How do I prove that the set $\{0,1\} ^{\Bbb N} $is equivalent to the power set of natural numbers $P(\Bbb N)$?
How to show equinumerosity of  the powerset of $A$ and the set of functions from $A$ to $\{0,1\}$ without cardinal arithmetic?
Continuous and bounded variation does not imply absolutely continuous How to setup two completely separate partitions on one SSD I have a MacBook with 1TB SSD. I want to have two separate partitions with two independent macOS installations. I want to make sure that none of the partitions have any kind of possibility to interact with the other, I want them to be as separated as if they were physically separate. Is this achievable with, for example, FileVault for each of them? Edit: My question is neither &quot;how can I install two different macOS versions on two different partitions on one SDD&quot;, neither &quot;how do I use FileVault&quot;, etc. I know how to do all of that. I want to make sure that there is no interaction between the partitions possible. Can this be made sure with the above approach? Is it sufficient? Suspended user cannot award bounty I asked a question on Stack Overflow: . I want to award my bounty to  by user  However, my Stack Overflow account is suspended recently for 7 days, and the suspension still has 5 more days to end.And I know I can't vote, ask, answer or comment as pointed on  by  with the sentence below.  You will be unable to vote, ask, answer, or comment.  I couldn't find anything about not being able to accept an answer or award bounty so I tried to accept the mentioned answer above and award my bounty to it and I got a  Suspended users cannot vote  rejection which can be seen below with a screenshot.  I know this  If you do not award your bounty within 7 days, the highest voted answer created after the bounty started with at least 2 upvotes will be awarded half the bounty amount.  from . None of the answer given to my question is upvoted and the answer I mentioned is not the oldest one after the bounty started so it won't be awarded by community. What can I do? (except being a good guy and not get suspended :)) short view distance in 3d view? pgfplots and using axis cs to add points I am trying to mark two inflection points using a foreach loop in a pgfplot, but it isn't working.  \documentclass[tikz]{standalone} \usepackage{pgfplots} \pgfplotsset{compat = 1.10}  \begin{document} \begin{tikzpicture}   \pgfmathsetmacro{\infx}{1/sqrt(2)}   \pgfmathsetmacro{\infy}{exp(-.5)}   \begin{axis}[     xmin = -3,     xmax = 3,     ymin = -0.1,     ymax = 1.05,     domain = -4:4     ]     \addplot[blue, samples = 500, smooth] gnuplot {exp(-x^2)};      \foreach \x/\y in {\infx/\infy, -\infx/\infy}{       \draw[red] (axis cs: \x, \y) circle[radius = .025];     }   \end{axis} \end{tikzpicture} \end{document}   I am being told:  Undefined control sequence.^M &lt;argument&gt; axis cs: \x ^M                        , \y ^M l.34   \end{axis}^M In some text it is given that we can't assign constant values to a reference variable. When I executed such a program I could do it. Is there any condition we can't assign a constant value to a reference variable? How do I see that $$f_{n+1}f_{n-1} = f_n^2 + (-1)^n$$ for $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$? Dihedral group - elements of order $2$ If $D_{n}$ is the Dihedral group of degree $n$ (order $2n$) where n is an odd number, can I then conclude that the numbers of elements of order 2 in $D_{n}$ is equal $n?$  I suppose there are the following types of elements in $D_{n}$: b, $a^{i}$, $a^{i}b$ for some $0\leq i &lt; n$ . One of the assumptions I make as GM is that the players should always be in the spotlight.  Sometimes the players find themselves in a situation where they watch NPCs speak to each other.  I don't like using this kind of situation, because it forces the players to watch me talk to myself and that turns them from active participants into an audience.  Usually I handle it by avoiding the situation.  I try not to write scenes that the players watch.  Instead the players speak to the NPCs one at a time.  When I do come across an exchange between NPCs I opt to summarize the conversation instead of acting it out.    I don't like this limitation though.  It gets a little contrived when all conversations in the game feature 5 PCs and 1 NPC.  Any suggestions? Either how to avoid doing it, or how to effectively role play a conversation between two (or more) NPCs that the characters may or may not be interacting with?   How do you keep clear which character is speaking? How do you make sure that the conversation feels like a real one and not like one person playing both roles? How to correctly add a path to PATH? Find html element which id starts with Transitivity check vs. Transitive Closure Is checking transitivity of a digraph not easier than (in terms of asymptotic complexity) taking the transitive closure of the digraph? Do we know any lower bound better than $\Omega(n^2)$ to determine if a digraph is transitive or not? There have been  for a while, but this is ridiculous. It reminds me of , but this time it's network-wide.   The banner must be scrolled past (or dismissed) on every page unless both first-party JavaScript and first-party cookies are enabled. If first-party JavaScript is enabled, and the banner is visible, the rest of the page jumps up and down every so often.  Under certain circumstances (not quite sure what they are, yet – a direct link to an answer, maybe? I'm seeing it ) the entire page jumps up and down a line every few seconds even when the banner is off-screen.   The cookie banner keeps coming back unless both third-party JavaScript  are enabled (and  is solved). The cookie box occludes the “Learn more about Teams” link in the banner (with default Firefox window sizes) and the Job / Teams link in the sidebar.  As it stands,  – a site for hobbyists and researchers, whose audience is not very likely to purchase Teams – is much, much harder to use, for no benefit that I can see. Can you roll back this change, please? What is `$?` ? Is it a variable? The question is pretty straight forward. I can get the exit code of the previous command by using $?. But what exactly is it?  I don't think it is a variable, because you can't have any special characters in the name of variable other than _.  foo?foo=10   will result in the error: foo?foo=10: command not found  So if it is not a variable, what is it? Are there others like it? White space below footer How to set default style for arrow tips in TikZ? Is there any way to see the custom messages/remarks of resolved/expired bounties? Outlook: scheduling recurring out-of-office? In Outlook, is there a way (or an add-on that can do it) to schedule a recurring out-of-office message?  I work 4 10's, so I'm out of the office every Monday, and it's a bit of a pain to have to reset my out-of-office message every single week. I try to color the page numbers of my table of contents.  I'm using the package hyperref along with the options :  \hypersetup{   colorlinks,   linkcolor=wine-stain % custom color }      Here's the output :       The numbers remain black..  What should I do so the numbers become the same color as defined for the sections ?  I tried to \color{wine-stain}{\tableofcontents} but still that doesn't work since "Contents" is also colorized and the subsection numbers are remaining black. It is true that $P(X&gt;t+s|X&gt;t)=P(X&gt;s)$ for certain values $t$ and $s$.  However, how can I show that this still holds if:   $T$ is a continuous random variable. That is $P(X&gt;T+s|X&gt;T)=P(X&gt;s)$ Both $T$ and $S$ are continuous RVs: $P(X&gt;T+S|X&gt;T)=P(X&gt;S)$   All the random variables are independent.