroup /dev/sda2        96M   29M   68M  31% /boot/efi /dev/sda8       159G  1.1G  150G   1% /home tmpfs           1.6G   56K  1.6G   1% /run/user/1000   what is /boot ? Its not '/' or '/home' Can anyone can suggest where I can freely download ArcObjects SDKs for ArcGIS?  I am using version 10.2. Non-ASCII characters in external links Layers with same coordinate system should align/overlap in ArcMap but do not? We know that the maximum level a dweller can train to on any SPECIAL skill is 10, however this does not consider bonuses given from their outfit.    Is there any additional benefit for going over 10? For example, a dweller with an Intelligence skill of 10 wearing a lab coat that boosts Intelligence by 3?   If it is a hard cap, would it be more useful to use outfits that raise a secondary characteristic instead? Lagrange Multipler KKT condition I am looking at SVM, and reviewing Lagrange multiplier. Let's say with the constraint function $$g(x,y) = x^2 + y^2 -1 &gt; 0, $$ I am maximizing $$f(x,y) = x + y -1 .$$ Intuitively, since the constraint function does not provide a finite constrained region, there is no solution, and $f(x,y)$ can infinitely increase.  However, if I just plainly proceed with Lagrange multiplier calculation,  $$L(x,\lambda) = (x + y -1) +\lambda(x^2 + y^2 -1)$$ I get the solution: $$y,x = -\sqrt{2}/2, \lambda = 1/\sqrt{2}$$ And this still satisfies KKT conditions although the solution is wrong.  $$g(x) =&gt; 0 , \lambda =&gt;      0, \lambda g(x) = 0.$$  I expected that I would have some result that conflicted KKT conditions, but nothing was violated.   Does this mean that Lagrange multiplier under KKT conditions can produce a solution that cannot be solved, without giving any indication of intractability?  If functions are complicated, what's the systematic way to confirm that the optimization with inequality constraint functions is solvable using Lagrange multiplier? Restricing domains for universal and existential quantifiers I am currently using Rosen's "Discrete Mathematics and Its Applications" (7th ed.) for my discrete mathematics course. We recently talked about quantifiers, more specifically the universal and existential quantifier. What confuses me is when we started talking about restricting the domain (i.e. instead of all real numbers, only all real positive numbers), where does the conditional statement come from when restricting the domain of a universal quantifier and where does the conjunction come from when restricting the domain of a existential quantifier?  This link did help:   However, the part that confuses here is how are the domains restricted in the answer given in the link above? Aligning Vertices Diagonally Customizing citations and the list of publications depending on the author Say I use a bibliography file in my document, relying on Biblatex and Biber for citation management.  Given a citation command such as \cite{A, B, C} (or any other standard Biblatex citation command), I'd like the resulting citations to be typeset differently depending on whether a given field (e.g. the author field) in the corresponding bibliography entry contains a given string (e.g. someone's lastname).   Besides, I'd like to generate two different bibliography sections in the resulting document, depending on the same condition as above.  Typically this would be used to differentiate, both at the citation level in the main text and in the appendix, the publications that I co-authored from others.  EDIT: adding a MWE:  \documentclass{article} \usepackage{filecontents} \usepackage[backend=biber,style = alphabetic]{biblatex} \usepackage[colorlinks=true, breaklinks, allcolors = blue]{hyperref} \DeclareSourcemap{%   \maps[datatype=bibtex]{%     \map{%       \step[fieldsource=author, match=Doe, final]%       \step[fieldset=keywords, fieldvalue=doe]%     }%   }% } \begin{filecontents}{\jobname.bib} @Book{p1,   author =       "A. U. Thor and John Doe",   title =        "A Scientific Article 1",   year =         "1975"   } @Book{p2,   author =       "John Doe",   title =        "A Scientific Article 2",   year =         "1985"   } @Book{p3,   author =       "Joe Bar",   title =        "A Scientific Article 3",   year =         "1995"   } @Book{p4,   author =       "Jane Baz",   title =        "A Scientific Article 4",   year =         "2005"   } \end{filecontents} \addbibresource{\jobname.bib} \begin{document} % Requirement 1: should make citations of Doe bold In \parencite{p4, p1, p2, p3}, ``TD75'' and ``Doe85'' should appear in bold (the semicolon should remain plain).  % Requirement 2: separating bibliographies. IT WORKS. \section{Co-authored Publications} \nocite{*} \printbibliography[keyword=doe,heading=none]  \section{Other Publications} \nocite{*} \printbibliography[notkeyword=doe,heading=none]  \end{document} One-dimensional [Noetherian] UFD is a PID How are app sales/acquisitions taxed in the US?