Let $A^A$ be the set of all functions from $A$ to $A$. Prove $\mathcal{P}(A)$ and $A^A$ have the same cardinality for infinite sets $A$. ($\mathcal{P}(A)$ is the power set of $A$.)  Is it possible to show this without invoking axiom of choice? It can be proven with $A \times A \sim A$ for infinite sets $A$ but that is equivalent to the axiom of choice.
Usually, the equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ for infinite cardinal $\mathfrak{m}$ is proved like that: $$ 2^\mathfrak{m} \leqslant \mathfrak{m}^\mathfrak{m} \leqslant (2^\mathfrak{m})^\mathfrak{m} = 2^{\mathfrak{m} \cdot \mathfrak{m}} = 2^\mathfrak{m} $$ The last step relies much on the axiom of choice, because the fact $\mathfrak{m}\cdot\mathfrak{m} = \mathfrak{m}$ does. Moreover,  states that in $\sf{ZF}$ this fact is equivalent to $\sf{AC}$.  Note, by a cardinal I mean an equipollence class of an arbitrary set, not necessarily admitting a well-ordering.  That has motivated my question: what can we say about relationship of the initial equality and the axiom of choice in pure $\sf{ZF}$?    To avoid question about definition of cardinals without $\sf{AC}$, we may consider the following statement instead:     For any infinite set $X$ exists a bijection $\; F \, \colon X^X \rightarrow 2^X$   where $X^X$ is the set of all functions from $X$ to $X$, $\; 2^X = \mathcal P(X)$ is the power set, and an infinite set is such that cannot be bijectively mapped onto a finite ordinal.
Can we apply for a Schengen tourist visa while visiting another country on a short-stay visa? We are Indians and will be visiting UK on short-stay visas for about five months.  Can we apply for Schengen tourist visas while we are in London? iPhone makes shutter sound when taking photos despite being on mute How can I take pictures through a skyscraper, airplane, or train window? I find that I take a lot of pictures through skyscraper and vehicle windows (those that are impractical or impossible to open). There are all kinds of problems with taking photographs through windows: reflections from light inside, dirt on the window, etc. My technique mostly involves placing the lens directly on the window glass and shielding it from lights reflecting from above.   Are there any non-obvious gizmos / techniques for easily shooting through windows? I've heard that a polarizing filter might be used in this situation, but I don't carry one around most of the time. Besides, most of the pictures I am interested in taking are in low light situations (at sunset or in a moving train, etc). $a,b,m,n$ be positive integers such that g.c.d.$(a,b)=1$ and $a^m+b^m|a^n+b^n$ , then $m|n$? Let $a,b,m,n$ be positive integers such that g.c.d.$(a,b)=1$ and $a^m+b^m|a^n+b^n$ , then is it true that $m$ divides $n$ ? Number of genes required to sustain life Are there estimates of the minimum number of genes required to sustain life?   In what I mean by life here, I don't include viruses. "I had my house [be] burned down" I have found out that using the verb be in passive constructions such as:      I had my house be burned down    is incorrect, therefore it should be      I had my house burned down.    But is it possible instead of be to use get? E.g.      The wire is passed through the pliers in a specific way to avoid having it get bent.   The rope had its surface wear.   The rope had its surface get worn.   The rope had its surface worn.   The first two sound very odd to me, but just for clarification could you please point out the errors and which ones are correct? Or is using get, be it in -ing or the to infinitive, in any situation along with have incorrect? I have often heard people say "x of his" or "x of mine". But since his and mine point out ownership, isn't using "of" here doubles that ownership? Wouldn't it be more appropriate to say "x of him" or "x of them" etc.?  Example:     I was a good friend of your dad's. He told me all about you` Running a script during booting/startup; init.d vs cron @reboot I am currently trying to understand the difference between init.d and cron @reboot for running a script at startup/booting of the system.   The use of @reboot (this method was mentioned in  by hs.chandra) is some what simpler, by simply going into crontab -e and creating a @reboot /some_directory/to_your/script/your_script.txt and then your_script.txt shall be executed every time the system is rebooted. An in depth explanation of @reboot is   Alternatively by embedding /etc/init.d/your_script.txt into the second line of your script ie:  #!/bin/bash # /etc/init.d/your_script.txt  You can run chmod +x /etc/init.d/your_script.txt and that should also result for your_script.txt to run every time the system is booted.  Q1: What are the key differences between the two? Q2: Which is more robust? Q3: Is there a better one out of the two? Q4: Is this the correct way of embedding a script to run during booting?  I will be incorporating a bash .sh file to run during startup. I am having some issues when rendering an image, my rendering screen is turning into grey even when making a full render. What am I doing wrong? What does standard deviation tell us in non-normal distribution In a normal distribution, the  imparts standard deviation a lot of meaning, but what would standard deviation mean in a non-normal distribution (multimodal or skewed)? Would all data values still fall within 3 standard deviations? Do we have rules like the 68-95-99.7 one for non-normal distributions? Preparing CSV files for use in ArcGIS Desktop? When is the graph of the divisor relation on $[n]$ planar? A recently-deleted question asked to show that if $G(n)$ is the graph on $[n]=\{1,\dots,n\}$ where $x$ and $y$ are adjacent if $x|y$ or $y|x$, then $G(n)$ is planar.  This is false if $n\geq150$, for the subgraph induced by $\{2,3,5,90,120,150\}$ is isomorphic to $K_{3,3}$.  My question is, what is the largest value of $n$ such that $G(n)$ is planar?  Of course, various software packages would make short work of this, but I'd be interested in a verbal argument, if someone can give one. What are the rules of parallelism? I've been reading a lot about it since yesterday and all I encounter is "They must have the same grammatical form"? What does this exactly mean?  I know how to make parallel sentences with gerunds. The things get trickier when it comes to using modifiers.  For example:     I love using the phone that you bought me and the computer.   On the left side of the conjunction there is a noun phrase that consists of an article + noun + another modifier and on the right side there is an article + noun.  They are both noun phrases but the first one has additional modifier. Is that still considered a parallel sentence?  Another example:     I like to swim and to feel good.   Here on the right side of the conjunction there is an infinitive and on the left side there is an infinitive + complement? Is that parallel? Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$ Is there a nice presentation for the group $\text{PSL}(2,\mathbb{F}_q)$, for every prime $q$?  (I don't have any particular definition in mind for "nice", other than, say, a small number of generators and relations.) Are there any performance comparisons? Are there any reliable performance comparisons between vanilla Drupal 8 versus Drupal 7?  I have searched the web but only find the post "", which is far from encouraging. Proving that two matrices are congruent, in order to establish a rank and signature result. Limit point of the set $G= \{1/n: n \in \mathbb N\}$? In $\mathbb{R}$ , what is the limit point of the set $G= \{1/n: n \in \mathbb N\}$?  Definition:  A point $p \in X$ is a limit of of E if every neighborhood of $p $ contains a point $q \in E$ $q\neq p$  I do understand $0$ is a limit point but shouldn't it be the case that all the points in [0,1] are limit points by the definition? Why am I getting a NoClassDefFoundError in Java? I am getting a NoClassDefFoundError when I run my Java application.  What is typically the cause of this? Allow Documentation in other Stack Exchange communities I'm a Salesforce developer and would love Documentation to be available in other Stack Exchange communities. The reason is  a lot of Salesforce developers get help, especially newer ones like myself. The developer forums and documentation on  is mediocre. It would be nice to allow users to create their own and give pack on Can we get by as English speakers visiting Iceland? We're going to Iceland for the first time in September. Will we be able to get around ok just speaking English, or should we try to learn some rudimentary Icelandic? And is that even what their language is called?