Prove that the following statements are equivalent:   If $A$ is a non empty set, there exists a function $f:\mathcal{P}(A) \setminus \{\emptyset\} \rightarrow A$ such that $f(X) \in X$ for every $X \in \text{dom}(f)$. Every non empty family $\mathfrak{F}$ of sets such that any two distinct sets of the family are disjoint has a set $X$ such that $|X \cap F|=1$ for every $F \in \mathfrak{F}$.   Now, this is how I've done $1 \Rightarrow 2$:  We notice that for every non empty family $\mathfrak{F}$ of disjoint sets we have that: $$\mathfrak{F} \subseteq \mathcal{P} \left(\bigsqcup _{F \in \mathfrak{F}} F \right) \setminus \{\emptyset\}$$ If we set $A=\bigsqcup _{F \in \mathfrak{F}} F$, we have (for hypotesis) a function $f$ such that $f(Y) \in Y$ for every $Y \subseteq A$. So, in particular, we have that $f(F) \in F$ for every $F \in \mathfrak{F}$, and also that $f(F) \notin F'$ for every $F' \in \mathfrak{F}$ such that $F' \neq F$. So, if we consider the set $X=\{f(F) \mid F \in \mathfrak{F}\}$, we obtain the thesis.  For $2 \Rightarrow 1$ I really don't know what to do. Any suggestions?
This is the first part of the first half of the a proof in the Axiom of Choice. The proof that the Axiom of Choice is equivalent to the choice function. REsources: Keith Devlin's Joy of Sets: pg. 58. F.R. Drake and Singh Intermediate Set Theory: pg. 110).   I am confused on proving disjointness and will ask further questions of proof once I get this first part v. clear for myself.   AC1: Let $\mathcal F$ be a family of pairwise disjoint, non-empty sets. Then there is a set $\mathcal M$ that consists of precised one element from each member of $\mathcal F$  AC2: For every family $\mathcal F$ of non-empty sets, there is a choice function $\mathcal f$. (I.e. $\mathcal f(X)$ is a member of X, for each X in $\mathcal F$.   Disjointness: Two sets are disjoin if Q $\cap$ R = $\emptyset$. Meaning that they have zero elements in common. These sets do not overlap.   Show: AC1 $\leftrightarrow$ AC2   (1) ($\rightarrow$) Let $\mathcal F$ be a set of nonempty sets.   (2)For each X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X}) = {(x, X) | x $\in$ X})  (3) Show Disjointness.  An outline of how book proves  disjointness:  (a) if u = {(X $\times$ {X}) | x $\in$ X} then u is of the form $\lt  z, t \gt$ where z = X and t=x, i.e. x $\in$ X.   (b) then if u $\in$ ({X} $\times$ x) $\cap$ ({Y} $\times$ Y) $\rightarrow$ u = $\lt z, t \gt$ with z = X and z=Y, i.e. X=Y. Which means ({X} $\times$ x) = ({Y} $\times$ y).  Questions: (1) Apparently we cannot prove disjointness in $\mathcal F$ but must prove in a different set. If so, why is that?  (2) why is the case that if u $\in$ ({X} $\times$ X) $\cap$ ({Y} $\times$ Y) then z =X and z=Y? I am not following that step at all.   (3) why does the axiom of choice have to do with ordered pairs? Why are we setting X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X})?    Thank you!
Latexmk and the location of Asymptote's generated files I am trying out Asymptote with latexmk and I would like to automate the interaction.  Consider the following latex document test.tex:  \documentclass{minimal} \usepackage{asymptote} \def\asydir{asymptote}  \begin{document}  Testing Asymptote:  \begin{asy} settings.outformat = "pdf"; unitsize(2cm); draw((0,0) -- (1,0)); \end{asy}  \end{document}   In my latexmkrc I have the  for Asymptote integration:  sub asy {return system("asy -o asymptote/ \"$_[0]\"");} add_cus_dep("asy","eps",0,"asy"); add_cus_dep("asy","pdf",0,"asy"); add_cus_dep("asy","tex",0,"asy");   Running latexmk -pdf test.tex generates test.pdf in the current folder and test-1.asy and test-1.pdf in the asymptote folder -- but only if   I manually created the asymptote folder prior to running latexmk. The folder specification in the latex document (\def\asydir{asymptote}) and latexmkrc (-o asymptote/) are identical.   I would like latexmk to read asydir from the main tex file, create the directory if it does not exist and run asy with the read output folder.  Is this possible? Zooming in to inspect texture details in 3D view clips some closer faces like shown in the image. It's not the camera view so changing camera settings doesn't help.   Where can the distance to the clipping plane be adjusted? Why is the covariant derivative of the metric tensor zero? Countable sets have the same cardinality as the natural numbers (or a subset of the natural numbers, depending on who you ask).  Can one make the same claim about uncountable sets and real numbers?  In other words, do all uncountable sets have same cardinality as real numbers? Note: this post is different to , which just cites part from somewhere else, without any further analysis and thinking. Furthermore, none of the answers to that post answers my question - the situations where &quot;will (going to) be doing&quot; is more appropriate.   I will be participating in IELTS test next Tuesday afternoon.  In an online class, the teacher says the following at the beginning of the lesson.  I will be using NotePod (pointed out by red rectangle) to write relevant points we talk about today.   I guess &quot;will be using&quot; is more appropriate than &quot;will use&quot; there, right?  A  explains  This &quot;tense&quot; is used for reporting a scheduled activity with special focus on its timing— as if one is standing in the future and experiencing the flow of the time. The focus is on the temporary, ongoing, or repetitive nature of the activity.   I understand the grammar rules of those verb tenses and the difference between this group  I am going to drive home tomorrow.   I am going to be driving home tomorrow.  The former is just convey my activity in the near future while the latter probably makes an implication that &quot;don't disturb me as I am driving in that time&quot; However, in the following situations, I don't see the difference between 2 verb tenses.  Welcome back to this class. Today's lesson is for the IELTS students who are going to be taking the IELTS test. it's for everybody to be honest with you. this lesson is useful for anybody who needs to do some writing.   who are going to take the IELTS test ... Why exactly does diffraction occur? Why do waves that were traveling in a straight direction change direction when passing through an opening?  I thought that the waves (red arrow) when colliding with the wall bounce in the opposite direction (green arrow).  And the waves that pass through the aperture follow its path normally as shown in the image on the right.   The waves that go in a straight direction should follow traveling straight line like a car that goes under a bridge the car is straight on the road. But this is not so.  Why does the direction of the waves change?  How is the direction of the waves calculated? I received some text that is encoded, but I don't know what charset was used. Is there a way to determine the encoding of a text file using Python?  deals with C#. How to get past a forgotten iPhone passcode? Why isn't Derivative equal to NeumannValue? How are cellphone IP addresses assigned? Delete files except n files with the "bigger" name? How do you collect valuable information from regular "User Group" meetings? The scenario:   You work on a SaAS application.  Some of your users use your application heavily each day. Every 2 weeks (or month or whatever), a "User Group" of your most avid users gets together to talk shop and see how their progress is going in the context of a larger project they are working on. You can participate and potentially direct some group activities aimed at improving the application.   How do you gather valuable user information in this group setting? Is there a destructor for Java? Is there a destructor for Java? I don't seem to be able to find any documentation on this. If there isn't, how can I achieve the same effect?  To make my question more specific, I am writing an application that deals with data and the specification say that there should be a 'reset' button that brings the application back to its original just launched state. However, all data have to be 'live' unless the application is closed or reset button is pressed.  Being usually a C/C++ programmer, I thought this would be trivial to implement. (And hence I planned to implement it last.) I structured my program such that all the 'reset-able' objects would be in the same class so that I can just destroy all 'live' objects when a reset button is pressed.  I was thinking if all I did was just to dereference the data and wait for the garbage collector to collect them, wouldn't there be a memory leak if my user repeatedly entered data and pressed the reset button? I was also thinking since Java is quite mature as a language, there should be a way to prevent this from happening or gracefully tackle this. How Do I Allow A Standard User to Install Programs? Yes, I realize that defeats the purpose of standard user. But here is the situation. A friend of mine wants to set up time limits for his home schooled son, and his son needs to be a standard user in order to use Family Safety. But at the same time, his son (who I personally think is way too old to have time limits), needs to be able to update and install some of his games on his computer. So I guess it's just sort of "light" parental control (just something to help him focus and not get distracted with hours of gaming).  It is the basic version of Windows 8.1, so there is no Group Policy editor (gpedit.msc). However, I'm fairly good with computers and I figured out how to install it. But it doesn't have a lot of the features that the "real" gpedit seems to have. Here is an example of what is in my gpedit: Why can't we use entanglement to defy Heisenberg's Uncertainty Principle? In principle, it is possible to entangle any property of two particles, including speed and momentum. Surely then, this could be used to defy the Uncertainty Principle, which states that the momentum and position of a particle cannot be together measured with accuracy, but the more accurate one gets, the less accurate the other becomes:  $$\Delta x\Delta p=\dfrac{\hbar}{2}.$$  If one was to set up an experiment where two particles are entangled, one could then measure the position of one particle with great accurary at the same time as measuring the momentum of the other particle with great accuracy, which would mean that, as the two particles are entangled, one would then know the momentum of the particle they measured the position of.  Therefore, the Uncertainty Principle would have been defied, as both momentum and position have been measured with a high accuracy.  However, this can't be true, as someone would have thought of this before.  So why wouldn't this work? Need help in the longtable with the figure Unable to upgrade pip How to use "text" as a verb Closing changes: [on hold], unclear, too broad, opinion-based, off-topic reasons, bye-bye to Too Localized What would be the best approach to get the recordtype id I know of two approaches to get the recordtype id of a particular record type   Use SOQL   RecordType RecType = [Select Id From RecordType  Where SobjectType = 'Account' and DeveloperName = 'Business'];    Benifit here is if the admin changes the name of the record type, the code still works as we can use developer name   Using Describe  Map&lt;String,Schema.RecordTypeInfo&gt; rtMapByName = d.getRecordTypeInfosByName(); Schema.RecordTypeInfo rtByName =  rtMapByName.get('Business');    Saves on SOQL limits but uses a describe call.  What would be your choice and why   EDIT : Usage is for setting up recordtype while inserting records in apex or trigger