Why cardinalities inequality defined that way? This is the definition of $|A| &lt;= |B| $: there exists an injective function from $A$ into $B$. Why it is defined that way and not like the following proposal, which is equivalent under the assumption of the axiom of choice: there exists a surjective function from $B$ into $A$.
In ZF, how would the structure of the cardinal numbers change by adopting this definition of cardinality? In ZFC, a good way of ordering sets by cardinality is by leveraging the notion of an injection. We define: $$X \lesssim Y \leftrightarrow \mbox{ there exists an injection } X \rightarrow Y.$$  Alternatively, we can leverage the notion of a partial surjection, by which I mean a partial function that hits every point of its codomain at least once.  $$Y \succsim X \leftrightarrow \mbox{ there exists a partial surjection } Y \rightarrow X.$$  Now in ZFC, these approaches are equivalent, in the sense that $$X \lesssim Y \iff Y \succsim X.$$  But if we just consider ZF (that is, we drop choice) then the equivalence breaks down. However, we've still got the forward implication, because the converse of an injection is always a partial surjection.  A specific question. Define that two sets are weakly equipotent iff $X \succsim Y$ and $Y \succsim X$. Question: does weak equipotence imply genuine equipotence? I'm guessing "no."  A general question: In broad terms, how would the structure of the cardinal numbers in ZF be altered by adopting $\succsim$ rather than $\lesssim$ as the fundamental order relation, and taking equality of cardinal numbers to correspond to weak equipotence of sets? I'm guessing this would hide a lot of the (usually very complicated) structure of the cardinal numbers in ZF.  Remark. The order symbols can be written \lesssim and \succsim.
Earlier, I voted to close  as off-topic.  The question has since been closed.  There is an , but it didn't show up in the list when I selected "off-topic".  The options were:   Meta Stack Overflow Server Fault Super User Webmasters Programmers   Questions:   Should Wordpress appear in this list? Are there any other sites that should appear in the list?   Should all non-beta sites appear in that list by default? How do you find the distance from a point to a plane? I am having trouble with this:      Find the distance from the point $(1,1,1)$ to the plane $2x+2y+z=0$.   Any ideas? Thanks. Are there any downsides to passing structs by value in C, rather than passing a pointer? Are there any downsides to passing structs by value in C, rather than passing a pointer?  If the struct is large, there is obviously the performancd aspect of copying lots of data, but for a smaller struct, it should basically be the same as passing several values to a function.  It is maybe even more interesting when used as return values. C only has single return values from functions, but you often need several. So a simple solution is to put them in a struct and return that.  Are there any reasons for or against this?  Since it might not be obvious to everyone what I'm talking about here, I'll give  a simple example.  If you're programming in C, you'll sooner or later start writing functions that look like this:  void examine_data(const char *ptr, size_t len) {     ... }  char *p = ...; size_t l = ...; examine_data(p, l);   This isn't a problem. The only issue is that you have to agree with your coworker in which the order the parameters should be so you use the same convention in all functions.  But what happens when you want to return the same kind of information? You typically get something like this:  char *get_data(size_t *len); {     ...     *len = ...datalen...;     return ...data...; } size_t len; char *p = get_data(&amp;len);   This works fine, but is much more problematic. A return value is a return value, except that in this implementation it isn't. There is no way to tell from the above that the function get_data isn't allowed to look at what len points to. And there is nothing that makes the compiler check that a value is actually returned through that pointer. So next month, when someone else modifies the code without understanding it properly (because he didn't read the documentation?) it gets broken without anyone noticing, or it starts crashing randomly.  So, the solution I propose is the simple struct  struct blob { char *ptr; size_t len; }   The examples can be rewritten like this:  void examine_data(const struct blob data) {     ... use data.tr and data.len ... }  struct blob = { .ptr = ..., .len = ... }; examine_data(blob);  struct blob get_data(void); {     ...     return (struct blob){ .ptr = ...data..., .len = ...len... }; } struct blob data = get_data();   For some reason, I think that most people would instinctively make examine_data take a pointer to a struct blob, but I don't see why. It still gets a pointer and an integer, it's just much  clearer that they go together. And in the get_data case it is impossible to mess up in the way I described before, since there is no input value for the length, and there must be a returned length. I have a USB flash drive which is no longer recognized by my computer. Windows Disk Management and DiskPart report No Media with no storage space (0 bytes) on the drive and I cannot partition or format the drive:     If the drive appears in Windows Explorer, trying to access it returns an error message indicating that there is no disk inserted, such as the following:     Please insert a disk into drive X:.   Various disk partitioning and data recovery utilities don't recognize the drive or only give a generic name for the drive and cannot access the contents of the drive.  What can I do? How do I recover data from the drive?     This question comes up often and the answers are usually the same. This post is meant to provide a definitive, canonical answer for this problem. Feel free to edit the answer to add additional details. After some searching, I find very little on the incorporation of observation weights/measurement errors into principal components analysis. What I do find tends to rely on iterative approaches to include weightings (e.g., ). My question is why is this approach necessary? Why can't we use the eigenvectors of the weighted covariance matrix? We have a situation: someone saw the boy break the window. Can I make this passive sentence?     The boy was seen to break the window.   I use Murphy's Grammar and this structure is never discussed. Randomly populating procedural shape (or Image Texture) in Cycles I've got a qml file and I just want to ssh to the phone and do  qmlscene myfile.qml   Is it possible without doing all that stuff with SDK and packages? Just vim and some simple commands that can fit into person's memory. Cross-site AJAX requests I need to make an AJAX request from a website to a REST web service hosted in another domain.  Although this is works just fine in Internet Explorer, other browsers such as Mozilla and Google Chrome impose far stricter security restrictions, which prohibit cross-site AJAX requests.  The problem is that I have no control over the domain nor the web server where the site is hosted. This means that my REST web service must run somewhere else, and I can't put in place any redirection mechanism.  Here is the JavaScript code that makes the asynchronous call:  var serviceUrl = "http://myservicedomain"; var payload = "&lt;myRequest&gt;&lt;content&gt;Some content&lt;/content&gt;&lt;/myRequest&gt;"; var request = new XMLHttpRequest(); request.open("POST", serviceUrl, true); // &lt;-- This fails in Mozilla Firefox amongst other browsers request.setRequestHeader("Content-type", "text/xml"); request.send(payload);   How can I have this work in other browsers beside Internet Explorer? Indefinite integral of $ \int \frac{x^3}{\sqrt{x^2+1}} \,\text{d}x$. Can you please provide any sort of hint or suggestion on how to find the following indefinite integral?  $$\int\frac{x^3}{\sqrt{x^2+1}}\text{d}x$$  I tried substituting everything but it didn't work. I also tried trigonometric substitution but I couldn't find any valid trigonometric identity for $ \sqrt{\cos^2(x)+1} $ or $ \sqrt{\sin^2(x)+1} $. Graphing the amount of time a function takes How do I install Adobe Flash player? I have had significant problems with watching flash video in 64-bit Ubuntu.  Does anyone know of a good way to get flash running on the platform? How to create a page like this? I'm a LaTeX newbie. I want to create a page layout like in the picture below on every page.  Can someone help please? Thanks. How to navigate to C drive in bash on WSL-Ubuntu? For what real values of $a$ does $\sum_{n=1}^{\infty}{(\frac{an}{n +1})^n}$ converge? For what real values of $a$ does the series $$\sum_{n=1}^{\infty}\left(\frac{an}{n +1}\right)^n$$ converge?  At first, I wanted to do the root test, but the problem is that $a$ could be negative.  If I take it out of the fraction to get $$\left(\frac{a}{n+1}\right)^n$$ it doesn't meet the criteria of either the Dirichlet or the Abel tests. What is the standard team composition for a conquest game? I'm very new to Smite, and I'm wondering if there is some kind of standard team composition in high level play, comparable to the top, jungle, mid, dual lane meta in LoL.  What are the roles, and where should they go to start the game? Gmail App in my Android device shows correctly all the labels that I created using Gmail through web interface.  Can I create new labels in Gmail on Android device using Gmail App?   Also, ideally, once I create these new labels in Gmail App, I obviously should be seeing the same in Web Interface of Gmail too (Sync!!).  Any ideas? Can the ship's captain really perform your wedding when in international waters? On , Mark Mayo posted a comment that any marriage has to be on dry land or close to it.  But I'm sure all of us have heard of ships' captains marrying couples at sea.  From the movie Casablanca:     Say, why don't we get married in Marseilles?   ...   Well, let's see. What about the engineer? Why can't he marry us on the train?   ...   Well, why not? The captain on a ship can. It doesn't seem fair that...    So is this fiction after all? If not how does it fit with Mark's comment? What special rules and such should we know?  If I'm travelling at sea with a significant other and wish to tie the knot, can we do so with any captain on any ship on any sea? Must it be international waters? How to download all required Ubuntu drivers How can I get maximum DoF with no distance markers? I am aware of what hyperfocal distance is and the general principles behind it. However, most of the tutorials I have seen and read regarding it make use of lenses with focus distance indicators. Unfortunately, neither of my lenses (Nikkor 18-55mm &amp; 35mm f1.8) have indicators.  I have read that you can 'cheat' and simply focus a third of the way into the shot (which I have done previously simply by spot-focusing on the lowest focus point in my VF). I also assume you could pace out the distance and note an object to focus on when you get back to the camera (though that might be easier said than done).  Does anyone have any suggestions on how to achieve maximum sharpness from front to back without focus distance markers. A new lens is not an option; global economic crisis and all that!