If the proof is done by a basic matrix diagonalwise model array then there arise a question that what will happen if 2 elements are equal? How can one make a one to one correspondence with the set of natural numbers?
I am doing some homework exercises and stumbled upon this question. I don't know where to start.      Prove that the union of countably many countable sets is countable.   Just reading it confuses me.   Any hints or help is greatly appreciated! Cheers!
du shows completely different folder size for * and folder name I'm trying to move some data over and to start with I'd like to make sure that the size is ok. I run into a problem however, if I check the folder with   du -sh  myfolder/ 780M    myfolder/   but   du -sh * .. 71M     myfolder ..   I thought it might be something to do with hidden files but this particular folder does not have any inside.  Am I missing something? Answering a phone call with "Yes, XX, tell me" I have heard a lot of people pick their phone and go "Yes, XX, tell me" (highly used in India).  I think the right way should be "Hey, XX, what's up?" or "Hey, XX, what's going on"? But does this work in a professional environment? Can command substitution be nested in variable substitution? Show/Hide function folding margin on Notepad++ Is it possible to get rid of the function folding margin on Notepad++? It seems like the sort of thing that should be in the prefs but I'm not seeing it anywhere. Prove $\gcd(a+b, a-b) = 1$ or $2\,$ if $\,\gcd(a,b) = 1$ I want to show that for $\gcd(a,b) = 1$ $a,b \in Z$   $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ holds.      I think the first step should look something like this:      $d = \gcd(a+b, a-b) = \gcd(2a, a-b)$   From here I tried to proceed with two cases. 1: $a-b$ is even, which leads to $\gcd(a+b, a-b) = 2$ 2: $a-b$ is odd, which leads to $\gcd(a+b, a-b) = 1$  My main problem I think is, that I do not know how I should include $\gcd(a,b) = 1$ in the proof.  Any help is appreciated. Thx in advance. Cherio Woltan When & How often do 4e Monster powers recharge? How often should I roll power recharge rolls and at which point in the monster's turn do I re-roll them?  I'm trying to find the official definition of the recharge mechanic in the online compendium and I'm not seeing it there. 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. How can I determine how much bleed to use? Practical Uses for the "Curiously Recurring Template Pattern" What are some practical uses for the ""?  The "" example commonly shown just isn't a convincing example to me. Can someone please explain how I can install  in the following ways?   By using the package manager? By downloading and installing a pre-built package? How does Delta baryon decay conserve angular momentum? I'm a chemist so bear with me:  I understand the  $\Delta^{+}$ and $\Delta^{0}$ to be in some sense spin (and isospin) quartet states of the proton and neutron. These can decay straight to a proton or neutron through emission of a pion, which is spin-0.  I've been puzzling about the mechanism by which this decay conserves spin. I note via Wikipedia that  into either a lepton pair or a photon pair, for which it's easy to show spin is conserved, however pion production seems to induce a loss of $J=1$.  This seems to suggest momenta of:  $\Delta^{+} (J=\frac{3}{2}) \rightarrow{} p^{+} (J=\frac{1}{2}) + \pi^{0} (J=0)$  Am I missing a boson? Is my understanding of spin flawed? I assume that if this decay exists, similar things turn up everywhere else. Divergence of $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$, what is the 'paradox'? I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2} \equiv \frac{{\bf r}}{r^3}$, now from the book, Griffiths says:  Now what is the paradox, exactly? Ignoring any physical intuition behind this (point charge at the origin) how are we supposed to believe that the source of $\vec v$ is concentrated at the origin mathematically? Or are we forced to believe that because there was a contradiction with the divergence theorem? Also how would the situation differ if $\vec v$ was the same vector function but not for a point charge? Or is it impossible? Does Stack Exchange use caching and if so, how? Recently, I read this  about how they have some of their infrastructure laid out. I was curious to know, does Stack Exchange use caching (both at the application layer and/or database layer) and if so, how? What can i put into \setmathfont{&lt;...&gt;.otf} for the unicode-math package to make it all work (with XeLaTeX or LuaLaTeX)?  There are several suggestions on the  of unicode-math, but the first option is commercial, and the second and the third do not work with the following example:  % !TEX TS-program = xelatex  \documentclass{article} \usepackage{unicode-math} \setmathfont{latinmodern-math.otf} % \setmathfont{texgyrepagella-math.otf} % \setmathfont{xits-math.otf}  \begin{document} $X\setminus Y$ \end{document}   I do not get the \setminus character in the output.  The \setminus character is also missing if i do not set a math font at all (i do not know which font is used in this case).  It works with xits-math, but i do not like it.  Is it expected by the way that the \setminus character is left blank without giving any warning? Find all possible solutions of $$2^x + 3^y = z^2.$$    My approach.   First I substituted $x = 0$, and got the solution, then for $y = 0$. And for $x &gt;  0$ and $y &gt; 0$ , I just know the Pythagorean triple and got the solution as $(4,2,5)$ and $(4,2,-5)$. Please help me to solve it properly. How to find $\partial\chi^2/\partial a$ where $\chi^2=\sum_{i=1}^N\dfrac{D(x_i)-a-b(x_i)^2}{\sigma_i^2}$?  My attempt: \begin{align} \dfrac{\partial}{\partial a}\sum_{i=1}^N\dfrac{D(x_i)-a-b(x_i)^2}{\sigma_i^2} &amp;= \dfrac{\partial}{\partial a}\sum_{i=1}^N\dfrac{D(x_i)-b(x_i)^2}{\sigma_i^2}-\sum_{i=1}^N\dfrac{-a}{\sigma_i^2}\\ &amp;=\sum_{i=1}^N\dfrac{D(x_i)-b(x_i)^2}{\sigma_i^2}-\dfrac{\partial}{\partial a}\sum_{i=1}^N\dfrac{-a}{\sigma_i^2}\\ &amp;=\sum_{i=1}^N\dfrac{D(x_i)-b(x_i)^2}{\sigma_i^2}-\dfrac{1}{\sigma_i^2}\cdot\dfrac{\partial}{\partial a}\sum_{i=1}^N-a\\ &amp;=\sum_{i=1}^N\dfrac{D(x_i)-b(x_i)^2}{\sigma_i^2}-\dfrac{1}{\sigma_i^2}\cdot\dfrac{\partial}{\partial a}a^N\\ &amp;=\sum_{i=1}^N\dfrac{D(x_i)-b(x_i)^2}{\sigma_i^2}-\dfrac{1}{\sigma_i^2}\cdot N\cdot a^{N-1}\\ \end{align}  Is this correct?  EDIT: As per the accepted answer, this is how I got my final partial derivative. \begin{align}             \dfrac{\partial}{\partial a}\sum_{i=1}^N\dfrac{D(x_i)-a-b(x_i)^2}{\sigma_i^2} &amp;= \dfrac{\partial}{\partial a}\Bigg[\sum_{i=1}^N\dfrac{D(x_i)-b(x_i)^2}{\sigma_i^2}-\sum_{i=1}^N\dfrac{-a}{\sigma_i^2}\Bigg]\\             &amp;=\dfrac{\partial}{\partial a}\Bigg[\sum_{i=1}^N\dfrac{D(x_i)-b(x_i)^2}{\sigma_i^2}\Bigg]-\dfrac{\partial}{\partial a}\Bigg[\sum_{i=1}^N\dfrac{-a}{\sigma_i^2}\Bigg]\\             &amp;= 0-\dfrac{\partial}{\partial a}\Bigg[\dfrac{-a}{\sigma_1^2}+\dfrac{-a}{\sigma_2^2}+\cdots+\dfrac{-a}{\sigma_N^2}\Bigg]\\             &amp;= -\Bigg[\dfrac{\partial}{\partial a}\Bigg[\dfrac{-a}{\sigma_1^2}\Bigg]+\dfrac{\partial}{\partial a}\Bigg[\dfrac{-a}{\sigma_2^2}\Bigg]+\cdots+\dfrac{\partial}{\partial a}\Bigg[\dfrac{-a}{\sigma_N^2}\Bigg]\Bigg]\\             &amp;= -\Bigg[\dfrac{-1}{\sigma_1^2}\cdot\dfrac{\partial}{\partial a}[a]+\dfrac{-1}{\sigma_1^2}\cdot\dfrac{\partial}{\partial a}[a]+\cdots+\dfrac{-1}{\sigma_N^2}\cdot\dfrac{\partial}{\partial a}[a]\Bigg]\\             &amp;= -\Bigg[\dfrac{-1}{\sigma_1^2}+\dfrac{-1}{\sigma_2^2}+\cdots+\dfrac{-1}{\sigma_N^2}\Bigg]\\             &amp;= \sum_{i=1}^{N}\dfrac{1}{\sigma_i^2}\\         \end{align} How to find the joint PDF? Ley X,Y two random variables with density $f(x,y)=8xy \  \text{if} \   0&lt;x&lt;y&lt;1$  Find the joint distribution F(x,y).  I find that $F(x,y)=2x^2y^2-x^4\ \text{  if}  \ 0&lt;x&lt;y&lt;1$ but I dont know how to find the didtribution in other points. How do you prevent oatmeal from overflowing? Whenever I cook oatmeal (whether steel cut oats in a rice cooker or pan, or rolled oats in the microwave), it always overflows. How do you prevent this?   Do you just use a bigger container? Do you just take the pan off the heat or bowl off the microwave when it's about to overflow? I've also heard that if you put dried fruit or something like that along with the oatmeal while cooking, that helps prevent it from overflowing -- is this true? Azure Same Storage on VM I'm looking to create a simple short-lived reservation system, and I'd like to generate confirmation numbers that are   unique random-looking alphanumeric short-ish, at least much shorter than 32 character-long strings returned by sha1   I'm only looking to have ~500 reservations, so I don't imagine high likelyhood of collissions.  One idea I had is generate an sha1 hash based on a date-time stamp and username, then truncating it to its first 10 characters.  Would something like that be reliably unique enough for the purposes of processing ~500 reservations?