Let $X$ be a set. What is $X\times \emptyset$ supposed to mean? Is it just the empty set?
If we have a function $f:A \rightarrow B$, then one way to give meaning, I think, to this function, in terms of set theory, is to say, that $f$ is actually a binary relation $f=(A,B,G_f)$, where $G_f \subseteq A \times B$ is the graph of the function. Now my question is: what is $f$ if   $\bullet \ A=\emptyset, \ B\neq\emptyset$,?  $ \bullet \ B=\emptyset, \ A\neq\emptyset$ ?  $ \bullet \ B=\emptyset, \ A=\emptyset$ ?  (Another way to formulate this, I think, would be: How do the sets $\emptyset\times B,\ A\ \times \emptyset, \ \emptyset \times \emptyset $ look like? Are they all $\emptyset$ ?)
Cancel a vote so that the tally goes back to zero and not minus Indefinite Integral with "sin" and "cos": $\int\frac{3\sin(x) + 2\cos(x)}{2\sin(x) + 3\cos(x)} \; dx $ From  by Richard Brualdi  We have a chess master. He has 11 weeks to prepare for a competition so he decides that he will practice everyday by playing at least 1 game a day. To make sure that he's able to take a couple of breaks, he also decides that he wont play more than 12 games per week.  My question is, how do you prove/dis-prove that there will be a succession of (consecutive) days in which he will play k games, $k \geq  22$?  Specifically, I don't quite understand the use of the pigeonhole principle here.  Thanks in advance! :D  EDIT  I'll try to clarify what I'm asking:  The goal is to prove that for every possible practice schedule that the chess master makes, there will always be a consecutive sequence of days such that he plays exactly k games.  So for example, if k = 22, how do you prove that no matter what his schedule is like, he will always play exactly k games in r days, $1 \leq r\leq  77$.  My main question is: what is the general method for proving different values of k? Are there values of k (from 1 to 132 inclusive) that are impossible to obtain?  Note that he does not necessarily have to play 132 games within 77 days! Also, keep in mind the bounds of k. Downcasting in Java Upcasting is allowed in Java, however downcasting gives a compile error.   The compile error can be removed by adding a cast but would anyway break at the runtime.   In this case why Java allows downcasting if it cannot be executed at the runtime? Is there any practical use for this concept?   public class demo {   public static void main(String a[]) {       B b = (B) new A(); // compiles with the cast,                           // but runtime exception - java.lang.ClassCastException   } }  class A {   public void draw() {     System.out.println("1");   }    public void draw1() {     System.out.println("2");   } }  class B extends A {   public void draw() {     System.out.println("3");   }   public void draw2() {     System.out.println("4");   } } What lens characteristics are important in star photography? Similar to Lifepod 1993, but not it This film is very similar to Lifepod 1993, but it takes place in a large ship, not a pod.  I can remember three characters... a technical engineer, being the one sabotaging the ship. And also this military cut character (who looks a lot like Kane from Mortal Kombat) and another girl (I think it was long, wavy black hair).  The "plot" twist that sends this one apart, at least back in the 1990´s, is that the ending comes to the fact there is only enough food and oxygen for one survivor and no escape pods or cryotubes, only a device that "could" work as a cryotube. For some reason that I cant remember the "Kane" dude chooses to die instead, by taking poison (which had been previously used on other surviving members). But the girl "tricks" him and drugs him instead, then places him in the "cryotube" and she stays awake to await for the rescue team.  Of course, they get salvaged, and he is still okay, surprised that she tricked him and saved him. Rings a bell for anyone? Is definitely a B-movie. I'd like to get a notification when a question that I've voted to close (but isn't closed yet) is edited How do we know the universe is expanding, and not that its contents are shrinking? Is there a physical reason why not to think that instead of space expanding, all physical constants and parameters are shrinking (including of course the instruments we use to measure the constants) and space is static, or is it a case of Occam's razor? Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$. How to use early stopping properly for training deep neural network? I have a deep neural network model and I need to train it on my dataset which consists of about 100,000 examples, my validation data contains about 1000 examples. Because it takes time to train each example (around 0.5s for each example) and in order to avoid overfitting, I would like to apply early stopping to prevent unnecessary computation. But I am not sure how to properly train my neural network with early stopping, several things I do not quite understand now:   What would be a good validation frequency? Should I check my model on the validation data at the end of each epoch? (My batch size is 1) Is it the case that the first few epochs might yield worse result before it starts converging to better value? In that case, should we train our network for several epochs before checking for early stopping? How to handle the case when the validation loss might go up and down? In that case, early stopping might prevent my model from learning further, right?   Thank you in advance. How to find the sum of the series $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$? How to find the sum of the following series? $$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n}$$ This is a harmonic progression. So, is the following formula correct? $\frac{(number ~of ~terms)^2}{sum~ of~ all ~the~ denominators}$ $\Rightarrow $ if $\frac{1}{A} + \frac{1}{B} +\frac{1}{C}$ are in H.P. Therefore the sum of the series can be written as : $\Rightarrow \frac{(3)^3}{(A+B+C)}$ Is this correct? Please suggest. Every time I touch an object a full-render starts I don't know what setting I switched, or what happened, but for some reason now every time I move, scale or rotate an object in the 3d View - blender starts a render. I am not talking about the 3d View being in "Rendered" mode. I am talking about the same behavior as when you click the "Render" button (switch to image editor, begin full render). This is driving me crazy and I can't seem to figure out how to turn it off or what I did to start it. It happens whether I use the mouse or keys. If I open a new document the behavior goes away so it is probably a preference setting(?) If you can help, I will be eternally grateful.  Thanks in advance. Infinite Cartesian product of countable sets is uncountable Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets and let $S=E_1\times\cdots\times E_n\times\cdots $. Show $S$ is uncountable. Prove that the same statement holds if each $E_n=\{0,1\}$.  By the definition of Cartesian product of sets,   $$\displaystyle S=\Pi_{n\in\mathbb{N}} \{f\colon\mathbb{N}\rightarrow\cup_{n\in\mathbb{N}}E_n\mid\forall n, f(n)\in E_n\}$$  If $E_n=\{ 0,1\}$, then  $$\displaystyle S_{01}=\Pi_{n\in\mathbb{N}}\{0,1\}=E^{\mathbb{N}}$$, where $E=\{0,1\}$.  By a theorem, $\cup_{n\in\mathbb{N}} E_n$ is countable since the sequence is countable.  I'm not sure how to go on from here to show $S$ is uncountable. Can we say anything about the function $f$ that maps a countable $\mathbb{N}$ to another countable union of sequence of countable sets? I faced this issue when I want to use sudo command :  sudo: /etc/sudoers is owned by uid 1000, should be 0 sudo: no valid sudoers sources found, quitting sudo: unable to initialize policy plugin   then I tried a solution from a similar question in order to change the owner back to the root:   $ pkexec visudo   also I tried this one   $ pkexec chown root:root /etc/sudoers /etc/sudoers.d -R   but I have faced another issue with these commands:   Error getting authority: Error initializing authority: Could not connect: No such file or directory   So is there a solution to this issue?    Full shell issue snippets:   ubuntu@LAPTOP-D4T16P7J:~$ sudo -i sudo: /etc/sudoers is owned by uid 1000, should be 0 sudo: no valid sudoers sources found, quitting sudo: unable to initialize policy plugin ubuntu@LAPTOP-D4T16P7J:~$ pkexec visudo Error getting authority: Error initializing authority: Could not connect: No such file or directory ubuntu@LAPTOP-D4T16P7J:~$ pkexec chown root:root /etc/sudoers /etc/sudoers.d -R Error getting authority: Error initializing authority: Could not connect: No such file or directory     This is my /etc/sudoers file:   ubuntu@LAPTOP-D4T16P7J:~$ cat /etc/sudoers # # This file MUST be edited with the 'visudo' command as root. # # Please consider adding local content in /etc/sudoers.d/ instead of # directly modifying this file. # # See the man page for details on how to write a sudoers file. # Defaults        env_reset Defaults        mail_badpass Defaults        secure_path="/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/snap/bin"  # Host alias specification  # User alias specification  # Cmnd alias specification  # User privilege specification root    ALL=(ALL:ALL) ALL  # Members of the admin group may gain root privileges %admin ALL=(ALL) ALL  # Allow members of group sudo to execute any command %sudo   ALL=(ALL:ALL) ALL  # See sudoers(5) for more information on "#include" directives:  #includedir /etc/sudoers.d     OS info:    Ubuntu WSL2. Wireless antenna does not appear despite its driver being installed Correct way to render crepuscular rays in cycles? While trying to use the new cycles volumetric rending features to create  (with the intention of updating ), I have found this to be a little trickier than I anticipated.    When I create a basic setup with a spot lamp (sun lamps don't seem to work, not sure if that is a result of their one directional nature or what..) and a world material like this:    Which gives me this (500 samples):    The problems with this that make me think that I'm missing something (aside from the rendertime and noise, which I understand to be a part of using bleeding edge new features) are:   The background shader/sky texture isn't visible at all (volume is too dense?) The lamp needed to be set to 50000 strength (this is of course dependent on the distance from the camera, but the kinds of strengths required to get some nice defined rays will blow everything out if volumes are disabled)   I can't seem to get a straight render (no compositing tricks) to look anything remotely like the pretty image in the release notes:        (source: )     Is this just how it works? Or is there some trick I have missed (not counting compositing/renderlayer tricks)? I have an iPhone 6 (with proper iTunes backup) where all of my contacts where, but I bought a new iPhone 7 and had to initialize it as new iPhone, due to several time-related constraints, so now I have a few but new contacts on my new phone, but none of the older ones, but would be the best practice oriented, for just recovering the old contacts from the iTunes backups?  PS: I still have the old iPhone 6, so it could be used as "temporary storage" if needed. How do I get an operating system onto a new hard drive? What is proper way to update D8 DEV? Is there an exact command for restarting GNOME Shell in Ubuntu 20.04 corresponding to Alt + F2 + 'r' + Enter? Is there a way to restart GNOME Shell other than by using Alt + F2, R + Enter? With respect to the question, I have gone through , but I didn't get what I was looking for. I am looking for a command which exactly does what Alt + F2, r + Enter do. Please note that I am not willing to do the gnome-shell -- replace command... I did check both the things Alt + F2,r Enter and gnome-shell --replace... There is a lot of difference, I found...