My book "Introduction to SetTheory" says      $\{\emptyset\}\in\{\{\emptyset\}\}$ but $\{\emptyset\}\not\subseteq\{\{\emptyset\}\}$   When we say $\{\emptyset\}\not\subseteq\{\{\emptyset\}\}$, we mean that $\emptyset$ is not listed amongst the elements of $\{\{\emptyset\}\}$.  I was under the impression that every set contains the empty set. Hence, the elements of $\{\{\emptyset\}\}$ would be $\emptyset,\{\emptyset\}$. If that were true, wouldn't the assertion that $\{\emptyset\}\not\subseteq\{\{\emptyset\}\}$ be false?
$\{\emptyset\}$ 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 $\{\emptyset\}$".
FPDF error: Some data has already been output, can't send PDF I'm being asked to determine if $\displaystyle\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$ converges. So, using Cauchy's Condensation Test, I reduced the problem to one of determining the convergence of $\displaystyle\sum\limits_{n=3}^\infty\frac 1{n\ln (n\ln 2)}$. Am I on the right path, and how do I proceed from here? Find a quadratic equation with root $x_1^2$ and $x_2^2$ I'm struggling to solve this problem:  "If the equation $ax^2 + bx +c$ $(a \neq 0)$ admits real and not null roots of $x_1$ and $x_2$, obtain the equation which evaluates to roots $(x_1)^2$ and $(x_2)^2$"  I know the answer is $a^2x^2 -(b^2 - 2ac)x + c^2$ but I am not able to get to this answer, I was trying to derive it from the sum and product equations $S=-b/a$ and $P=c/a$ but the answer I'm getting to is a monstruous different thing.  Could someone explain to me how to get to this answer?  Thanks Proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain We're proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain, using the norm function $$\nu (a + b\sqrt{2} ) = |a^2 - 2b^2|$$ and the first part says that since $\nu (a + b\sqrt{2} ) = |(a + b\sqrt{2})(a - b\sqrt{2})|$ it's clear that $\nu (xy) = \nu(x) \nu(y)$? ... Can someone please explain to me how this is clear? so I am trying to show that $f(x)$ divides $g(x)$ for all polynomials $g(x)$ satisfying that $g(A)=0$ where $f(x)$ is the minimal polynomial of a square matrix $A$.  I know from my professor that $f(x)$ is the minimal polynomial of $A$ so $f(A)=0$ so then $g(A)=0$. Therefore, $f(x)$ divides $g(x)$, i.e. $g(x)=f(x)h(x)$ where $h(x)$ is some other polynomial.  My prof told us to solve this question we need to let $b(x)=gcd((f(x),g(x))$ and by using Bezout's Identity, we need to show that $b(A)=0$. I am just having a little trouble trying to prove this. I know that this identity says:     For two polynomials $f_1(x)$ and $f_2(x)$, there exist polynomials   $g_1(x)$ and $g_2(x)$ such that $g_1(x)f_1(x) + g_2(x)f_2(x) = &gt; gcd(f_1(x), f_2(x))$.   And the uniqueness of minimal polynomial theorem is:     For a square matrix $A$, if $f_1(A) = f_2(A) = 0$ for two polynomials   $f_1(x)$ and $f_2(x)$, then $g(A) = 0$ for $g(x) = gcd(f_1(x), &gt; f_2(x))$. So the minimal polynomial of $A$ is unique up to a scalar.   So since $f(x)$ is already the minimal polynomial, I know that the $deg(b(x))=deg(f(x))$. I know how to prove the Bezout identity using the Euclidean algorithm, I'm just not exactly sure how I would show all of this using the notation he gave us of $b(x)=gcd((f(x),g(x))$. Any help at all would be appreciated. I've got a DVD that I've burnt from an MSDN ISO. I'm not entirely sure if its Windows 7 64bit or 32bit.   How can I be sure? Does there currently exist a free online service that accepts a file, hashes it, takes an authoritatively chosen timestamp (from one or more time services), signs these and sends this signed message back? E.g. a digital notary, but only with regards to the timestamp and not the validity of the content as legal notaries might be obligated to guarantee.  Such a service might have an extremely simple API: Simply wget http://service/&lt;checksum&gt; and receive a signed message containing one or more timestamps and the checksum. The emphasis would be on the type of authority who might make such a service.  Edit: Although the original question did not emphasise the following, the Bitcoin-based timestamping services provided below do not rely on one authority, but rather the entire Bitcoin network. They can be seen as tamper-proof and verifiable without the help of the service that helped you sign it! $ echo $(( 255 )) 255 $ echo $(( 33 )) 33 $ echo $(( ~33 )) -34 $ echo $(( ~255 )) -256 $    and my kernel is:  $ uname -a Linux HOSTNAME 3.2.0-40-generic-pae #64-Ubuntu SMP Mon Mar 25 21:44:41 UTC 2013 i686 i686 i386 GNU/Linux     QUESTION: ~ is for negating the number AFAIK. But why does ~33 produce -34 and why does ~255 produce -256? Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$? Is there a measurable set $A$ such that $m(A \cap B)= \frac12 m(B)$ for every open set $B$?  Edit: (t.b.) See also  for further answers. I would like to typeset the top part of Pascal's triangle. To get the triangle with the names of the binomial coefficients, i.e., {n \choose k}, I used the following code  \begin{tikzpicture} \foreach \n in {0,...,4} {   \foreach \k in {0,...,\n} {     \node at (\k-\n/2,-\n) {${\n \choose \k}$};   } } \end{tikzpicture}   The result is this   Now I want to be equally lazy and do something like this for the values of the binomial coefficients, i.e., replace {\n \choose \k} in the node label with \CalculateBinomialCoefficient{\n}{\k} where \CalculateBinomialCoefficient is a hypothetical macro that calculates the binomial coefficient. Has anyone done something like that?  The result should look like this: Sort file by group of lines If I have a file with contents similar to:  FirstSection     Unique first line in first section     Unique second line in first section  SecondSection     Unique first line in second section     Unique second line in second section  ...  NthSection     Unique first line in Nth section     Unique second line in Nth section   Is it possible to use unix commands (e.g. sort, awk) to sort the file alphabetically by the first non-indented line in each three line group, whilst keeping the indented lines under their existing group? Google is counting the domain home page ("/") as a duplicate of "/index.html" For only my index pages, I am receiving this warning in Google Search Console:     Duplicate, submitted URL not selected as canonical   I know that this occurs because Google is attempting to track both the default landing "page" (i.e. www.domain.example/) and the page to which to which DirectoryIndex points (i.e. www.domain.example/index.html). They are, in fact, the same page, and so the warning is the result of a false positive.  Two questions for our community:   Why does Google not conflate these and count them as a single page? Alternately, is there something wrong with my set up if this has happened? I can easily force both pages to point to a single canonical URL. But which is preferable, "/" or "index.html"? How can I simplify the use of flags with Views Bulk Operations? I have an administrative display (made with views) that lists content that is pending approval.  In the case of my site, approval means flagging with the approval_for_pub flag (which triggers a custom rule).   allows nodes to be flagged/unflagged all at once by clicking a handy button:    However, upon clicking this button, by default it shows all possible flags and the flag/unflag option:    This is a problem on my site because some of the people responsible for approval get confused by the names of the flags and select the wrong option.  So, what I would like to do is create a VBO option that, when the button is pressed, automatically flags all applicable content with the approval_for_pub flag.  How can I approach this?  It seems like a small detail, but this step happens several times a day, so the chance of making a mistake is quite high. Hide Accesibility icon from Panel I have Ubuntu 18.04 and set the Scaling Factor in Tweaks to 1,10 because the 1920x1080 resolution makes stuff on my screen look a bit too small for my taste. That toggled this stickman icon, which now seems to not want to go away from my Panel unless I disable "Large Text".    Is there any way to hide this icon while keeping my settings? Understanding 74LVC245 datasheet and current source/sink I'm using a SN74LVC245A () to shift 5V TTL down to 3.3V so I can read the signals with a Raspberry Pi.  However I am getting really weird results (a 60Hz input signal is picked up by the Pi at anything between 110Hz and 320Hz) and I'm really confused about how to read the datasheet to figure out whether I need pullup or pulldown resistors.  The datasheet says on page 4 the "high-level output current" is -24mA, and the "low-level output current" is 24mA.  So to me this means that when the device is outputting a high signal, it is able to sink 24mA of current (since outputting a negative amount of current suggests the current is going into the device instead.)  But then if you're outputting a positive voltage, isn't that sourcing current?  Why is the figure listed as a negative value?  Is anyone able to explain what these figures actually mean?  When I disconnect the 74LVC245 output from my GPIO pin, I still get weird signals from the floating input pin so I am assuming this means the 74LVC245 output is also floating in one state, but I don't know in which state it is (i.e. is it left floating on a high output signal or a low one?)  It doesn't really make sense to be floating on a high output signal since it's a level converter and the whole point is to maintain the correct 'high' voltage, so do I need pulldown resistors to handle the low output?  If so, why does the datasheet seem to suggest the chip can both source and sink current?  (In my mind, source current means the device can output a current with a positive voltage, while sinking current means the device takes in anything and connects it to GND, so perhaps that is where I am going wrong.)  Anyway I'm quite confused so any clarification would be much appreciated! Define the content of a polynomial (over an arbitrary commutative ring $A$) to be the ideal generated by its coefficients, denoted $c(f)$. I want to show that   $$c(fg) = c(f)c(g).$$      (I was told this is true.)    What I was able to show was that $c(fg) \subseteq c(f)c(g)$ (this is obvious), and that their radicals are the same. My reasoning for the latter was as follows: let $f = a_0 + \dotsb + a_nx^n,$ $g = b_0 + \cdots + a_mx^m,$ and consider the matrix  $$ \begin{pmatrix}  a_0 b_0 &amp; \cdots &amp; a_0b_m\\ \vdots &amp; \ddots &amp; \vdots\\ a_nb_0 &amp; \cdots &amp; a_nb_m \end{pmatrix} .$$  Then $c(f)c(g)$ is generated by the entries of this matrix, while $c(fg)$ is generated by the sums along the diagonals. Let $\mathfrak p$ be a prime ideal of $A$ not containing $c(f)$ or $c(g)$, and let $i,\,j$ be minimal such that $a_i,\, b_j \notin \mathfrak p$.  Then all the terms in the generator of $c(fg)$ corresponding to the coefficient of $x^{i+j}$ are in $\mathfrak p$ except for $a_ib_j$, so that $c(fg) \not\subset\mathfrak p$. It follows that $c(fg) \subseteq \mathfrak p \Longleftrightarrow c(f)c(g) \subseteq\mathfrak p$, and hence $\sqrt{c(fg)} = \sqrt{c(f)c(g)}$.     Can I go all the way and show that in fact $c(fg) = c(f) c(g)$? Tachyons and Photons Is there a particle called a "tachyon" that can travel faster than light?  If so, would Einstein's relativity be wrong?  According to Einstein no particle can travel faster than light. How to create a filter in Gmail to identify messages forwarded from Zimbra? Prove that if a number $n > 1$ is not prime, then it has a prime factor $\le \sqrt{n}$. Find Total number of ways out of N Number taking K numbers every M interval I have been stuck in a problem, that has thrown my brain out of the coding. This problem is at very high priority and I need the solution as early as possible. Problem is as :  There are exactly N advertising boards on the highway. Now a company want to advertise on some of these advertising boards(each advertising board costs some money) .  Company strategy is that, they want at least 'K' advertisement should be there among M consecutive advertising boards. But at the same time Company want to pay minimum for its advertisement .  Now, what is the total number of ways, different ways Company can advertise meeting its minimum cost strategy.  As for Example: N= 3, M = 2, K=1 ==> there is only one way for minimum cost, ie. 0C0 , where '0' denotes No company advertisement, and 'C' denotes company advertisement board.  Similarly, for N =4, M =2, K =1 ==> there is 3 possible ways, ie. C0C0, 0C0C, 0CC0.  Hope to get a reply from someone soon. Thanks.