I tried to prove this question but without a success: Let $K_1 \text{and } K_2$ be definable sets, prove that $K_1∪K_2$ is definable.  What I tried to do is to assume: $K_1=\text{Ass}(X)=\{v\mid v\models X\}$ and $K_2=\text{Ass}(Y)=\{v\mid v\models Y\}$ and to say that $K1\cup K2=\text{Ass}(X\cap Y)$. But I couldn't prove that: $K_1\cup K_2\supseteq\text{Ass}(X\cap Y)$.  I would like to get help with this proof
I tried to prove this question but without a success: Let $K_1$ and $K_2$ be definable sets, prove that $K_1\cup K_2$ is definable.  What I tried to do is to assume: $K_1=Ass(X)=\{ v|v \vDash X \}$ and $K_2=Ass(Y)=\{ v|v \vDash Y \}$ and to say that $K_1\cup K_2=Ass(X \cap Y)$. but I couldn't prove that: $K_1\cup K_2 \supseteq Ass(X \cap Y)$ .   I would like to get help with this proof
Inverse Green's Function identity in derivation of Hedin's equations I'm trying to work through a derivation of Hedin's Equations in Effect of Interaction on One-Electron States by Hedin and Lundqvist (1969) and I've come across an identity that is given without much explanation that I'd like to understand.   We start with an equation of motion with single particle Green's functions G:   $\left[ i\frac{\partial}{\partial t_1} - h(1) - \phi(1) - V_H(1)\right] G(1,2) - i\int v(1^{+},3) \frac{\delta G(1,2)}{\delta \phi(3)} d3 = \delta(1,2)$  where $1=(x_1,t_1)$, $h$ is the one-electron kinetic energy and ion interaction, $\phi$ is a small perturbing field that is set to zero at the end, $V_H$ is the Hartree potential, and $v$ is the Coulomb interaction.  They then insert the identity  $\frac{\delta G(1,2)}{\delta\phi(3)} = -\int G(1,4) \frac{\delta G^{-1}(4,5)}{\delta\phi(3)} G(5,2) d(4,5)$  which is then used to define the self energy  $\Sigma(1,2) =  -i \int v(1^+,3) G(1,4)\frac{\delta G^{-1}(4,2)}{\delta\phi(3)} d(3,4)$.  I know this identity is related to the definition of the inverse Green's function  $ \int G(1,3) G^{-1}(3,2) d(3) = \delta(1,2)$  but I can't figure out how to get from this definition to the identity that is inserted into the equation of motion. Any ideas?  Thank you! Why is my StackOverflow with an indent and doesn't have reputation Prove that if $A$ is transitive, then $\mathcal{P}(A)$ is transitive too. My book on set theory says     Prove that is a set $A$ is transitive, then $\mathcal{P}(A)$ is also transitive, where $\mathcal{P}(A)$ is the power set of $A$.   I don't know understand how. Let $A=\{a,b\}$. Clearly it is transitive. Now $\mathcal{P}(A)=\{\{a\},\{b\},\{a,b\},\emptyset\}$. This is clearly not transitive. The union of a strictly increasing sequence of $\sigma$-algebras is not a $\sigma$-algebra How many physical degrees of freedom does the $\mathrm{SU(N)}$ Yang-Mills theory have? The $\mathrm{U(1)}$ QED case has two physical degrees of freedom, which is easy to understand because the free electromagnetic field must be transverse to the direction of propagation. But what are the physical degrees of freedom the $\mathrm{SU(N)}$ Yang-Mills theory?   I believe that the $\mathrm{SU(N)}$ Y.M. theory has four degrees of freedom, and it is easy to see that by gauge fixing (such as the Lorentz condition or the Coulomb condition) that we can always remove one redundant degree of freedom. But then we still haven't completely fixed the redundant degrees of freedom. Thus my question is:     How many physical degrees and how many are redundant degrees of freedom does the Y.M. theory have?    I would be interested to understand how we can determine this mathematically, and also to understand what the physical intuition behind this is? Consider I've set variable site and needs to be printed by echo or printf, but If I use single quote to write something and want to use variable then how?  Example:  $ site=unix.stackexchange.com $ echo "visit:$site" visit:unix.stackexchange.com   But If I use single quote:  $ echo 'visit:$site' visit:$site   Then we know that '' is strong quote and will not expand the variable   I've tried something:  $ echo 'visit:"$site"' visit:"$site"   but do not succeed. So, I am looking for way to print value inside variable while using single quote. I am a US citizen and have recently visted the UK  with simply a passport which allowed me to vist for 6months I have heard rumors that I am obligated to stay in the US for a Equal amount of time that I have visited the UK before I am able to return, but I am struggling tour no any proof of this obligation .. Am I missing somthing or is it simply not true I use Microsoft Outlook 2013 for work and Google Calendar for everything else. I'd like to find an easy way to sync my work calendar to my personal calendar. Ideally, this would be a software application that sits in the background and automates the syncing.  Requirements   Syncing should be one-way only: Outlook to Google. Must support repeating calendar items. Must allow me to specify which calendar in Google to sync to. I will set up a separate calendar for work items. No intermediate server. Export is direct.   Nice to Haves   Automated. Syncs in real time or regularly (say, every fifteen minutes or so). I've just installed Mir and I'm wondering if I'm actually running it or if it fell back to X. How can I tell if Mir is running? In English, the number 0 is treated as plural. It is then:   0 seconds 1 second    1.2 seconds   2 seconds     Shouldn't it be "nobody want to go there," instead of "nobody wants to go there"?  I also saw in TOEFL that "any __" should be used with a singular.  But I see it very common that it is a plural also.  Why is that?  (updated: example, "we don't have any apples any more" vs "If you get any apple, please let me know.") What are some typical LuaTeX gotchas? You start flipping cards over one at a time and holding them in one hand. When you have two cards of the same suit that are separated by two other cards (e.g. 4♣ 2♦ 7♥ 10♣), you discard the middle two. When two cards of the same value are separated by two other cards (e.g. 4♣ 2♦ 7♥ 4♥), you discard all four.  This game is neither Accordion nor Royal Marriage, but bears similarities to both. Somebody else  on Yahoo Answers five years ago and the response was unsatisfactory.  If someone can identify this game, I would also like to know what happens when you finish with two cards left: do you win, as in Royal Marriage, or not? Minecraft crashes on launch with EXCEPTION_ACCESS_VIOLATION, Problematic frame: ig4dev32.dll or ig4dev64.dll or ig4icd32.dll or ig4icd64.dll When I try to launch Minecraft 1.7.4 or newer, the game crashes on startup, and the game output screen says this:  #  EXCEPTION_ACCESS_VIOLATION (0xc0000005) at pc=*, pid=*, tid=* # # JRE version: Java™ SE Runtime Environment (7.0_51-b13) (build 1.7.0_51-b13) # Java VM: Java HotSpot™ Client VM (* mixed mode, sharing windows-* ) # Problematic frame: # C  [ig4dev32.dll+0x3e88] OR # C  [ig4dev64.dll+0x55ec] OR # C  [ig4icd32.dll+0x104a77] OR  # C  [ig4icd64.dll+0x84930a]   This happens with both Java 7 and 8.  What causes this error, and what can I do to fix the issue? This occurs in the  released recently to the public. When someone posts a code block in their question or answer, the code block is cut off on the right hand side, and I can't scroll to the right to view the rest of it. How can I view these code blocks? 4 GB of RAM installed but only 2.97 GB usable. What gives? Possible Duplicate:        I just bought a used MacBook on  that came installed with 4 GB of . The first thing I did was installing Windows&nbsp;7 using . Here's where it gets weird. When I finally booted it up in Windows&nbsp;7, I checked the Computer properties, and it said that there were 4&nbsp;GB of RAM, but only 2.97&nbsp;GB is usable. Why is that? Is there anything I can do about this? Are dual US citizens affected by President Trump's ban on Iran, Iraq, Libya, Somalia, Sudan, Syria, Yemen? C# datetime parse issue What is the best option for South America Brazil to configure: provider.get_backend (''), because melbourne_16 is very slow. Proof of the First Isomorphism Theorem for Groups The statement is the First Isomorphism Theorem for Groups from Abstract Algebra by Dummit and Foote. This proof was left as a exercise, so I'd like to check if all is ok. In particular, I'm a bit worried about the (*) line. It looks a bit awkward, but is it too bad? Any suggestions?    Theorem: If $\varphi : G \rightarrow H$ is a homomorphism of groups, then $ker \varphi \trianglelefteq G$ and $G/ker \varphi \cong \varphi (G)$.  Proof: Let $\varphi:G \rightarrow H$ be a group homomorphism. If $g \in G$ and $h \in ker \varphi$, then we have that $ghg^{-1} \in ker \varphi$ because $$\varphi (ghg^{-1})= \varphi (g) \varphi (h) \varphi (g^{-1})= \varphi (g) \varphi (g^{-1}) = \varphi (gg^{-1}) = \varphi (1) =1,$$ so that $ker \varphi \trianglelefteq G$. Now, define $\phi: G/ker \varphi \rightarrow \varphi (G)$ by $$\phi (g(ker \varphi))=\varphi(g),$$ for each $g(ker \varphi) \in G/ker \varphi$, for some $g \in G$. This is well defined because if $g' \in g(ker \varphi),$ then $$\phi (g'(ker \varphi))= \varphi(g')= \varphi(g)= \phi (g(ker \varphi)).$$ Also, this is a group isomorphism because for each $\varphi (h) \in \varphi (G),$ for some $h \in G$, we have $$(*) \ \phi^{-1}\{\varphi(h)\}= \phi^{-1} \varphi[g(ker \varphi)]= \phi^{-1} \varphi(\pi^{-1}\{g(ker \varphi)\})= \{g(ker \varphi)\},$$ a set with a single element of $G/ker \varphi$, so that it is a bijection, and for every $g(ker \varphi), g'(ker \varphi) \in G/ker \varphi$, for some $g,g' \in G$, we have $$\phi [g(ker \varphi) g'(ker \varphi)]=\phi [(gg')(ker \varphi)]=\varphi(gg')=\varphi(g) \varphi(g')=\phi [g(ker \varphi)] \phi [g'(ker \varphi)],$$ so that it is a group homomorphism. How to auto-start unprivileged lxc containers? On Ubuntu 14.04, I have created an unprivileged container that I can manually start and stop.  But I would like this to start and stop along with the system.  I have added the following to the container's configuration:  lxc.start.auto = 1 lxc.start.delay = 5   However, the system scripts don't seem to pick unprivileged containers.  There is  related to this on linuxcontainers.org, but the solution seems to be restricted to root user.  Is there a clean way to do this for a non-root user (with root user's consent)?