What is the proof that the total number of subsets of a set is $2^n . WİTH İNDUCTİON.
What is the proof that the total number of subsets of a set is $2^n$?
Any way to add CSS for a single page/node? I'm cleaning up my big crazy style sheets (possibly pertinent to a future question) and I'm wondering the best way to add custom CSS to a specific node or page. In particular, my work site's home page is a Panel Page and it has a bunch of different styling. Right now the CSS is just included with the main theme style sheet. Is there a way to say, &quot;if this is node Foo, then add foo.css&quot;? Is  what I'm looking for? I might be interested in generalizing this to other nodes/sections/etc, but at the moment I just want to handle this one item. Limit of $L^p$ norm Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$?    I don't know where to start. The following integral should be doable using the residue theorum:     $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$ Meaning of "stackexchange" What does stackexchange mean? I have checked out this  to read about the Stack Exchange Network. $\bigodot=?$ for $(\sqrt 2) ^ \sqrt 3 \bigodot (\sqrt 3) ^ \sqrt 2$ where $\bigodot is \lt or \gt$ $\bigodot=?$ $(\sqrt 2) ^ \sqrt 3 \bigodot  (\sqrt 3) ^ \sqrt 2$  where $\bigodot is either \lt or \gt$   In other words, which one is larger $(\sqrt 2) ^ \sqrt 3$ or $(\sqrt 3) ^ \sqrt 2$? The best I could do was ending up with $\sqrt \frac {3}{ 2} = \frac {\ln 3}{\ln 2}$  side question : is there a symbol like encircled $?$ for when what we are trying to find is the correct order symbol like one of $\lt,\leq,=,\geq , \gt $ ?  Hopefully using just juxtopisition of symbols without resorting to real analysis. is it possible to do with precalculs algebra? Possible Duplicate:          So, in my calculus class (one I'm teaching, not taking), the sum $\sum_{n=1}^\infty \frac{\sin(n)}{n}$ has come up a few times.  Unfortunately, as someone not exactly an expert in the convergence of sums, it seems to resist the few techniques I know.  Certain none of the usual first year calculus tricks (integral test, alternating series test, ratio test, etc.) work, and the only more tricky technique, partial summation, I can think of doesn't seem to work either (one would need that $\sum_{n=1}^N\sin(n)$ is bounded, which I believe is false).  It seems like it should converge, since it switches sign quite often, but on the other hand, the harmonic series can mess with your intuition, so I don't have much trust in that.  So, I ask to you:     Does this series converge? Since one or two months ago, my laptop with ubuntu 12.10x64 hangs during shutdown. There's just a black screen, no harddisk activity, it justs sits there. I feel it happens near the end of a normal shutdown cycle, judging from time elapsed.  Only resolution is hard power-off (holding power key for several seconds) - neither Ctrl-Alt-Backspace (which normally works, I have it switched on in Settings) nor Ctrl-Alt-Del help, nor can I get into a text console with Ctrl-Alt-F1.   How do I diagnose that? It does not happen on every shutdown, but often enough to be annoying (as it forces a diskcheck on reboot, of course).   Which log files can tell me what's going wrong? More or less blindly searching in the sys log viewer reminds me of needles in a haystack.  I imagine I could pinpoint the time it happens via log timestamps, but how do I read timestamps like e.g. in kern.log: [36169.146130]? How do I get this human-readable?  Lastly, a bit of a btw, I've never managed to execute an Alt-Sysrq-REISUB to force a cleaner shutdown, but that's probably because I'm on a laptop, PrintScr is only accessible via an Fn key, and I'm not sure of the exact key sequence I need. Is there a "harmless" but user-visible magic command I could practice this with (during normal operation), so I can find out how to do a REISUB next time?  Thanks all! Comment if you need additional data... Bug in editor for numbered lists? I noticed what seemed to be a bug in the way the stackoverflow editor handles numbered lists. I did a search on meta for this and it didn't come up, but if it is a duplicate, I'll delete the question.  Consider the following code:  Here are some points:  1. One 2. Two 3. Three  Here are some more points which continue the theme:  4. Four 5. Five   When you actually type this into the text editor, this is the result:  Here are some points:   One Two Three   Here are some more points which continue the theme:   Four Five   In this case, the ordered list resets after the break. This is probably just an edge case (and not necessarily worth fixing), but I ran into the problem today and figured this would be a good place to raise the topic! My workaround was to use back-ticks around the numeral, but then you lose the hanging indent. (ie, 1. One)  Might this be worth fixing? Is it expected behavior? Or am I doing it wrong? Avoiding labeling features if overlapped by another layer in QGIS? With QGIS 2.12.2, how can I set up layer labeling to avoid placing labels where features from another layer already exist?  For example, if I have a stream/river polyline layer that contains lake "centerlines", and I place a "lake" polygon layer above it in the drawing order, I don't want the river layer to place a label inside the lake.  Instead, I would rather have the river labeled outside of the lake (as needed).  That way, I can place labels from the lakes layer and I don't run into label collisions.  Here is an example, where (I have intentionally put the lines on top for visual purposes) what I am hoping to achieve is no river center-line labels shown inside the lake polygon: How to delete pre-installed apps in Catalina? I want to delete the pre-installed apps, like Music, Books, Siri etc. from my iMac running Catalina. How do I do this?  (NOTE:  asks the same thing but was closed as a duplicate of . Annoyingly, the supposed duplicate doesn’t give an obvious answer to the question, and the original was locked can’t be answered, hence a new question. I can’t add an answer to  either) Hint:  $a^2 -ab +b^2 = (a+b)^2 -3ab.$  I know we can say that there exists an $x,y$ such that $ax + by = 1$.  So in this case, $(a+b)x + ((a+b)^2 -3ab)y =1.$  I thought setting $x = (a+b)$ and $y = -1$ would help but that gives me $3ab =1.$  Any suggestions? Arranging GUI elements in QGIS 3 processing model user interface Using QGIS 3 graphical modeler I created a model with a variety of input parameters (> 10). When starting the model, a nice user interface is generated. But the arrangement of the GUI elements is totally confused and apparently unpredictable during creating the model. It does neither seem to depend on the arrangement of input parameters in the graphical modeler nor on the chronological order in which the input parameters are added to the model.   Is there a way that let's me arrange the GUI elements in the order I desire? bash: syntax error near unexpected token `}' I tried using this:  $ if [$a == 1] then { echo 'yes'; } fi;   but I get an error:  -bash: syntax error near unexpected token `}'   What is the correct format?  I tried several with no luck. Going through a text file line by line in C On my Windows 7 system I installed via  from Fonts.com:   The font does not appear in C:\Windows\Fonts. However, it shows up in certain desktop applications, for example  as: Sabon Next Com Display  My intention was to use the font in a LaTeX document test.tex:  \documentclass{article} \usepackage{polyglossia} \setromanfont{Sabon Next Com Display}  \begin{document} Hello World! \end{document}   Unfortunately, xelatex test.tex fails:  […] Couldn't find `Sabon Next Com Displ.cfg' miktex-maketfm: No creation rule for font "Sabon Next Com Display".   !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! fontspec error: "font-not-found" !  ! The font "Sabon Next Com Display" cannot be found. !  ! See the fontspec documentation for further information. !  ! For immediate help type H &lt;return&gt;. !...............................................    l.4 \setromanfont{Sabon Next Com Display}                                            % Sabon Next Com Display, SabonNex...  ?    Can XeLaTeX even work with fonts installed via SkyFonts? If not, what could be a workaround? Can I color text by using an image, so that the color isn't flat? I'm using tikz to attempt to make a book cover, and I have a "wallpaper" image that looks like leather (not that I've handled that many leather-bound books). I'm using the node command to place text over the top of it, but none of the colors I've experimented with look decent.  Can I use an image of gold-leaf as the fill/stroke for the text? Or for that matter, for the path lines as well?  I've tried googling for this, but I can't figure out a set of keywords non-generic enough to pull back useful results. How do you prove that $\sqrt{2} + \sqrt{5}$ is irrational?    I tried to prove it by contradiction and got this equation:  $a^2/b^2 = \sqrt{40}$. How can I buy apps in the App Store without a credit card? While I'm waiting for my credit card to arrive, I still want to be able to buy some apps for my iPad!  Are there alternative ways to pay for apps, without resorting to someone else's credit card? I've been reading up on STL containers in my book on C++, specifically the section on the STL and its containers. Now I do understand each and every one of them have their own specific properties, and I'm close to memorizing all of them... But what I do not yet grasp is in which scenario each of them is used.  What is the explanation? Example code is much prefered. Why does evolution not make our life longer? Why does evolution not make life longer for humans or any other species?  Wouldn't evolution favour a long life?