Let B be a countable set, and let A be an uncountable set. Prove that A-B is uncountable
Problem statement  Prove that if $A$ is an uncountable set and $B$ is a countable set, then $A\setminus B$ must be uncountable.  What I think  The statement does not mention $A$ and $B$ relationship. I think there are two possibilities:   If $A \cap B = \emptyset $, then $A\setminus B$ is trivially uncountable If $A \cap B = B$, then $B \subset A$ and as a bijection can not be made between $A\setminus B$ and $\mathbb{N}$, $A\setminus B$ is uncountable.   And there is where I'm stuck. How can I prove that a bijecton can't be made?  TIA.
Baking with oven without bottom heating element My mom wants to do baking but her oven doesn't have bottom heating element. It only has an element on top, with a fan at the back.  She told me that she can't make me a good pizza or cake without it.  Is there any way around? Does the Divergence Theorem Work on a Surface? The divergence theorem in $\mathbb{R}^3$ says that the integral of the divergence of a vector field over a solid $\Omega$ in $\mathbb{R}^3$ equals the flux through the surface of $\Omega$ denoted by $\partial \Omega$.   My question is that can you use the divergence theorem in $\mathbb{R}^3$ where you're integrating the divergence of a vector field over a surface $S$ in $\mathbb{R}^3$ to get the flux through the edges ($\partial S$) of the surface? So if the surface $S$ is closed in $\mathbb{R}^3$ such as a sphere, then clearly $\partial S = 0$ so the integral of the divergence over this particular $S$ is $0$? Python rounding error with float numbers Prove that $\left | \frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a} \right | < \frac{1}{8}.$ LED as power generator Real Size Image Printing Is any divergence-free curl-free vector field necessarily constant? Search engines and browser accept-language I'm building a web portal where language content will generally depend on the "accept-language" sent by the browser. The same content-URI will thus serve different content to different users depending on their browser setting.  I'm very curious to know how this will affect search indexing. Does Google index using all languages, and is it handled well? What is the exact difference between a 'terminal', a 'shell', a 'tty' and a 'console'? I think these terms almost refer to the same thing, when used loosely:   terminal shell tty console   What exactly does each of these terms refer to? Resolve $\lim_{x \to 0} \frac {\tan 3x}{\tan 5x}$ I started resolving this limit:  $$\lim_{x \to 0} \dfrac{\tan 3x}{\tan 5x}$$  so:  $$\begin{align}\lim_{x \to 0} \dfrac{\tan 3x}{ \tan 5x} &amp; = \lim_{x \to 0} \dfrac{\frac{\sin 3x}{\cos 3x }}{\frac{\sin 5x}{\cos 5x}} \\ \\ &amp; = \lim_{x \to 0} \dfrac{ \sin (3x) \cdot \cos (5x)}{ \sin(5x) \cdot \cos (3x)}\\ \\ &amp; = \quad?\end{align}$$  Could someone help me please I know that a similar question has been asked in:        although I didn't really understand the answer and couldn't really make it work.  However, I have tried something different. I decided to read the manual page for ls and it mentions different environment variables that can be set when dealing with the ls command. If one goes to the -G option it says to look at the CLICOLOR environment variables. I did and that one links you to documentation for the LSCOLORS environment variable. I went to it and it mentions:     "The value of this variable describes what color to use for which attribute when colors are enabled with CLICOLOR. This string is a concatenation of pairs of the format fb, where f is the foreground color and b is the background color. "   It also mentions that the default value of LSCOLORS is "exfxcxdxbxegedabagacad" and that each part specifies the color of some specific thing in the ls command. For example, the first pair ex in example specifies the color of directories. the second fx the color of symbolic links etc... e means blue and x means default "foreground".  1) First what does this mean: "when colors are enabled with CLICOLOR"?  2) What does "foreground" mean?  What must environment variables be for the ls command to color the output however I desire?    I also tried:  ls --color   But it threw me the following error:  ls: illegal option -- - usage: ls [-ABCFGHLOPRSTUWabcdefghiklmnopqrstuwx1] [file ...]   The same error happened when I did ls --color=auto: it alone also throws an error. I have no idea why, and am not sure if it was due to iTerm2 or OS X or why that happened. It seems it works on other systems... Use grep --exclude/--include syntax to not grep through certain files Can I open Microsoft Word and Excel files? Can I open Microsoft Word and Excel files with Ubuntu? Quad tree vs Grid based collision detection An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$ how to disable system apps without root? An awful identity We take place on $\mathbb C(x_1,...,x_r,x'_1,...,x'_p,u_0,...,u_r,u'_0,...,u'_p)$ with $r,p\in \mathbb N$     Show that :   $$\displaystyle{\sum_{i=1}^r \left(   \frac{\prod_{j=0}^r (u_j-x_i) \prod_{j=1}^p (x'_j-x_i) }{  \prod_{1\leq j\leq r, j\neq i     }(x_j-x_i) \prod_{j=0}^p (u'_j-x_i)}      \right)}-\displaystyle{\sum_{i=0}^p \left(   \frac{\prod_{j=0}^r (u'_i-u_j) \prod_{j=1}^p (u'_i-x'_j) }{  \prod_{0\leq j\leq p, j\neq i     }(u'_i-u'_j) \prod_{j=1}^r (u'_i-x_j)}      \right)}=\displaystyle{\sum_{i=1}^p x'_i +\sum_{i=0}^r u_i -\sum_{i=1}^r x_i-\sum_{i=0}^p u'_i}$$   I have seen this exercise in an old book at the library. Unfortunately there was no indication how can I solve the problem.   Anyway if someone can give me some ideas it will be greatful.   NB: Sorry for the title, I have not had other ideas How to create nested loops in x86 assembly language As far as I know, it's possible to create a radially polarised ring magnet, where one pole is on the inside, and the field lines cross the circumference at right angles.    So imagine if I made one which was shaped like a sector of a torus.    Then I forced a load of these magnets into a complete torus.     Clearly this magnet is impossible because there's no way for the field lines to get back into the middle. So what happens to the field in this case? Does it disappear completely? Do the magnets blow up? What does $\dot\sim$ notation (dot over tilde) mean, in the context like $x \mathrel{\dot\sim} \mathcal N(0,1)$?  Turns out it is easier to find how to typeset it correctly:  that one should type \mathrel{\dot\sim} instead of simply \dot\sim to fix the spacing issue -- than to find what it actually means. It has only been used  on CV until now; is it standard?