In the book "mathematical logic", it is said that $(a,b)$ is an abbreviation for $\{ \{a,a\},\{a,b \}\}$.  I don't understand this, since firstly, $\{a,a\}=\{a\}$, and secondly, even if this weren't the case, how does  $\{ \{a,a\},\{a,b \}\}$ capture the ordered relation of $(a,b)$?  i.e. how do we see a tuple as a set?
My book says  \begin{equation} (a,b)=\{\{a\},\{a,b\}\} \end{equation}  I have been staring at this for a bit and it doesn't make sense to me. I have read several others posts on this, but none made any sense to me.  For example,   Based on how my ignorant brain is viewing this, I don't see why the definition could not be.  \begin{equation} (a,b)=\{\{a\},\{b\},\{a,b\}\} \end{equation}  aka the power set. What is the significance of the {a} in that definition? Please keep things simple if possible. Normally definitions have a valid and clear reason for being defined that way.    Clarification  First, I understand what an ordered pair is. I just don't see how the set notation says that.   Second,  \begin{equation} (a,b) = \{\{a\},\{a,b\}\}=\{\{a,b\},\{a\}\} \end{equation} Sets don't preserve order, but ordered pairs do. How does the third part of the equality apply to the definition?  Third, another issue with the notation that I have starts with the Product Property of Sets  \begin{equation} \text{Let $X$ and $Y$ be sets} :\ X=\{a,b,c\}\text{ and }Y=\{a,d,e\}. \end{equation} \begin{equation} \text{Then }X \times Y = \{(a,a),(a,d),(a,e),(b,a),(b,d),(b,e),\dots,(c,e)\} \end{equation} If we look at the first ordered pair and our given definition we have  \begin{equation} (a,a)=\{\{a\},\{a,a\}\} \end{equation}   How can this be so, you can't have duplicates in sets? I guess what I am looking for in an answer, is not a proof or a definition of ordered pairs, but rather something like, "This notation says what it says because...". Except for the second to last point I get the terminology, I just don't get the connection between the two different uses of notation.
Can't connect to wireless networks on a Dell Latitude D430 As stated above, I can't connect to my home wi-fi with natty but can with Windows 7. The strange thing is I that I have gotten it to connect to other password protected networks with natty, it's just mine that it dosn't like.   I am using a WPA2 pass-code with AES Encryption and all other devises work fine on my network. Laptop used is a Dell Latitude D430. Complex Exponent of Complex Numbers How does one find the algebraic solution of a Complex number raised to the power of another Complex number?  Here is the work I have done so far, if there are any mistakes please inform me.  A real number with a Complex Exponent: $$A \in \Bbb R, \space z \in \Bbb C,\space z = x + iy$$ $$A^z = A^{x+iy} = A^xA^{iy}$$ $$\implies e^{iy\ln A} = A^{iy}$$ $$\exp(iy\ln A) = \cos (y\ln A)+i\sin(y\ln A)$$ $$A^z = A^x[\cos (y\ln A)+i\sin(y\ln A)]$$  A Complex number with a Complex Exponent: [Using previous variables] $$C \in \Bbb C,\space C = a +bi\space|\space re^{i\theta}, \theta =\arg C$$ $$C^z = (a+ib)^z$$ [After previous mistake the following notes are inaccurate] $$C^z = a^x[\cos (y\ln a)+i\sin(y\ln a)] + ib^xib^{iy}$$  And this is as far as I could go as the expansion of $ib^xib^{iy}$ is being problematic. Could I have some help with this expansion? Show $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \|f\|_{\infty}$ for $f \in L^{\infty}$ I have a question that I need help with getting started (possibly I would be back for more help).     I have a measure space $(X,A,\mu)$ that is finite, and $f \in L^{\infty}(\mu)$. Also, defined is $a_n = \int_X\,|f|^n\,d\mu$. I need to show that the limit is: $$\lim_{n\to\infty}\,\frac{a_{n+1}}{a_n} = \|f\|_{\infty} .$$   I am stuck on getting started, anybody have any suggestions?  thanks much Why does signing on to one site not sign me on to all of them? I've seen this question asked 3 times, but in none of those threads did I find a satisfactory answer.  I'm not asking about reputation: it makes sense that different sites should have different reputations.  However, consider the StackExchange drop-down menu at the top left of the screen.  When looking through the "Hot questions", the links are often to sister websites of StackExchange (programmers.stackexchange i.e.).  It doesn't make sense to me that if I go to one of those sister sites, I'm no longer logged in.  I feel like it's counterintuitive to click a link to go answer a question, only to find that you can't answer that question until you log in.  What is the reason for this?  Is there an option to do as I'm suggesting? There was the following paragraph in the article that came under the title, “GOP and the rise of Anti-knowledge” written by  (October 29, 2015):     “English unfortunately doesn’t have a precise word for the German   “Fachidiot,” a narrowly specialized person accomplished in his own   field but a blithering idiot outside it. In any case, a surgeon is   basically a skilled auto mechanic who is not bothered by the sight of   blood and palpitating organs. We need the surgeon’s skills on pain of   agonizing death, and reward him commensurately but that does not make   him a Voltaire.”    In Japanese we have a word, 専門バカ, of which literal translation is “specialist idiot” meaning exactly same as defined the above - a narrowly specialized person accomplished in his own field but a blithering idiot outside it.  I’m really curious about the absence of short descriptions of the specialist who knows nothing more than his specialty area in English like German “Fachidiot,” and Japanese "専門バカ."  Isn’t there really any short word counterpart to “Fachidiot,” specialist’s ignorance, for instance, in English?  P.S.  I found "narrow-minded specialisit" as a German-to-English translation of "fachidiot" in the web site, Interglot translation dictionary. Ratio of Boys and Girls In a country where everyone wants a boy, each family continues having babies till they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of having a boy or a girl is the same) Can you clone a larger drive onto a smaller drive (used space fits on the smaller) I have a 500GB drive which is about 50GB full. I got a 80GB SSD, which has 74.5GB of available space. I want to clone the system drive onto the SSD.   DriveImage XML will not let me copy a larger drive to a smaller one. Determining relationship (e.g. number of removes) with cousin? How do I figure out if someone is a first cousin, second cousin, cousin once-removed, etc? Possible Duplicate:          I am trying to troubleshoot my webcam and make it work with skype.  One workaround that is often suggested is to LD_PRELOAD the v4l1compat.so file. But when trying that, I get an error message:  fdr@maggie:~$ LD_PRELOAD=/usr/lib/x86_64-linux-gnu/libv4l/v4l1compat.so skype ERROR: ld.so: object '/usr/lib/x86_64-linux-gnu/libv4l/v4l1compat.so' from LD_PRELOAD cannot be preloaded: ignored.   and I am rather sure the path is correct:  fdr@maggie:~$ ls /usr/lib/x86_64-linux-gnu/libv4l/v4l1compat.so  /usr/lib/x86_64-linux-gnu/libv4l/v4l1compat.so   and the kernel and skype have matching architecture:  fdr@maggie:~$ uname -a Linux maggie 3.2.0-29-generic #46-Ubuntu SMP Fri Jul 27 17:03:23 UTC 2012 x86_64 x86_64 x86_64 GNU/Linux fdr@maggie:~$ dpkg -s skype | grep Architecture Architecture: amd64   So my question is:  What is causing the "... from LD_PRELOAD cannot be preloaded: ignored" error message? I'm using a common charger for both of my laptops.  The charger is specified at 20 Volts and 3.25 Amperes. My other laptop has a specification of 19 V and 3.42 A.  Is it safe to charge my laptop with chargers whose specifications exceed or are not equal to the mentioned specifications?  I have read somewhere that it may cause some damage if the voltage or amperage is not matching the specs. Let $a_1,\dots ,a_n \in \mathbb{R}^n$ with $a_i = (a_{i1}, \dots , a_{in})$.  Show that when $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid \; \forall \; j\in \{1,\dots ,n\}$, then $a_1,\dots ,a_n$ are linearly independent.  I want to show that the matrix \begin{matrix} a_{11} &amp; a_{21} &amp; \cdots &amp; a_{n1} \\ a_{12} &amp; a_{22} &amp; \cdots &amp; a_{n2} \\ \vdots &amp; &amp; \ddots &amp; \\ a_{1n} &amp; a_{2n} &amp; \cdots &amp; a_{nn} \end{matrix} can be transformed into an upper triangular matrix with all entries in the main diagonal $\neq 0$.  For the cases $n=1$ and $n=2$ this is easy. However, I fail to show this for the general case; the problem being that the tranformations mess up the coefficients such that I can't use the assumption on the $a_{jj}$'s anymore.  I'm pretty sure that this has already been solved somewhere on here, but I don't know how to search for it. How do I delete files that are locked when no other app can delete them? So I found out that I had a Trojan on my external hard drive. My anti-virus app killed it, but it has created a bunch of random files on my computer which cannot be deleted. These are image files. They cannot be renamed - using any method - nor can they be moved, cut, copied, permanently deleted or even opened (in any program).  How can we rid these files from our system?  I have tried:   Right-click > Delete Shift+Delete (Permanently Delete) Swearing   and then:   FileASSASSIN from MalwareBytes Eraser program sdelete program several commands in cmd   but none of them have been successful. How can I completely obliterate these files? Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$? Moment of inertia of a cylinder about its base I've tried to find the moment of inertia of a cylinder rotating about an axis parallel to its base (i.e about the 'End diameter') as one can see  . But when I checked my results with different references ,I've found that it's incorrect!.I need a help to figure out where I did it wrong. since $$I=\int\limits x^2.dm$$ $$dm = \rho.dv$$ where $$dv= r.d\theta . dr .dh$$ &amp; $$\rho=\frac{M}{\pi R^2 H}$$ $M$:cylinder total mass. $H$:total height of cylinder. $R$:the radius of the cylinder. The distance of each infinitesimal element from the axis of rotation would be : $$\sqrt{r^2+h^2}$$ Therfore, $$I= \frac{2M}{ R^2 H}\int\limits_{0}^{R} \int\limits_{0}^{H}\, r(r^2+h^2)dh\,dr$$ and this gives a result of : $$\frac{M~H^2}{3}+\frac{M~R^2}{2}$$ Which is obviously wrong since the second term should be multiplied by a factor of $1/2$. What's a good back-up strategy for 1 desktop PC? What noun or adjective would one give to a person that is intolerant of stupidity? A person that becomes annoyed, angry, or of very short temper when conversing or interacting with people of a lower intelligence?  For example:       The man is very xyzxyz; his son can't do multiplication very well.     or     She is a(n) xyzxyz. She gets very frustrated when people forget things.   I'm looking almost for a scientific/medical/psychological term. One that would fall into the same kind of group as words like haemophiliac, masochist, nihilist, etc. Let $\Omega=\mathbb{N}$. For each $E\subset\Omega$ let $N_n(E)$ be the cardinality of the set $E\cap \{1,2,\ldots,n\}$. Define $C=\left\{E: \lim_{n\rightarrow \infty} \frac{N_n(E)}{n} \text{ exists}\right\}$. Show that $C$ isn't a field.  I already know that it isn't closed to finite union of non-disjoint set but I can't see why. I saw a post that somebody said &quot;the set of natural numbers whose first digit is 1 doesn't have this limit&quot;. I'm in trouble to see why this limit wouldn't exists in finite unions. Thanks in advance. What does the notation $S^n$ mean? I see this notation a lot, but I can't seem to find an answer in my google searches. In particular I am looking at the question      Let $\alpha:S^n\rightarrow S^n$ be the antipodal map. Prove that $n$ is odd, then $\alpha\simeq \operatorname{id}.$    If it is an arbitrary symmetric group, then I understand, but I have seen $S^1$ be used for the circle. If $f$ is continuous and $f'(x)\ge 0$, outside of a countable set, then $f$ is increasing I've been using mkvmerge to merge a video file and subtitles, I'm using the following command  mkvmerge -o output.mkv video.mp4 subtitles.srt   Which is working fine but as you can probably guess this is really slow operation   I've to open terminal in every one of those directories where movies are stored. Then I've to copy and paste the names of the files in the command Then I've to rename the output file Then delete the original files which I used   This is a lengthy process.     Ideal situation that I want.   I'll go into the folder which contains a movie file and a subtitle file with the same name. I select a movie file and and a subtitles file, right click and run a nautilus script on them. They get merged Original file gets deleted. Rename the merged file with the name of the orignal file with mkv extension      This doesn't need to be recursive, I don't want my HDD creating and deleting 100's of files at ones. Movie files and Subtitle files generally have the same name but some times have "_en" post fixed to subtitle files. If you have another approach of doing it I'm on board with that too.