Say that $\Bbb N \times \Bbb N$ is the set of all pairs $(n_1, n_2)$ of natural numbers. Is it countable? My hypothesis is yes it is countable because sets are countable. But I am unable to come up with a proof. Is my hypothesis correct? How is it countable?
How can I prove that the Cartesian product of two countable sets is also countable?
I have a directory which contains numbered image files, something like this:  01.png 02.png 03.png 03.svg 04.png 05.png 06.jpg 07.png 08.png 09.png 09.svg 10.png   Sometimes there may be multiple versions of a file in different formats (eg. a png and svg version of the 03 and 09 files above) but the numbers are otherwise consecutive.  Typically there are 40-80 such files in each directory.  The numbers correspond to the order these images appear in a manuscript (a Word document, but that's not important).  There is no other way to determine the order of the images.  If I add a new image to the manuscript I need to place a copy of the image in this directory with the correct numbering.  So if the new image is the fifth in the manuscript I need to rename the files in the directory to this in order to make room for it:  01.png 02.png 03.png 03.svg 04.png 06.png 07.jpg 08.png 09.png 10.png 10.svg 11.png   What is the most straightforward way from the command line, or from a script or macro to renumber all the files starting at a certain number?  I have a standard Fedora Linux install using bash. Deriving the conditional distributions of a multivariate normal distribution We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning $\boldsymbol\mu$ and ${\boldsymbol Y}$ into  $$\boldsymbol\mu = \begin{bmatrix}  \boldsymbol\mu_1 \\  \boldsymbol\mu_2 \end{bmatrix} $$ $${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\  {\boldsymbol y}_2 \end{bmatrix}$$  with a similar partition of $\Sigma$ into $$  \begin{bmatrix} \Sigma_{11} &amp; \Sigma_{12}\\ \Sigma_{21} &amp; \Sigma_{22} \end{bmatrix} $$ Then, $({\boldsymbol y}_1|{\boldsymbol y}_2={\boldsymbol a})$, the conditional distribution of the first partition given the second, is  $\mathcal{N}(\overline{\boldsymbol\mu},\overline{\Sigma})$, with mean $$ \overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\Sigma_{22}}^{-1}({\boldsymbol a}-\boldsymbol\mu_2) $$ and covariance matrix $$ \overline{\Sigma}=\Sigma_{11}-\Sigma_{12}{\Sigma_{22}}^{-1}\Sigma_{21}$$  Actually these results are provided in Wikipedia too, but I have no idea how the $\overline{\boldsymbol\mu}$ and $\overline{\Sigma}$ is derived. These results are crucial, since they are important statistical formula for deriving Kalman filters. Would anyone provide me a derivation steps of deriving $\overline{\boldsymbol\mu}$ and $\overline{\Sigma}$ ? Thank you very much! In his first lesson (transcripted in "Tips on Physics"), Feynman talks about math for physicists in a very cool and practical way. And at the end of the section he talks something like "so the first thing to do is to learn to learn derivative, integral and algebra" (I don't know how much precise I'm being because I've read it in Portuguese). I would to know if there is some book that deals with math as Feynman did it in this lesson (respecting formalities, but teaching how to use the practical rules)? Also, someone have any recommendations for algebra book (college level)? I am trying to update Xcode to 8.0. My OS is 10.11.6 El Capitan. However I am getting:     We could not complete your update   There is not enough disk space available to install the product.   I have got 14.22 GB free space. How much do I need?  UPDATE: I do not know the exact figure, though I did manage to update with 17.21 GB free space later on. How do I get a tourist visa (or a tourist card) for Cuba?  I live in the Netherlands, but I'm not a Dutch citizen, I have an Italian passport. Faculty positions: What to say in a statement on diversity? I was trying to prove this inequality using induction, but couldn't do.  Question: Suppose $a$ and $b$ are real numbers with $0 &lt; b &lt; a$. Prove that if $n$ is a positive integer, then:  $$a^n-b^n \leq na^{n-1}(a-b)$$ One-sided Fisher's exact test and its complement What is the significance of the name Bleach? The name "Bleach" seems to be having no relevance to the plot unlike most other series. Was it just chosen at Kubo-sensei's whim or does it have some significance? Maybe some cultural significance associated with shinigami, etc. that I am now aware of? Ubuntu Software doesn't open in Ubuntu 20.04 If i click on Ubuntu Software, it doesn't open. I have a fresh install of Ubuntu 20.04. I tried try to upgrade all packages, but it doesn't fix this problem.  If I try to run killall software-center there is no process with this name, and software-center returns command not found. If I try to reinstall it with apt I have these results:  $ sudo apt-get --purge --reinstall install software-center Reading package lists... Done Building dependency tree        Reading state information... Done Package oracle-java8-installer is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source E: Package 'software-center' has no installation candidate The TweakUI powertoy for Windows XP had an option to auto-login. It stored your password and did the login for you. This was useful, since you kept password protection, but had login programs loaded when you first booted.  When you switched users or hibernated, it would still ask for a password, which is the desired behavior.  Is there a way of doing this in Windows 7? I'm a US passport holder. What documents do I need when I fly from the U.S. to Germany?  Assume I'm going for a few weeks of vacation.  Obviously I need my passport.  What about a "visa"?  Anything else? How do I coat popcorn with flavor? Omitting "and" in a sentence He called her, emailed her, texted her, tweeted her—all to no use.   Strictly speaking, I would need to write texted her and tweeted her, but I'm omitting and to convey a rhythm and sense of urgency. What do you call this kind of construction, and is it accepted grammar (at least for creative writing, if not in an academic context)? Has the concept of non-integer $(n+m)$-dimensional spacetime ever been investigated by theoretical physicists? The following image serves to aid the reader in understanding the "privileged character" of $3+1$-spacetime.    The wikipedia article on , and the  "The priveleged character of $3+1$-dimensional spacetime" in particular, made me think a bit about the possibility that we might live in a non-integer amount of spatial and/or time dimensions.   The notion of attaching a non-negative real number to a metric space has at least mathematically already been described by such concepts as "" and "".   This may sound silly/ignorant/absurd to professional practicing theoretical physicists. To me (a layman), however, it doesn't sound much stranger than the idea of wrapping up six extra dimensions (which is, from what I understand, considered to be a serious possibility by those who study $10$-dimensional String Theory) into intricate shapes called "Calabi-Yau Manifolds".   Has any research on $(a +b)$-dimensional spacetime (where $a,b \in \mathbb{R}_{\geq 0} $) ever been done? If so, what where the findings? If not, why not? Punctuation in an indirect quotation Consider the brief passage:     "I love your work, but calling you 'the artist' {1} just doesn't seem to be   cutting it anymore. What shall I call you?"      "OK. Call me George." {2}   So my instincts tell me to put {1} in single quotes as opposed to no punctuation (or perhaps even italics) but to do the opposite for {2}. Is this right? Why?  Let me expand this one further iteration:     "Wow, that's a great name, especially with this crowd. Maybe for your next book    you should call yourself Ringo."{3} In Python, is there a concise way of comparing whether the contents of two text files are the same? I don't care what the differences are.  I just want to know whether the contents are different. how to make winforms webbrowser control to work very similar to targeted Internet explorer If $p,q,r$ are all primes,and $p|qr-1$,$q|pr-1$ and $r|pq-1$,find all possible values of $pqr$. If $p,q,r$ are all primes,and $p|qr-1$$~~~~~~~~~~~~~$$q|pr-1$$~~~~~~~~~~$ and $~r|pq-1$. Find all possible values of $pqr$.    My work: $qr-1=pk_1$ $pr-1=qk_2$ $pq-1=rk_3$ From the above equations, we can conclude that either $k_1,k_2,k_3$ are all even or one of the primes is $2$. I can also get that, $p^2q^2r^2=(pk_1+1)(qk_2+1)(rk_3+1)$ Now, I cannot proceed. Please help! Center of a free product