How is $\mathbb{Q}$ countably infinite?   The definition says all elements of the set must have a one-to-one relation to the natural numbers. I do not understand this.  How do the elements in $\mathbb{Q}$ have a one-to-one relation with natural numbers?
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such a bijection?
How do I control Macbook volume when a DP display is connected? When I plug my Mac  into a DisplayPort monitor that has loudspeakers, the Mac's volume controls become disabled.    Is Apple assuming that if I have external speakers I am so much better using their own volume control that I have to be forced to do it?   That is annoying because the monitor's volume controls are buried in terrible menus accessed by terrible buttons on the back of the monitor.  Also, if I am inside an app where I can select the audio device and I select the Mac's internal speakers, the volume controls are STILL disabled.  Is there no way to control the volume of an app when an external speaker is connected even if the app is not using the external speaker?????  Some additional info: It's a Macbook Pro 13" 2018 version.  Using a Thunderbolt to Displayport cable, plugging in to a Dell monitor with speakers.    In another location I use the same cable to plug in to Dell monitors with Displayport but without built in speakers in the monitors.   At that location the volume controls remain active and they control the internal speaker in the laptop as they should. How to parse a month name (string) to an integer for comparison in C#? I need to be able to compare some month names I have in an array.  It would be nice if there were some direct way like:  Month.toInt("January") &gt; Month.toInt("May")   My Google searching seems to suggest the only way is to write your own method, but this seems like a common enough problem that I would think it would have been already implemented in .Net, anyone done this before? Find the limit: $\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$ Callback after the record was created using e.force:createRecord Are there a way to add callback after the user created a record and clicked the Save button?  I am using a custom tab and custom components and open the Create a record modal window using e.force:createRecord, but I was not able to find out how to attach callback to the event.  I even found :     Currently force:createRecord event does not accept any callback method or any after save event to handle custom action. Current force:createRecord always redirects to the newly created record. But in case if you want to just close the force:createRecord popup after save and refresh some other related component using the refresh is not possible.    So, is it really impossible? If so, what are current workarounds here? How to use a flyback diode on a +/-12V motor I've got a motor hooked up to +12V on one side and -12V on the other, being driven in both directions by a Motor controller (). We were getting voltage spikes in the common ground after sending big impulses to the motor, so we put a flyback diode from the negative cable to the positive cable. However, this caused the motor to be less responsive in the reverse direction. How do we correctly prevent voltage spikes in the ground from this type of motor? On average, how many times must I roll a dice until I get a $6$?  I got this question from a book called Fifty Challenging Problems in Probability.   The answer is $6$, and I understand the solution the book has given me. However, I want to know why the following logic does not work: The chance that we do not get a $6$ is $5/6$. In order to find the number of dice rolls needed, I want the probability of there being a $6$ in $n$ rolls being $1/2$ in order to find the average. So I solve the equation $(5/6)^n=1/2$, which gives me $n=3.8$-ish. That number makes sense to me intuitively, where the number $6$ does not make sense intuitively. I feel like on average, I would need to roll about $3$-$4$ times to get a $6$. Sometimes, I will have to roll less than $3$-$4$ times, and sometimes I will have to roll more than $3$-$4$ times.  Please note that I am not asking how to solve this question, but what is wrong with my logic above.  Thank you! Many C++ books contain example code like this...  std::cout &lt;&lt; "Test line" &lt;&lt; std::endl;   ...so I've always done that too. But I've seen a lot of code from working developers like this instead:  std::cout &lt;&lt; "Test line\n";   Is there a technical reason to prefer one over the other, or is it just a matter of coding style? How to unlock gungeoneers I've seen on a wiki that there is a ghost gungeoneer and a robot gungeoneer but it didn't say how to unlock them, so how do I unlock more gungeoneers ? When used like this:  import static com.showboy.Myclass;  public class Anotherclass{}   what's the difference between import static com.showboy.Myclass and import com.showboy.Myclass? I’m not sure What it is? I’m trying to get a glossy shade to building in my scene as soon as I put roughness to zero I see these trees in glass. I have no trees in my scene anywhere, background is black. Even when I render I see these trees. How can I get rid of them and still keep my glossy building material. How to call a space in which every G$_\kappa$-set is open Recall that in a topological space $X$, we say that a subset $A\subset X$ is a G$_\delta$-set if $A$ is the intersection of countably many open sets of $X$. Also, we say that $X$ is a  if G$_\delta$-sets are open.  Now, call as G$_\kappa$-set the intersection of $\kappa$ many open sets, where $\kappa\geq\aleph_0$ is a cardinal. I would like to know if there is some standard name for those spaces in which G$_\kappa$-sets are open, for a fixed cardinal $\kappa\geq \aleph_0.$  Motivation: this kind of space arose in my PhD thesis, but I do not know any references in which they appear. I'm temporary calling them as $P_\kappa$-spaces. Properties of p-adic valuations Let $n$ be a integer not equal to zero, $p$ a prime and $$|n|_p = p^{-\operatorname{ord}_p(n)}; |0|_p=0$$  Proposition: $|xy|_p=|x|_p|y|_p$  We want to show: $\operatorname{ord}_p(ab)=\operatorname{ord}_{p}(a)+\operatorname{ord}_p(b)$ So if we suppose there is a largest integral $v$ such that $p^{v}|(ab)$, then we can split up v such that $p^{v_1}|a,p^{v_2}|b$. If we suppose that there exist $v_1,v_2$ with them both being the largest integers such that $p^{v_1}|a, p^{v_2}|b$,  with $v_1+v_2= : v $ then $p^v|ab$. (I don't believe this to be a formal proof but rather my best own "idea" I can come up with...)  The script also mentions that the padic valuations' sums are bounded: $|x+y|_p\le \sup\{|x|_p,|y|_p\}$, how to show this?   Edit:   According to Robert Israel and Bill Dubuques answers, the first proposition is proven, if I understood correctly:   Given $ord(a)= v$ and $ord(b) = u$ which is the same as $a:= p^{v}m, b:= p^{u}n$ with $gcd(m,p)=gcd(n,p)=1$, because $ab=p^{u+v}mn$ with $gcd(mn,p)=1$ it follows that $ord(ab)=ord(a)+ord(b)$ and so $|xy|_{p}=p^{-ord(xy)}=p^{-(ord(x)+ord(y))}=|x|_{p}|y|_{p}$  Now to show: $|x+y|_{p}\le sup\{|x|_{p},|y|_{p} \}$  and the second can be tackled like this: set $a:=p^{v}m. b:=p^{u}n$, then checking two cases $1:(u=v)$ and $2:(v&lt;u)$: $(a+b)=p^{v}m+p^{u}n= p^{v}(m+p^{u-v}n)$ so in case of (u=v) it follows that  $ord(a+b)= u = sup\{ord(a),ord(b)\}$  and in case of $(v&lt;u)$ because the biggest v is $v:= u-1$: $ord(a+b)\le p^{u-1}(m+pn) \le p^{u}(m+n)= sup\{ord(a),ord(b)\}$  It seems that I did something wrong in both cases... This question exists as it fills a specific criterion.  While you are encouraged to help maintain its answers, please understand that "big list" questions are not generally allowed on Ask Ubuntu and will likely be closed per the . More information on the .   What kind of desktop environments or shells are there for Ubuntu users to install?  Please list a desktop environment or shell for each:   a description on why you like or suggest it (features, performance, etc.). a good screenshot, preferably of it running on Ubuntu and showing off some of its features. the minimal requirement for it to be usable, If there's any setting to lower it's requirement (Like kde's low-fat profile); how to enable the said setting some instructions on how to set it up if in the repositories, please provide a software centre link "Screaming" noise while steaming milk? Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$.  Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} $$  Show that:  1) $f$ is strictly increasing  2) $f$ is continuous at $x$ iff $x \in \mathbb{R} \setminus \mathbb{Q}$.  My work:  Let $x, y \in \mathbb{R}$, $x &lt; y$. Then there exists $q_n \in \mathbb{Q}$ s.t. $x &lt; q_n &lt; y$. Now $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} &lt; \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} + 2^{-n} \leq \underset{q_k \leq y}{\sum_{k \in \mathbb{N}}} 2^{-n} = f(y) $$ so $f$ is strictly increasing.  I think my proof of 1) is ok, but how can I prove 2)? I just did a clean install of 20.04 and I noticed that no key shortcuts for files manipulation work when my focus in on Desktop (not in Nautilus).   When I have copied a file to clipboard, I cannot Ctrl+V it to Desktop, when I click on a file on Desktop, I cannot Ctrl+C it, or I cannot Delete it with keys. Only ways to do it is either navigate to the Desktop folder in Nautilus, or right-click the file on the Desktop and copy/move to trash/paste...  Sorry if I'm missing something obvious, all the key shortcuts worked normally for me in 18.04. Word for resembling the truth How do we refer to the act of resembling the truth, regardless if it is indeed the truth?  Some context: suppose you are watching a film and the story bears a striking resemblance to "real life". Perhaps you remember similar events from your own background or the story has a strong internal coherence. So you feel that the story could be true, even though you know it is fictional. Or, as the legend has it, when Michelangelo finished his "Pietà" and the sculpture seemed so "real" and "natural" to him, that he was amazed that the sculpture couldn't talk. One last example: you hear the promises of a politician in election campaign and he seems to be telling the truth, but of course you have no way to figure out, beforehand, if he is being honest or not.  The word "likelihood" comes to mind, but this word suggests some kind of numerical (even statistical) evaluation, which is akin to "probability". The word I am looking for describes the feeling you get, which leads to a subjective "binary" judgement (either it resembles the truth or not).  In Portuguese, such word is "verossimilhança" (from Latin verum, truth + similis, similar). Showing this metric space is complete Let $X=(0,1]$ and $d(x,y)=\left|\frac{1}{x}-\frac{1}{y}\right|$. I've proven $(X,d)$ is a metric space but I don't know how to show its completeness. How can I do that? How to create an interactive bash script to collect arguments for a command in linux? I'm trying to create an interactive bash script where I can collect user input variables to be used as arguments in another command. Can someone provide me an example of this?  Currently i want to do:   what type of content you want to delete? what is the status?   then feed it into a mysql delete query command. In Python, how do I get a function name as a string, without calling the function?  def my_function():     pass  print get_function_name_as_string(my_function) # my_function is not in quotes   should output "my_function".  Is such function available in Python? If not, any ideas on how to implement get_function_name_as_string, in Python?