How to show $A \times (B \cup C) = (A \times B) \cup(A \times C)$, where $A$, $B$, and $C$ are sets. i.e. If $A=\{1,2\}$, $B=\{2,3\}$ then $A \times B = \{(1,2), (1,3), (2,2), (2,3)\}$.   I started with the proposition, suppose $x$ is an element of $LHS$, i.e. $A \times (B \cup C)$. Help is appreciated! Thanks.
How do I prove or disprove $X \times (Y \cup Z)= (X \times Y) \cup(X\times Z)$  for all sets $X$, $Y$, and $Z$? I'm lost on the steps here.
How is money held in escrow for a home purchase from a IRA treated by the IRS? YA coming-of-age fantasy series, protagonist keeps pigs I lost a series books during a move and I'm thinking I might like to get them for my son.  They are about a teenage orphan who lives with a (sort of) wizard who's primarily known for his prize pig. There is a "bad guy" but he doesn't do much - he starts off the action in one by scaring the pig, and the boy has to chase after it. He goes on several such adventures, notably one to "find his true nature" that ends with him staring into a pool.  I suppose it could be the early books of , but I'm pretty sure I read them in the 1980s, not 90s, and title seems wrong. And unlike the Wiki description, in the series I'm thinking of, there's no real magic to speak of - its "out there" but does not seem to be part of the action that I recall. Things to focus on in a model lesson A model lesson is a usual element in interviews for the faculty level positions. Normally this is not a full lesson but a 15-minute version which is performed in absence of actual students for just a few professors from a hiring committee. It is supposed to demonstrate "your teaching style", but by design this is clearly a different enterprise: you should "act normal" doing "the same" things    in a different time-frame,  for a different audience,  with completely different motivation and  under exorbitant cost of failure.   Having said this, I also acknowledge the model lesson as indeed a much needed element of the interview, which helps to assess a candidate's set of relevant skills.  The question is how one can ideally prepare to give such a model lesson. Putting aside obvious things like "structure of the talk," "clear slides," "projection of the voice," and "body language", which normally should be already trained by experience, are there specific things that should be taken into account for the model lesson only? What about techniques like jokes, questions to the audience (e.g. how many of you are familiar with the definition of the derivative), work in pairs, which you probably use in a real classroom &mdash; is it a good idea to demonstrate them in a room full of senior professors? I am a little confused. Using Pigeonhole Principle to prove two numbers in a subset of $[2n]$ divide each other Let $n$ be greater or equal to $1$, and let $S$ be an $(n+1)$-subset of $[2n]$. Prove that there exist two numbers in $S$ such that one divides the other.   Any help is appreciated! If $G$ has no proper subgroup, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.     I know that since $G $is a group with no proper subgroups, $g \in G$ is not just the identity. I don't know where to go from there. Are there any examples of $\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$? I am writing a short discussion of Farey numbers and was wondering if there are any examples of when the mediant function is ever actually equal to the sum of the two fractions in the usual sense? (Not to produce Farey numbers, I just thought it might be an amusing way to introduce Farey addition).   Explicitly: Are there any examples of fractions $\frac{a}{b}$ and $\frac{c}{d}$ where  $$\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$$ for positive integers?   Currently I have found the example  $$\frac{1}{1}+\frac{1}{i} =\frac{1+1}{1+i}$$ if we remove the integers requirement but it would be nice to find a case that didn't involve complex numbers though! Typesetting Chemical Formulas Is there any way to deserialize JSON into an object using JSON.deserialize if some of the property names in the JSON are reserved words in apex? I want to do something like this:  string jsonString = '{"currency" : "ABC"}'; public class JSONResult {     public string currency; } JSONResult res = (JSONResult) JSON.deserialize(jsonString, JSONResult.class); system.debug(res);   but of course I get an error that the identifier name is reserved. How do I update Android on my device? I'm trying to remember a film I saw years ago (1980's?).  The only part I clearly remember is a man and girl being bricked up alive (in a tower?) and escaping by turning his (magically large) coat or cloak into a rudimentary hang glider.  The film had a medieval sword/sorcery setting. Is there any way to tinker with the iPhone SDK on a Windows machine? Are there plans for an iPhone SDK version for Windows?  The only other way I can think of doing this is to run a Mac VM image on a VMWare server running on Windows, although I'm not too sure how legal this is. I have my Git repository which, at the root, has two sub directories:  /finisht /static   When this was in , /finisht was checked out in one place, while /static was checked out elsewhere, like so:  svn co svn+ssh://admin@domain.com/home/admin/repos/finisht/static static   Is there a way to do this with Git? first post between 12/26 and 1/1   Does that mean your very first post as a user should be made in those days?  Or that you should be the first user to post any post on a site starting 12/26?  Or something else?  The first option seems somewhat more likely but I'm not sure I'm parsing the description right. Escape velocity from a rotating black hole I have Windows 8 Pro activated with kms. when I open windows store, it doesn't show "win8.1 available". clicking on ms-windows-store:WindowsUpgrade just brings me to windows store main screen  slmgr /dli shows volume_kmsclient channel  I can upgrade it to WMC. Laptop is W8Pro WMC and update is available, it's retail channel  Is W8 for KMS clients delayed or what? I designed a poster in CorelDraw X5 for printing purposes. I was told by the Printshop guy to use colors only in CMYK. For the most part I did, I don't know why I missed it but I used one color in RGB. I liked this color so much that I've used it at many places. So, current situation is that some colors are in RGB and some are in CMYK. So, I want to convert all RGB to CMYK and thought the problem would be solved. But now I'm actually facing a lot of trouble. The colors are not at all same. I hate the colors that it is showing me.  This is the color I used in RGB:    When I changed the "Model" to CMYK, the color it shows is really different. As you can see in the picture below, it's too dull:    The RGB values of both colors are not the same. What can I do to get the same color in CMYK? I did a little bit of reading here and there. They say, that you can't get same. In that case, what is the closest color that I can get? Make back/forward work on paginated lists How can I remove duplicate icons for "launched" java programs in the launcher? When launching java programs (like IntelliJ IDEA and Crashplan) in Natty's Unity launcher, duplicate icons are shown (see image).  For IntelliJ I created the .desktop file, for Crashplan the .desktop file is supplied with the application.  Is there something that can be changed in the .desktop files (or somewhere else) that can prevent this from occurring?  I couldn't find a bug report for unity itself but programs like Gnome-Do/Docky have   and had to make internal changes to their applications to prevent this.   In this image the 1st icon is the one created from the .desktop file and the second icon is after launching it.  Second icon disappears when closing the application.  Custom IntelliJ .desktop file  #!/usr/bin/env xdg-open [Desktop Entry] Version=1.0 Type=Application Terminal=false Icon[en_US]=/opt/idea/bin/idea128.png Name[en_US]=IntelliJ IDEA Exec=/opt/idea/bin/idea.sh Name=IntelliJ IDEA Icon=/opt/idea/bin/idea128.png StartupNotify=true   Crashplan provide .desktop file  [Desktop Entry] Version=1.0 Encoding=UTF-8 Name=CrashPlan Categories=; Comment=CrashPlan Desktop UI Comment[en_CA]=CrashPlan Desktop UI Exec=/usr/local/crashplan/bin/CrashPlanDesktop Icon=/usr/local/crashplan/skin/icon_app_64x64.png Hidden=false Terminal=false Type=Application GenericName[en_CA]= How to get more complete references Sometimes when I work I write something like  \begin{proposition} \label{my new result} ... \end{proposition} ... blah blah in Proposition \ref{my new result} blah...   but then I realize that the Proposition should actually become a Lemma. So I change it to  \begin{lemma} \label{my new result} ... \end{lemma}   The problem is that I then have to hunt down all the places where I wrote Proposition and change it to Lemma, which is tedious and error-prone. Well, we have search facilities in editors, so it is not really that complicated, but still it feels the wrong way to do this. I would be happier to write  \begin{proposition} \label{my new result} ... \end{proposition} ... blah blah in \labelNameFor{my new result} \ref{my new result} blah...   where labelNameFor is a hypothetical magic command which outputs "Proposition". Then I could forget about all the places where I reference the result and just make any changes I wish.  Is there a way to do something like this? As far as I understand the label/ref mechanism is simply based on counters, so I do not see a straightforward way, but maybe someone more clever and industrious then I am has given the problem some thought. I thought I remembered hearing the food agency  say that you should never reheat chicken meat after the initial cooking. However, I cannot find the statement of it anymore. Now I'm starting to suspect my memory.  Could this statement be true?  Is it unsafe to reheat chicken?