Subsets of sets containing empty set Why is $\{\emptyset\}$ not a subset of $\{\{\emptyset\}\}$?  It contains this element, but why is it not a subset?
Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$? $\{\emptyset\}$ is a set containing the empty set. Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?  My hypothesis is yes by looking at the form of "the superset $\{\{\emptyset\}\}$" which contains "the subset $\{\emptyset\}$".
Uniqueness in existence of a bilinear form Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$)   $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\  \ , \alpha \text{ is alternating multilinear map}\}$ $\Omega^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\  \ , u \text{ is alternating multilinear map}\}$  I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon \Omega^p(E^*)\times\Omega^p(E)\rightarrow\mathbb R$   such that for all $ f_i\in E^* \ , u_j\in E^{**}\simeq E \qquad  i,j=1,\cdots,p$, it has the following rule:  $B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.  $\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$  How can I do this? dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) nginx The Software up-to-date Option automatically download the upgrade for nginx. but While installing it's giving me the following error:  Reading package lists... Done Building dependency tree        Reading state information... Done The following NEW packages will be installed:   nginx 0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/434 kB of archives. After this operation, 977 kB of additional disk space will be used. (Reading database ... 291344 files and directories currently installed.) Unpacking nginx (from .../nginx_1.4.7-1~precise_amd64.deb) ... ----------------------------------------------------------------------  Thanks for using nginx!  Please find the official documentation for nginx here: * http://nginx.org/en/docs/  Commercial subscriptions for nginx are available on: * http://nginx.com/products/  ---------------------------------------------------------------------- dpkg: error processing /var/cache/apt/archives/nginx_1.4.7-1~precise_amd64.deb (--unpack):  trying to overwrite '/etc/logrotate.d/nginx', which is also in package nginx-common 1.4.6-1+precise0 dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Errors were encountered while processing:  /var/cache/apt/archives/nginx_1.4.7-1~precise_amd64.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   Actually that file /etc/logrotate.d/nginx does not exist in this folder. Hide Command Window of .BAT file that Executes Another .EXE File Does the rank-nullity theorem hold for infinite dimensional $V$? The rank nullity theorem states that for vector spaces $V$ and $W$ with $V$ finite dimensional, and $T: V \to W$ a linear map,  $$\dim V = \dim \ker T + \dim \operatorname{im} T.$$  Does this hold for infinite dimensional $V$?  According to , the statement is false.  But according to , page 4, the statement is still true.  I'm thoroughly confused. My GNU/Linux system (Arch) now has libncurses.so.6 rather than libncurses.so.5. But I have upstream's vanilla TeX Live (fully updated) and xindy wants libncurses.so.5:    xindy --version   produces the following error  xindy release: 2.5.1 xindy script version: 1.18 /usr/local/texlive/bin/xindy.run: error while loading shared libraries: libncurses.so.5: cannot open shared object file: No such file or directory   And ldd can confirm the problem:  ldd /usr/local/texlive/bin/xindy.run    gives      linux-vdso.so.1 (0x00007ffc07997000)     libm.so.6 =&gt; /usr/lib/libm.so.6 (0x00007fab4d1b1000)     libcrypt.so.1 =&gt; /usr/lib/libcrypt.so.1 (0x00007fab4cf79000)     libncurses.so.5 =&gt; not found     libdl.so.2 =&gt; /usr/lib/libdl.so.2 (0x00007fab4cd75000)     libc.so.6 =&gt; /usr/lib/libc.so.6 (0x00007fab4c9d1000)     /lib64/ld-linux-x86-64.so.2 (0x00007fab4d4af000)   How can I use xindy without changing my operating system, while using vanilla TeX Live?  2 strategies have occurred to me:   compile xindy from sources against libncurses.so.6; compile and install libncurses.so.5 alongside libncurses.so.6.   Option (2) can be dodgy or can be fine, depending on the details. My suspicion is it may be dodgy for this library, simply because no packages are available for the old version of the library, which might mean there are problems having both installed. Moreover, since everything else is compiled against the new version, I'm not sure whether this would be enough for xindy anyway.  Option (1) I have no idea about. I don't know if this is even vaguely doable or, in fact, where the sources are or what might be required to compile them. (I don't suppose there's any chance a pre-compiled binary is available somewhere? But that is probably hoping for too much ;).)  What is the most practicable approach? Given  $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$,  and $\mu_X \approx 0$, looking for:   accurate closed form distribution approximation of $Y_N=\prod\limits_{1}^{N}{X_n}$ asymptotic (?) approximation of same product    This is a special case $\mu_X \approx 0$ of a more . How do I prove if $\sup\{ (n!)^{1/n} | n \in\Bbb N\} = +\infty$?  I don't know how to properly explain it How do I get HTML5 h.264 Video working on Firefox? According the the , h.264 video should now be working in Linux.  It does also say the appropriate gstreamer plug-ins need to be installed.  Does anyone know what these are?    I used this  and the  to verify that h.264 support works in Google Chrome on my Ubuntu 12.04 64-bit, but does not work on Firefox 26. Usually in the morning I brush my teeth after breakfast. But my dentist told me that it's better not brush them right after eating food, because eating makes them soft.  So now I brush my teeth before breakfast. Is this the correct procedure?  PS. Just found , the author has the same opinion.  Just want to get a additional confirmation. Meta-analysis of means and medians in R? I've been asked to review the literature for all studies that estimate the mean and median Blood Lead Level (BLL) in a certain country, and then perform a meta-analysis to come up with one overall value for the mean and median level.   Is such a thing typically done? I'm not interested in any "effect size," just a univariate measure.   I'd like to create a forest plot and do some of the standard tests of heterogeneity across studies. Is there a package and function in R that will allow me to do a meta-analysis of mean and median BLLs?  Just to clarify, this would be a separate meta-analysis for means, and a separate meta-analysis for medians. Why do proof authors use natural language sentences to write proofs? I haven't read very many proofs. The majority of the ones that I've read, I've read in my first-year proofs textbook. Nevertheless, its first chapter expatiates on the proper use of English in mathematical proofs, so I suspect that most proof authors do use both English (or another natural language) and formal proof systems for writing their proofs. However, I can't imagine many circumstances where mathematical notation and a formal proof system wouldn't suffice to convey the author's pertinent thoughts.   I suspect that proof authors have good reasons for making use of both English and formal expressions in their proofs, but I can't think of what those reasons are. Why do proof authors use English and formal expressions to write proofs? Are there stylistic rules (or guidelines) that state how, and in what parts of the proof, an author should, and shouldn't, use English? Analogue of closed graph theorem This is the analogue of closed graph theorem for compact space          Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset of $X \times K$. Prove that $f$ is continuous.      Because metric space is first countable, all we need is to prove that if $x = \lim_{n \rightarrow \infty} x_{n}$, then $f(x) = \lim_{n \rightarrow \infty} f(x_{n})$. Suppose $x = \lim_{n \rightarrow \infty} x_{n}$, then because $K$ is compact, sequence $\{f(x_{n})\}$ has a subsequence $\{f(x_{n_{i}})\}$ converge. Because the graph of $f$ is closed, we must have $\lim_{n \rightarrow \infty}\{f(x_{n_{i}})\} = f(x)$. All we need to prove now is $\{f(x_{n})\}$ converge to $f(x)$, but I can't prove that. Can any one help me. I really appreciate. Thanks The most efficient way to implement an integer based power function pow(int, int) What intuitively is "bias"? I'm struggling to grasp the concept of bias in the context of linear regression analysis.    What is the mathematical definition of bias? What exactly is biased and why/how? Illustrative example? Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$ Can anyone give a proof of the derivative of this type of function? Specifically showing that  $\dfrac{d(x^w)}{dx} = wx^{w-1}$ for a real $w$?  I tried to use the Taylor series expansion for $(x+dx)^w$ and got the correct result. However, the proof of the Taylor series requires knowledge of the derivative of these functions. So this is essentially circular reasoning. I know that the same series is also given by the binomial expansion, but that's not entirely satisfactory either, because where's the proof that the binomial expansion works for all reals (isn't it only apparent for integers)? So far all of the arguments I've come across involve circular reasoning. I was thinking of showing that the binomial expansion is true for all reals using some form of proof by induction e.g. something like this.  I'm really not sure. Is the notorious $n^2 + n + 41$ prime generator the last of its type? Use of "though" versus "however" Consider the sentence      E-books are on the rise, but they haven't suppressed paper books though.    This usage seems to be quite common, but when I learned English  I was taught to use "however" where I now read "though". My questions on this use of "though" are:     Is this a new trend? Is it restricted to American English? Is "though"  considered more colloquial than "however" ?    And on a syntactic level: Do you put a comma before "though"? How to Delete Elements from List1 appearing in List2? $f(f(x)f(y))+f(x+y)=f(xy)$ Find all functions $f\colon \mathbb R\rightarrow \mathbb R$ such that for all reals $x$ and $y$:   $$f(f(x)f(y))+f(x+y)=f(xy).$$   It was six hours ago in IMO 2017 (problem 2).  I tried  the standard way: $x=0$, $x=y$, $x=1$,etc. but without any success.   Let $f(0)=a$. Hence, for all $x\in\mathbb R$ we have $f(a(f(x))+f(x)=a$. What is the rest?   Also we have $(x-1)(y-1)-1+x+y-1=xy-1$ and  $1-(1-x)(1-y)+1-x-y=1-xy$. Rules when using a Joker I had on my rack among other tiles, a black 9, 1 x blue 10, and a black 12.  On the table was a set of 3 x 10s including a Joker(no blue 10), and other sets of tiles including an available black 11.  Can I replace the joker with my blue 10 and then set out :  my black 9, the joker(representing a black 10), a black 11 from the table, and my black 12?