Why does a rainbow only form with a 42° angle?
Why do rainbows have distinct colors?
Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally:     $$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$   If this statement is true, what is the proof ? Find all functions $f$ such that $f:\mathbb{N}\rightarrow\  \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$   Let us plug in $n=1$  $f(f(1))+f(2)=3$  Since the function is from $\mathbb{N}$ to $\mathbb{N}$, $f(2)$ can only take the values $1,2$. Now we divide the problem into cases.  Case-1: $f(f(1))=2,f(2)=1$  We can assume that $f(1)=c$ for the time being. Then plugging in $n=3$ and using $f(2)=1$ gives $$f(3)=4-c$$ and again since the range of the function is positive integers,then $4-c$ has to be positive and hence $c$ belongs to  {$1,2,3$}. Now, $$f(1)=c$$$$\implies f(f(1))=f(c)$$$$\implies 2=f(c)$$ by the assumption that $f(f(1))=2$ . Now,since $c$ can only take the values $1,2,3$,we start treating cases. If $c=1$,$$f(c)=2$$$$\implies f(1)=2$$ but we know from the deinition of $c$ that $f(1)=c=1$,a contradiction.If $c=2$,then $2=f(c)=f(2)$ but $f(2)=1$ by assumption. Finally,if $c=3$ $$2=f(c)=f(3)$$ but $$f(3)=4-c=4-3=1$$ which is once again a contradiction. Therefore there are no such functions in this case.  Case-2: $f(f(1))=1,f(2)=2$  Again assuming $f(1)=c$ and using $f(n)\le n$ along with plugging $n=c-1$ will give us that $f(1)=1$ and then it is easy to prove that such a function exists by recursion. I can only give a "sort of recursive" way to define the function. Here it goes $$f:\mathbb{N}\rightarrow \mathbb{N}$$$$f(1)=1$$$$f(n)=n+1-f(f(n-1))$$  But this case is harder to deal with.Some help will be appreciated. Where does the question mark go — inside or outside the parentheses? Is it possible to get a decimal output from doing division in Bash? Basically in Bash, what I'm trying to do is take a input of seconds from a user and find the hours. So basically if the user enter 35 the output should be 0.00972222.  But bash gives me is a zero.  heres the command I have:  echo "Enter the seconds you wish to convert to hours: " &amp;&amp; read sec &amp;&amp; echo " $((sec/3600)) is the amount of hours "   Is there a way I can make it so it prints out 0.00972222 if I enter 35.  Thanks! Can someone please explain the difference between the protected and protected internal modifiers in C#? It looks like their behavior is identical. The product of a normal and Rademacher variables, independent from each other Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that:  \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} &amp; \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 &amp; \mbox{otherwise}. \end{cases} \end{equation}   Define $Y = ZX$. Show that $Y \sim N(0,1)$. Determine $\operatorname{Cor}(X,Y)$. Are $X$ and $Y$ independent?   Progress:  I'm basically stuck at the beginning. I got as far as $P(ZX &lt; y) = P(Z &lt; y/X)$, which I'm not even sure is correct. I'm also not clear on what it means by defining $Y = ZX$ - does it mean I need to state the probability density function or something? And how do I show that $Y \sim N(0,1)$? Can a coin with an unknown bias be treated as fair? Why can other users see the files in my home folder? Is there a word for happiness made possible by a tragic situation? I frequently find myself needing a word to express happiness that acknowledges a tragic or unfortunate circumstance underpinning that happiness. Preferably the word could have both adjective and verb forms. For example, "Gregory [verb-ed | felt adjective about] the balmy February day, suspecting it was yet another consequence of catastrophic and irreversible climate change."  "Bittersweet" is close, but not precisely right. It implies ambivalence rather than genuine gladness. "Schadenfreude" is also in the neighborhood, but I don't want to imply that the tragic part of the situation is what causes the pleasure. (Also, I recognize that the word may not quite exist in English, and I don't know if this is forbidden for this forum, but I'd welcome a loan word or phrase if there's a better one.) Reference: Why are my "special" Unicode characters encoded weird using json_encode? How do I write a correct micro-benchmark in Java? How do you write (and run) a correct micro-benchmark in Java?  I'm looking for some code samples and comments illustrating various things to think about.  Example: Should the benchmark measure time/iteration or iterations/time, and why?  Related: Suppose I wish to allow only users who have scored at least 15 in the particular tag to answer my question, Is it possible to set a condition like this while asking a question? So that I can avoid strangers/answers from dummy users. Magic quotes in PHP According to the , in order to make code more portable, they recommend using something like the following for escaping data:  if (!get_magic_quotes_gpc()) {     $lastname = addslashes($_POST['lastname']); } else {     $lastname = $_POST['lastname']; }   I have other validation checks that I will be performing, but how secure is the above strictly in terms of escaping data? I also saw that magic quotes will be deprecated in PHP 6. How will that affect the above code? I would prefer not to have to rely on a database-specific escaping function like mysql_real_escape_string(). Leading zero in umask 0022 In the magical world (in Harry Potter) can ghosts use magic?  Can magic be used on ghosts?  Or, once they become ghosts, can they interact with magic at all? Plotting graphs using bash Removing columns in a SpatialPolygonsDataFrame in R? My spatial polygon data frame (SPDF) contains too many columns (variables) and I want to remove most of the columns entirely.   I know how to do this with a regular data frame in R, but I am unsure how to do this when dealing with object of class SpatialPolygonsDataFrame? I know that an easy to handle formula for the CDF of a normal distribution is somewhat missing, due to the complicated error function in it.  However, I wonder if there is a a nice formula for $N(c_{-} \leq x &lt; c_{+}| \mu, \sigma^2)$. Or what the "state of the art" approximation for this problem might be. Should I keep 32bit or 64 bit Windows 10 on my x64 based Intel Atom Z3735F cpu on 2gb ram non-upgradable laptop I have an Acer Aspire Switch 10 SW5-012 detachable notebook which has 2GB of RAM based on an x64 based processor. I got it with a 32 bit version of windows 8 but upgraded to windows 10 recently. But before I do anything else, I was considering if I'd be having Better performance with the 64 bit version of windows 10. Would it make my notebook perform better since it's based on an x64 based Intel Atom Z3735F? Or should I just stick to the 32 bit version of windows 10? How do I debug permissions? How do I debug permissions in Drupal 7?  I use core reporting, error logs, devel module error logs, backtrace and node access permission block but sometimes it is not enough. What else should I check to find out why certain fields, views or blocks are not showing to a user?  I also found there is a  for reporting on permissions called  but it is not available for Drupal 7.  I am using various permission related third-party modules:   - devel node access 7.x-1.x-dev   - fast permissions administration 7.x-2.0  - organic groups access control 7.x-1.x   - profile2 group access 7.x-1.x-dev   - ur-node access 7.x-1.x-dev