We know for any $n\in \mathbb{N}$ ,we can find find n+1. Because we wants to prove it with ınduction. For example$ A=\{a_1,a_2,...a_n\}$ and I proof with Induction $B=\{a_1,a_2,...a_{n+1}\}$ . How should I begin?
What is the proof that given a set of $n$ elements there are $2^n$ possible subsets (including the empty-set and the original set).
Is it possible to redirect a command's output to cd? For example, I searched for a directory using locate and got the path to it. Now, instead of writing a cd path, can I redirect the locate output to cd?  I tried this:  $ locate Descargas | grep Descargas$ /home/oliver/Descargas $ locate Descargas | grep Descargas$ | cd $ locate Descargas | grep Descargas$ &gt; cd $ locate Descargas | grep Descargas$ &lt; cd /home/oliver/Descargas $    No luck. This probably isn't particularly useful, but I'm curious. How to Install Ubuntu on separate hard drive in a dual boot? I have already installed Windows 7 on C: drive. Now I want to install Ubuntu on E: drive which is a free partition. How do I proceed? How can I show that $\frac{1}{1+|z|^2}$ isn't analytic at $z=1$? Very new to Tex. Trying to get the most basic doc to work. Not matter what I do I always get an "Undefined control sequence" at line 1 of my document. As of now, I am only trying to build the following:  \documentclass{article} \begin{document} This is a test of the article \end{document}   Full error is:  This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)  restricted \write18 enabled. entering extended mode (./example.tex ! Undefined control sequence. l.1 \documentclass                   {article} ?   I have tried many other variations, including \usepackage directives that people mentioned - nothing every varies. Questions like:   ♦ ♦-should-discriminate-more-when-bumping-unanswered-questions ≠-users-who-want-to-write-good-posts   well, you can see there.  The diamond (and other problematic Unicode characters) in the title breaks the automatic prettification of internal question links.  This becomes a bit more of an issue when one wants to link to an answer.  For example ♦-break-the-automatic-hyperlink-parser/242572#242572 should link to Martijn's answer, but, it doesn't (tangent, but a different issue is that  which is the share link doesn't auto-prettify unless the userid part is removed - one keeps having to manually remove things to get the auto prettify to engage).  If one looks at the html generated for  for the title and the url linked in the &lt;h1&gt; of the page, we get:  &lt;title&gt;Users who can&amp;#39;t write good posts ≠ users who want to write good posts - Meta Stack Exchange&lt;/title&gt;     &lt;h1 itemprop="name"&gt; &lt;a href="/questions/238644/users-who-cant-write-good-posts-%e2%89%a0-users-who-want-to-write-good-posts" class="question-hyperlink"&gt;Users who can&amp;#39;t write good posts ≠ users who want to write good posts&lt;/a&gt; &lt;/h1&gt;   And while that's a long link title, I will point out that the quote as part of the word "can't" was removed from the slug.  The bug is that this should be done for other characters too. Lightning component scripts loading issues since Spring 17 update When is it correct to use "yourself" and "myself" (versus "you" and "me")? How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition) I'm doing exercise on discrete mathematics and I'm stuck with question:     If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f \circ\ g)$ is given by $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$.   I've no idea how to prove this, please help me by give me some reference or hint to its solution. I'm trying to  working on Linux. I have PAINSTAKINGLY installed NVidia drivers, Cuda 5.0 and incidentally I'm failing on the last step when I run make:  make  all-recursive make[1]: Entering directory `/var/progs/CudaMiner' Making all in compat make[2]: Entering directory `/var/progs/CudaMiner/compat' Making all in jansson make[3]: Entering directory `/var/progs/CudaMiner/compat/jansson' gcc -std=gnu99 -DHAVE_CONFIG_H -I. -I../..     -g -O2 -MT dump.o -MD -MP -MF .deps/dump.Tpo -c -o dump.o dump.c /bin/bash: gcc: command not found make[3]: *** [dump.o] Error 127 make[3]: Leaving directory `/var/progs/CudaMiner/compat/jansson' make[2]: *** [all-recursive] Error 1 make[2]: Leaving directory `/var/progs/CudaMiner/compat' make[1]: *** [all-recursive] Error 1 make[1]: Leaving directory `/var/progs/CudaMiner' make: *** [all] Error 2   I'm pretty sure I have gcc, in fact I have multiple versions of it. I installed version 4.6 (which is what cuda requires) but it doesn't seem to find it. I'm so close!  Thanks in advance. A function $f: \mathbb R \to \mathbb R$ satisfies equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb R $ In quantum field theory, scalar can take non-zero vacuum expectation value (vev). And this way they break symmetry of the Lagrangian. Now my question is what will happen if the fermions in the theory take non-zero vacuum expectation value? What forbids fermions to take vevs? Reducing voltage with resistors I have been looking around for an easy way to convert 12V to 5V. I have seen some people saying that a simple resistor is all that is needed.  $$ Volts = Ohms \cdot Amps $$ $$ Amps  = \frac{Volts}{Ohms} $$ $$Ohms  = \frac{Volts}{Amps}$$  So applying a resistor will diminish the voltage of the circuit. That should mean that an appropriately sized resistor could simply be placed in the path of a 12V circuit, converting it to 5v.   If this is the case how would one reduce amps? Would series vs parallel make a difference in this area?   I have seen designs that include a regulator IC and some capacitors, but if a simple resistor/fuse/diode setup will do the trick, I would really prefer that. How do I sudo a command in a script without being asked for a password? I want to turn my system on automatically every day. So I use the below code in my Python script, but sudo asks me for a password every time:  os.system('sudo sh -c "echo date \'+%s\' -d \'+ \        24 hours\' &gt; /sys/class/rtc/rtc0/wakealarm"')   How can I run this script without sudo asking for the password every time? How to prove that $4^m + 5^m$ is divisible by 9 when m is an odd number I am trying to use congruence theorems, specifically Euler's Theorem for a proof. let's suppose that I cast a Silent image of a wall on an airship. Let's suppose that I don't move it with concentration.  Does the image of the wall move with the airship or it will progressively go far? How to unmap TAB and do not make Ctrl-I invalid in VIM Is there any Windows installer (wubi) for Ubuntu 14.04 LTS?  When I mount the .iso file with Daemon Tools and run wubi.exe, it just asks me if I want to reboot (I think that's useful only when the .iso  is burned on CD), so is there no "real" installation? Is const_cast safe? I can't find much information on const_cast.  The only info I could find (on Stack Overflow) is:     The const_cast&lt;&gt;() is used to add/remove const(ness) (or volatile-ness) of a variable.   This makes me nervous.  Could using a const_cast cause unexpected behavior?  If so, what?  Alternatively, when is it okay to use const_cast? How do you minimise hand washing between every step to keep tools clean? Say you cooking beef steak:  You have to unwrap the beef from its packaging, lay it down, get blood on your hands, wash hands, sprinkle salt, wash hands, grind pepper, rub the seasoning into the beef, wash hands, grab the tongs etc..  Is it normal to wash hands after every step or is there a trick to avoid getting all the tools dirty?  If you watch TV, they don't ever wash as much as I do. How to know if Mac OS X machine is being remotely viewed?