What is wrong with the followig proof that $\mathbb{Z}$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$
Cancellation Law for Direct Sums - What is wrong with this argument?
If $2^n - 1$ is prime from some integer $n$, prove that n must also be prime. I know that there are questions asking like "intersection of a infinite collection of sets" and I can understand that the answer for that one is a null set, but I got a question here, in which all sets are finite and nonempty. Please take a look at the pic below.  It's a True/False question.  What I'm confused about is that the "infinity" sign in the interception. Since all sets are finite, does it mean that there are many sets in the chain are the same? (because the symbol used is for subsets not for proper subsets). Please give me some hints how I should think about this question.  Thank you SEO scope for multiple domains hosting on the same IP address Debugging errors of SAGA interpolation algorithms in QGIS Processing? I encounter recurrent problems with the geoalgorithms that perform interpolations. No matter what technique I use I end with the following errors. As stated in the OGR tab QGIS 2.0 is unable to open a file in C drive. I have very little IT background so the message error is meaningless.   Is there anybody able to decrypt the message and solve the problem?      here this is the algorithm i used How to enable fast user switching when Login Options are disabled? Which is the rarest badge awarded/the least popular one/the one hardest to get? About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$ A Terms of Service update restricting companies that scrape your profile information without your permission Taking Seats on a Plane Are there any examples of Humans and Aliens producing offspring in Star Wars? The title is pretty self explanatory, although to expand on the question further I am looking for any examples of inter-species breeding in the Star Wars universe, not just inter-species (ahem) "relations". What are the criteria for questions to be selected for Hot Network Questions? I really like the , where certain questions from the Stack Exchange Network are presented, along with a hotness rating that is described as &quot;arbitrary&quot; in its tooltip. Such questions also appear randomly on the sidebar on sites across the network, under the heading &quot;Hot Network Questions&quot;. How do these arbitrary hotness points work, and how are they used to select questions to appear in those lists? Proof that $\inf A = -\sup(-A)$ Probability of success on third and fourth trials? I have a pretty basic probability question, but I'm just having difficulties remembering what distribution this is.  A coin is tossed until a head appears two times in a row. Given that we are using a fair coin, what is the probability that we toss the coin exactly 4 times such that the two consecutive heads are the 3rd and 4th trials?  I know that this would be fairly easy to solve by brute force and listing out the different probabilities, but I'm a little confused as to how to use a "shortcut" such as identifying this as a binomial distribution. I also don't know how to account for the probability that we are using 4 tosses.  Any help is appreciated. Thanks! Im confused about fire. The way I see it :   Heat creates (kinetic) energy in mass and this creates stronger vibrations of atoms. When those vibrations are strong enough the electrons interact stronger due to electromagnetic forces. This causes the electrons to fly away. This is the creation of a plasma and the direction of most electrons is the direction of the flame. the shape and heat of the flame depends mainly on the amount of electrons and their velocity ( apart from the source of heat and the considered gas and solid material) the electrons move freely ( but not random ) in the flame. there is a chemical reaction going on that usually produces smoke or such. the entropy increases in all the above 6 steps and the process is repeated. the electromagnetic energy is converted into kinetic energy of the electron and conductivity in the plasma (flame). the moving electrons create an electromagnetic field.   That is more or less how I see it. What steps are ok ?  And the main question , why is there light involved ?  I mean why cant we just have kinetic energy of the electrons and a changing magnetic field ? Why are there photons ? If the electrons are in a high energy state why cant they just move faster instead of emitting a photon ???  Why is the light usually visible ? is that because of our structure of air ?  If the light is an electromagnetic wave why does the flame go up ? does that imply all electromagnetic waves go up ?? But electromagnetic waves are not influanced by gravity or buoyancy are they ??  What is the final destiny of the electrons and photons if the heat is gone ??  I am puzzled by this. If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$? Does $\{f_ng_n\}\to fg$ uniformly? Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on I have a scenario. (Windows Forms, C#, .NET)   There is a main form which hosts some user control. The user control does some heavy data operation, such that if I directly call the UserControl_Load method the UI become nonresponsive for the duration for load method execution. To overcome this I load data on different thread (trying to change existing code as little as I can)   I used a background worker thread which will be loading the data and when done will notify the application that it has done its work.   Now came a real problem. All the UI (main form and its child usercontrols) was created on the primary main thread. In the LOAD method of the usercontrol I'm fetching data based on the values of some control (like textbox) on userControl.    The pseudocode would look like this:  CODE 1  UserContrl1_LoadDataMethod() {     if (textbox1.text == "MyName") // This gives exception     {         //Load data corresponding to "MyName".         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage.     } }   The Exception it gave was     Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on.   To know more about this I did some googling and a suggestion came up like using the following code  CODE 2  UserContrl1_LoadDataMethod() {     if (InvokeRequired) // Line #1     {         this.Invoke(new MethodInvoker(UserContrl1_LoadDataMethod));         return;     }      if (textbox1.text == "MyName") // Now it wont give an exception     {     //Load data correspondin to "MyName"         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage     } }   BUT BUT BUT... it seems I'm back to square one. The Application again  become nonresponsive. It seems to be due to the execution of line #1 if condition. The loading task is again done by the parent thread and not the third that I spawned.  I don't know whether I perceived this right or wrong. I'm new to threading.  How do I resolve this and also what is the effect of execution of Line#1 if block?  The situation is this: I want to load data into a global variable based on the value of a control. I don't want to change the value of a control from the child thread. I'm not going to do it ever from a child thread.   So only accessing the value so that the corresponding data can be fetched from the database. How can I explain to people that the phrase “off of ” is grammatically incorrect? How can I explain to people that the phrase off of is grammatically incorrect?  I‘ve heard this phrase used a lot, especially by Americans (though they aren't the only ones).    In my understanding, off of should usually be replaced by off, as in,     I took the book off the shelf   as opposed to     I took the book off of the shelf   Am I wrong?  Or is there perhaps some simple way I can explain this to most people? If I was at point $A$ and I wanted to walk directly to point $B$, I would have to walk half way to point $B$, but before that I would have to walk half way to halfway to halfway to point $B$ and half of that again and so on and so fourth. if I halved this distance an infinite amount of times then there would be an infinite amount of actions I would need to perform in order to cross from $A$ to $B$. Therefore, if each action took any quantity of time at all, then it would take me an infinite amount of time to cross from $A$ to $B$, even if $A$ and $B$ were only a few centimetres apart!   Because I am not infinitely old and I can move, there must be a flaw in this logic. Where is it? Units and Nilpotents If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit.   I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) = 1 = (u+a)x$. After tried several elements and manipulated $ua = au$, I still couldn't find any clue. Can anybody give me a hint?