Let $A,B,C$ be sets. Show that $A\subseteq C$ and $B\subseteq C$ if and only if $A\cup B \subseteq C$ Let $A,B,C$ be sets. Show that $A\subseteq C$ and $B\subseteq C$ if and only if $A\cup B \subseteq C$.  MY ATTEMPT  We shall prove that $A\cup B\subseteq C$ iff $A\subseteq C$ and $B\subseteq C$. Let us consider the implication ($\Rightarrow$) first. If $x\in A\cup B$, then $x\in A$ or $x\in B$. In both cases, $x\in C$, once $A\cup B \subseteq C$. Thus we conclude that $x\in C$ implies that $x\in A$ as well as $x\in C$ implies that $x\in B$. Therefore $A\subseteq C$ and $B\subseteq C$, as desired.  Conversely ($\Leftarrow$), if $x\in A\subseteq C$, then $x\in C$. Similarly, if $x\in B\subseteq C$, then $x\in C$. In other words, if $x\in A\cup B$, then $x\in C$, which means that $A\cup B\subseteq C$, as desired.  Could someone check if I am solving it correctly?
Proof of A ⊆ C and B ⊆ C ⇔ A ∪ B ⊆ C I know that this is valid because by thinking about it, you can see it's true, but I'm just not sure how I would go about actually proving it mathematically. I've kind of tried proof by contradiction, since that's what I use most of the time, but it doesn't seem to work.    In my mind, A ∪ B ⊆ C kind of just implies that A ⊆ C and B ⊆ C.
Replacing whiteness with transparency in Photoshop I want to replace the whiteness in pixels with transparency in the same way as described in  - i.e. for lighter pixels to be made proportionally as transparent. All solutions found via google are based on magic wand or other methods that do not work for my needs.  I'm sure there used to be a way to do this in earlier version of Photoshop (a simple functions to replace a colour with transparency), but I can't seem to do it in CS6.  Replace Colour has no alpha options.. I'm consistently receiving significant delays on Stack Overflow when opening the flag dialog on questions. Delays do not occur on any other StackExchange site I have checked. This started a couple days ago (as far as I can tell).  Delays occur not only when loading the initial screen, but also when loading the submenus. This is slightly confusing for a couple seconds (until you're used to it) since it appears as if the flag dialog has simply vanished.   What's going on here? I demand to know, otherwise somebody will be very upset.  Not me. I'm not going to get that upset over something like this.  But someone will.    Specs (though I know of several others who have this issue):  Windows 7 64-bit Chrome Version 30.0.1599.69m     Ping: 31ms Download: 6.88mbps   Upload: 1.07mbps Android OS without Google Services? I have a Sony Xperia Z with version 4.1.2 running.  My question is about removing all of the Google baggage from the phone.  I have no Google account tied to my phone and would really like to get rid of all traces of Google from my phone for a few reasons:   Battery performance: Google services ranks in the top 3 of what uses up my battery Privacy: I try to minimize what Google siphons from my phone with the exception of Google maps Less Dependency: I don't want to be dependent on one company for all my needs.   So are there any Android versions out there that are minimalistic and do not invade your privacy? How can I re-enable backspace navigation in Chrome? Forms: Modal dialog or separate page in single page app? In my application, I have pages that show lists of some objects, for example "People" or "Contracts".  From these pages, the user may create a new object.  Often these objects only have a handful of properties, maybe 4 or 5.  My question is about the best approach for the forms used for creation of new objects.  Sometimes I have used a separate page, especially when the number of properties is greater than 5-6.  Sometimes I have used a modal dialog.  I have even seen some applications that insert an empty row into a table and you fill-in the values.  (I will not be doing that!)  Any UX-preferences or suggestions for what approach to take?  Also- is it important to be consistent between different types of objects?  Or is a mix of approaches OK?  The technology of my application (Ember.js, single-page app) allows me to navigate to a different page instantly, so any timing difference between approaches is not a factor.  In fact, the animation of the Bootstrap dialog takes longer than the page transition.  The application is a back-office administrative system for insurance companies. Prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$. $f(x)$ is a differentiable function and $g(x)$ is a double differentiable function such that $|f(x)|\leqslant 1$ and $f'(x)=g(x)$. If $$f(0)^2+g(0)^2=9$$ then prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)&lt;0$.  Attempt: Let us define a function $h(x) = g(x) g~'(x)$. Then $$h'(x) =  g(x)g''(x) + \left( g'(x) \right)^2 \tag 1$$ If we prove that for some $c \in (-3,3), h~'(c) &lt; 0,$ then $$g(c)g''(c) &lt;0 \tag 2$$ Also, $$\left|f(0)\right| &lt; 1 \implies f'(0) \in (-3,-2\sqrt 2 ) \cup (2\sqrt 2,3) $$ Could someone please advise me how do I move forward from here. Thank you very much for your help in this regard. When playing with the complete elliptic integral of the first kind and its Fourier-Legendre expansion, I discovered that a consequence of $\sum_{n\geq 0}\binom{2n}{n}^2\frac{1}{16^n(4n+1)}=\frac{1}{16\pi^2}\,\Gamma\left(\frac{1}{4}\right)^4 $ is:     $$\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \tfrac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2\tag{A}$$   which might be regarded as a sort of Ahmed's integral under steroids.  I already have a proof of this statement (through Fourier-Legendre expansions), but I would be happy to see a more direct and elementary proof of it, also because it might have some consequences about the moments of $K(x)$ of the form $\int_{0}^{1}K(x)\,x^{m\pm 1/4}\,dx$, which are associated with peculiar hypergeometric functions. Is "this Monday" or "next Monday" the correct way to refer to the very next Monday in the future? Possible Duplicate:          When I refer to the very next Monday that will occur in the future, I say "next Monday". Some colleagues refer to it as "this Monday", with "next Monday" meaning the second Monday which will occur in the future (I would refer to that as "Monday week", "this Monday" to me would mean the most recent Monday in the past).  Are these both acceptable usages, or is one more correct than the other? Consider the closed interval $[0,1]$, there is $\frac{2}{3} \in [0,1]$ where $p=2$ and $q=3$. Similarly consider $[2,3]$, one can have $\frac{5}{2} \in [2,3]$ where $p=5$ and $q=2$. Does every interval of the form $[a,b]$, where $a,b \in \mathbb{R}$ contain a rational of this kind. If yes, how can we prove it? Where can I find a Java to C# converter? I needed to convert a Java 1.5se app to C# 2.0.  Does anyone know of a tool (preferably free/open source) to do this? Color gradient for bars in pgfplots I want that each bar in my diagram has a color gradient from black to white (from left to the right). However, for the following code, the gradient exists for the entire diagram such that the left bar is black, the middle one is gray, and the right one is white.   \documentclass{report}  \usepackage{pgfplots}   \begin{document}  \begin{tikzpicture}      \begin{axis}[ybar]          \addplot[left color=black,right color=white] coordinates {(1,2) (2,3) (3,4)};      \end{axis}  \end{tikzpicture}  \end{document} I'm trying to use push with AppInbox/Cloud Pages. I want the Cloud Page that is linked to the push message to be able to display dynamic content based on an attribute for that contact. This page -  - says that it is possible to personalise Cloud Page content but doesn't say how. I've already had a Salesforce support case opened and closed with no luck - apparently it's not their job to assist me with how to do something even if they say something is possible but back it up with no documentation for how it might be done. I've tried the normal way you'd do it from an email, using both the _subscriberkey personalisation string as well as the RequestParameter function with fields from the entry source DE, but neither of these work. Given you're selecting a static Cloud Page when you set up the send and there's no visible way to add attributes, how do you actually get a Cloud Page to pick up these values? I have read answers to questions like  but I couldn't find a general rule for using "you" or "yourself", "him" or "himself" that could explain things I see in some examples.  For example, I think that one normally say (according to examples I see in ):   You bring something with you. You have something in you. You have something on you. You pressed her to you.   But on the other hand one would say:   You draw attention to yourself. He killed himself. You can try it out for yourself.   As a general rule, it seems that the reflexive pronoun is used when the object is the same as the subject of the verb. But for prepositional phrases, it seems more complicated. Do some prepositions (like "with", "in", "on") always come with the accusative form of the pronoun, and some other (like "for") don't? What about "to", that seems to be followed by both in different circumstances? Does Kylo not know the fate of Darth Vader? It seems Kylo has an obsession of Darth Vader and has an interest in the dark side because of it.   Does he not realize that Darth Vader turned good and helped destroy the Empire (Spoilers?)? I feel like Kylo would've seen how much better the Light Side was as it was able to turn Vader from the Dark Side if he had known this. What are logic gates? std::auto_ptr to std::unique_ptr Login to Stack Exchange account doesn't use HTTPS/SSL? My C# application is returning 0xE0434352 to Windows Task Scheduler but it is not crashing setting up subdomain on ubuntu server I have to maintain a large number of classic ASP pages, many of which have tabular data with no sort capabilities at all. Whatever order the original developer used in the database query is what you're stuck with.  I want to to tack on some basic sorting to a bunch of these pages, and I'm doing it all client side with javascript. I already have the basic script done to sort a given table on a given column in a given direction, and it works well as long as the table is limited by certain conventions we follow here.    What I want to do for the UI is just indicate sort direction with the caret character ( ^ ) and ... what?  Is there a special character that is the direct opposite of a caret?  The letter v won't quite cut it.  Alternatively, is there another character pairing I can use?