I am trying to prove this (it looks true to me) :  Let $A,B $ be two sets. If there is a surjection $A\to B $ and a surjection $B\to A$ then $A $ and $B $ are in bijection.  I showed that is it equivalent to the following statement :  If there is an injection $A\to B $ and an injection $B\to A$ then $A $ and $B $ are in bijection.  But I am stuck, I don't see how to prove either.
Let $A,B$ be two sets. The  states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$.  I was wondering whether the following statements are true (maybe by using the AC if necessary):   Suppose $f \colon A\to B$ and $g\colon B\to A$ are both surjective, does this imply that there is a bijection between $A$ and $B$. Suppose either $f\colon A\to B$ or $g\colon  A\to B$ is surjective and the other one injective, does this imply that there is a bijection between $A$ and $B$.
Is there a streaming API for JSON? Declaring static data members of normal class and class template The positive integer solutions for $2^a+3^b=5^c$ What are the positive integer solutions to  the equation     $$2^a + 3^b = 5^c$$    Of course $(1,\space 1, \space 1)$ is a solution. I saw this keyword for the first time and I was wondering if someone could explain to me what it does.   What is the continue keyword? How does it work? When is it used? Can I read and write to the same file in Linux without overwriting it? Can the default font be changed? I have been playing with the settings for the text object recently, and every time I add a new text object, I need to change the font to my favorite one. (Eurostile LT ExtendedTwo) Is there any way to change the default font so I do not need to change it every time I add a new text object?   Example: Prove a graph Containing $2k$ odd vertices contains $k$ distinct trails How to debug a Custom Timer Job in SharePoint 2010? I have a geo-tiff created by Pix4D which is about 375 Mb in file size.   I have clipped it to a smaller area using Qgis 2.8(using Raster ‣ Extraction ‣ Clipper), and the output file is about 900 MB in size.  Why did the clipping process generate such a large file, when the source file was smaller?   How do I run the clipper tool so that it produces a smaller size? When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes? I've been considering the rings $R_1=\mathbb{F}_p[x]/(x^2-2)$ and $R_2=\mathbb{F}_p[x]/(x^2-3)$, where $\mathbb{F}_p=\mathbb{Z}/(p)$.   I'm trying to figure out if they're isomorphic (as rings I suppose) or not for primes $p=2,5,11$.   I don't think they are for $p=11$, since $x^2-2$ is irreducible over $\mathbb{F}_{11}$, so $R_1$ is a field. But $x^2-3$ has $x=5,6$ as solution in $\mathbb{F}_{11}$, so $R_2$ is not even a domain.  For $p=5$, both polynomials are irreducible, so both rings are fields with $25$ elements.  I know from my previous studies that any two finite fields of the same order are isomorphic, but I'm curious if there is a simpler way to show the isomorphism in this case, without resorting to that theorem.  For $p=2$, neither ring is even a domain as $x=0$ is a solution of $x^2-2$ and $x=1$ is a solution for $x^2-3$, but I'm not sure how to proceed after that. Thank you for any help. Why do the Walking Dead Zombies only show intelligence in the very first episode? I enjoy The Walking Dead, however one question keeps nagging me. The 'Walkers' are always shown to not have any intelligence or memory of having been human. This is for all seasons except the very first one.  In the very first scene of Season One, Episode One, we see a little zombie girl stop to deliberately pick up a teddy bear from the ground.   Later in the same episode we see a zombie apparently try to turn the door knob to get into Morgen Jones' house.  Also at Atlanta they can climb fences and used rocks to break glass.  After season 1 I do not recall seeing any such intelligence demonstrated by a zombie again.  My question is: are these initial displays of intelligence a discrepancy in the story, or is there a plausible explanation why the Walkers performed these actions? Given:  DateTime.UtcNow   How do I get a string which represents the same value in an -compliant format?  Note that ISO 8601 defines a number of similar formats. The specific format I am looking for is:  yyyy-MM-ddTHH:mm:ssZ Do I need to change my ESTA address when visiting the US? How can I search for special characters? Situation: It started out as me  for the literal string C-c in  but it ended up omitting the - sign and showing results which contained the strings C/C++ and C &amp; C++.  is an example of a post which contains that string of characters (yet fails to show up in the results).  Things I've tried:   post suggests that I can use the undocumented code: operator, but when I do that, it just wraps the whole term in double quotes like .  As @terdon in Unix &amp; Linux chat has , using this  does yield some relevant results, but it doesn't show me  that is supposed to be there. edit: It appears that the Symbolhound query no longer returns that result.  Finally,  says that you can search for literal characters, but as you can see:    Quoted phrases are exact matches except for case-sensitivity, for example, you can search  or .  The second example (symbols) leads to 0 results.  So by now I assume that the search engine has changed (but the old documentation still lingers on this site). Is there a method for me to search for strings that include special (or is it literal?) characters? Fonts larger than \Huge? How can I make text that is larger than the size of the output of {\Huge ...}?   I would like to be able to make text arbitrarily large (even if that is done by some suboptimal scaling routine). I have no idea how to prove it by using only what the book has talked about so far. Can anyone help? The proof shouldn't be using set theory as set theory is only mentioned in the following chapter. The proof should only make use of the addition of natural numbers, order properties of natural numbers, the trichotomy of order for natural numbers and principle of induction.  Proposition 2.2.14 (Strong principle of induction). Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number $m$. Suppose that for each $m ≥ m_0$, we have the following implication: if $P(m')$ is true for all natural numbers $m_0 ≤ m' &lt; m$, then $P(m)$ is also true. (In particular, this means that $P(m_0)$ is true, since in this case the hypothesis is vacuous.) Then we can conclude that $P(m)$ is true for all natural numbers $m ≥ m_0$. Exercise 2.2.5. Prove Proposition 2.2.14. (Hint: define $Q(n)$ to be the property that $P(m)$ is true for all $m_0 ≤ m &lt; n$; note that $Q(n)$ is vacuously true when $n &lt; m_0$.)  Thanks! prove inequality through mean value theorem put (a) and (b) to one in subfigures Protected Question does not show Add an Answer button This question is protected. To answer it, I must have at least 10 reputation.        Ignoring the association bonus of 100, I should .  Yet, the Android app does not show the blue "Add an Answer" button at the bottom. The app also does not show the protection status.  If I view the question in the desktop or mobile browser (and I'm signed in), I can add an answer there. How to iterate over a comma separated list? How can I iterate, loop, process, or otherwise do something, with each element from a comma separated list of things?  This question       , and the are already lots of very good answers everywhere. However I think it would also be valuable to have a single reference point where all such alternative techniques are explained in detail, with their advantages and disadvantages discussed.  All of plain-TeX, LaTeX, LaTeX3, LuaTeX, Context, etoolbox, pgffor, etc. answers are invited. Please consider explaining, for example:   How does the command/method deal with whitespace around items? How does the command/method deal with empty/missing items? (e.g. a,b,,c) How does the command/method deal with a trailing comma? Can the command/method work of a list stored on a macro? Is each iteration evaluated globally or on a local group? (i.e., if a command is defined during an iteration, does the command survive the loop?) Do items get expanded or somehow mangled before being processed? What about lists where items are key=value or some other kind of pairs? Does the method work for lists separated with something other than a comma? Any other thing I should know before deciding to use this method?