Is Cartesian product of two uncountable sets uncountable? Suppose we have a set of real numbers $R$, Can't it be shown that $R$ is uncountable by Cantor's diagonalization method, so it follows that the Cartesian product of two sets of real numbers in uncountable?
My hypothesis is that $\mathbb{R \times R}$,  the set of all pairs $(r_1, r_2)$, of real numbers is uncountable. I understand that the set of all pairs of natural numbers is countable. But could someone explain why the set of all pairs of real numbers uncountable? I am having trouble proving it using diagolization
Does python optimize modules when they are imported multiple times? If a large module is loaded by some submodule of your code, is there any benefit to referencing the module from that namespace instead of importing it again?  For example: I have a module MyLib, which makes extensive use of ReallyBigLib. If I have code that imports MyLib, should I dig the module out like so  import MyLib ReallyBigLib = MyLib.SomeModule.ReallyBigLib   or just  import MyLib import ReallyBigLib I am using the below script to move two days back when script runs at starting two days of the year and also check first and second days of every month and move two days back. if [$month=&quot;01&quot;] &amp;&amp; [$day=&quot;01&quot;]; then         date=&quot;$last_month/$yes_day/$last_year&quot;         fulldate=&quot;$last_month/$yes_day/$last_year&quot; else         if [$month=&quot;01&quot;] &amp;&amp; [$day=&quot;02&quot;]; then                 date=&quot;$last_month/$yes_day/$last_year&quot;                 fulldate=&quot;$last_month/$yes_day/$last_year&quot;         else                 if [ $day = &quot;01&quot; ]; then                         date=&quot;$last_month/$yes_day/$year&quot;                         fulldate=&quot;$year$last_month$yes_day&quot;                 else                         if [ $day = &quot;02&quot; ]; then                                 date=&quot;$last_month/$yes_day/$year&quot;                                 fulldate=&quot;$year$last_month$yes_day&quot;                         else                                 date=&quot;$month/$yes_day/$year&quot;                                 fulldate=&quot;$year$month$yes_day&quot;                         fi                 fi         fi fi  But my bad am getting the below error message Etime_script.sh: line 19: [06=01]: command not found Etime_script.sh: line 24: [06=01]: command not found how to adapt the sorting of a custom type `@standard` in using biblatex/biber Are objects in PHP passed by value or reference? What does it mean for something to hold "up to isomorphism"? When I try to install any package through the command line, I get an error.   $ sudo apt-get install &lt;package&gt; Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package &lt;package&gt;   Can anyone help me on this? How to preserve custom number formatting after closing a workbook? I have created a custom format ##,##0.0,,, CTRL+J %% in Excel 2010 to display  10000 as 0.1 100000 as 1.0 1000000 as 10.0 10000000 as 100.0 100000000 as 1,000.0 and so forth.  The problem I am facing is that once I apply this format, save and close the workbook, the next time I open it the format gets changed by multiplying by 10 (100000 is displayed as 10.0).  I don't understand how it is getting changed automatically the next time I open it. However, when I apply other formats they remain unchanged. Can someone use DC generator along with electric motor to make electricity along with mechanical energy? I know this might sound stupid and maybe stupid totally, but I need some clarification....  I came through electric motor principle.  and then I came across the following link on DC generators    Now my question is, can't we combine them both to provide DC current with little DC used in motor side? By "combine," I mean using a DC generator coil only at the other side of the coil of the motor in the motor itself. The condition is that the 2 coils are connected via some apparatus that causes the 2nd coil to move along with the 1st coil.   In this way we can use both the mechanical energy produced by the 1st coil and the electrical energy produced by the 2nd coil, right?   This is a great concept if it works, right? Please throw some light on this. Prove that for any square matrix, an invertible matrix B exists, so that BA is triangular I'm given a matrix A, its dimensions are n x n.  I am required to prove that an invertible matrix B exists, such that the product of the matrices BA is triangular.  Any help? Difference between "per" and "a" Is there any way to set a system wide memory limit a process can use in Windows XP? I have a couple of unstable apps which do work ok for most of the time but can hit a bug which results in eating whole memory in a matter of seconds (or at least I suppose that's it). This results in a hard reset as Windows becomes totally unresponsive and I lose my work.  I would like to be able to do something like the /etc/limits on Linux - setting M90, for instance (to set 90% max memory for a single user to allocate). So the system gets the remaining 10% no matter what. number of ways to make $2.00 How many different ways can you make $2.00 using only 1 cent, 5 cent, 10 cent, and 25 cent pieces, and 1 and 2 dollar bills (there are 100 cents in a dollar)? I have worked out an equation:  $$p + 5n + 10d + 25q + 100l + 200t = 200$$  Where all variables are non-negative integers. But I don't know what to do from here to figure out how many different sets of non-negative integers there are that satisfy this equation. What do I do next? Why can we always drop any constant term in a Lagrangian density in quantum field theory?  This issue is somehow related to the constant term being some kind of cosmological constant.  Can you please explain this issue? How to run adb and fastboot commands from Termux? UK Visa has been refused for false document I applied for a marriage visitor visa, to get married to my boyfriend of 2 years.   We met while we were students in the UK and have had vacations together in the two years but not in the U.K.   With the application I had attached an offer letter from a company stating that I have a job offer open starting 3 months later.   This however has been tagged as non genuine, although the company is registered and is run by someone I know.  How can I appeal to this? Let's say we have $2$ particles facing each other and each traveling (almost) at speed of light.    Let's say I'm sitting on #$1$ particle so in my point of view #$2$ particle's speed is (almost) $c+c=2c$, double light speed? Please say why I am incorrect :)    EDIT: About sitting me is just example, so in point of view of #1 particle, the second one moves at (almost) $c+c=2c$ speed? Is there any hint that the MoM or anywhere else got support in combating voldemort the first time around? Or is everything completely UK-centric?  I can't remember anyone in the second war being specifically sent from europe to help, even though we see international wizards in book 4. But I don't think all the events of the first war are covered in the books.  So are other bodies like the MoM involved in the first war, or was it isolated in the UK? (a) Solve the equation $yu_x+xu_y=0$  with the condition $u(0,y) = e^{-y^2}$. (b) In which region of the xy plane is the solution uniquely determined?     I did the first part but I don't understand how to do part b.  I read   But I don't understand how the argument is based. A more clear explanation is appreciated There is an UI issue for the ranking div when a user crossed 200 reputation When a user just crossed 200, or 200+ reputation, the ranked div (i.e., top 10% overall) is trying to display. But rank calculation can be done on end of the UTC day. So instead of the ranking div, there is a blue colored hyphen is visible to users, it may be the initial loader for the ranking div.  I saw this UI issue previously too for the same scenario. Can this be fixed? Algorithm for offsetting a latitude/longitude by some amount of meters