Prove $X\times Y$ is an equivalence relation
If $X,Y$ are equivalence relations, so is $X \times Y$
Does Change Data Capture (CDC) work in a SQL Server 2012 AlwaysOn Failover Cluster Instances setup? Does  work in a  setup?  There would be two nodes in the AlwaysOn Failover Cluster Instances setup and two Domain Controllers.  We are not using AlwaysOn Availability Groups.  Will CDC work? and will it failover? Cyclic vectors in a real vector space Let $V$ be an n-dimensional vector space over $\mathbb{R}$ and $T:V \rightarrow V$ be linear. Call a vector $v \in V$ cyclic if $V$ is spanned by $\{v, \ Tv, \ T^2v,..\}$.  Question:  Show that $v$ is cyclic for $T$ then there is some $k \geq 1$ such that   $$B=\{v, \ Tv,..T^{k-1}v \}$$ is linearly independent in $V$ and that $T^kv$ lies in  the span of $B$.  Thoughts:  If we suppose that no such $k$ exists, then every subset of the form $\{v, \ Tv,..T^{l-1}v \}$ is linearly dependent, or it is linearly dependent with $T^lv$ not lying in it span.  There are in effect then two cases to consider.  Suppose every subset of the form $\{v, \ Tv,..T^{l-1}v \}$ is linearly dependent. Then $T^{l-1}$ can be expressed as a linear combination of $v,...,T^{l-2}v$. Now, $\{v, \ Tv,..T^{l-2}v \}$ is also linearly dependent by assumption and so we can express $T^{l-2}$ as a linear combination of  $v,...,T^{l-3}v$ etc... Continuing in this way, we show that $T^{l-1}v= \lambda v$ for some $\lambda \in \mathbb{R}$. Thus $$B=\{ \mu v \ | \ \mu \ \text{is an eigenvalue of} \ T^mv \ \text{for some} \ m \}$$ This is 'at best' countably infinite, and therefore could not span $V$ even if it were 1-dimensional, wince our base field is $\mathbb{R}$, which is uncountable.  Suppose then that $B$ is linearly independent for some $k$ but that $T^kv$ does not lie in its span. I can't see how to progress this argument.  Note: This question is for second year Undergraduates, and I think there must be some more simple approach; even the first part surprises me if we need to argue using the uncountability of $\mathbb{R}$, so any alternative approaches are also appreciated. Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$ Show that the set of functions $\mathbb{N}\to\{0,1\}$ is not countable Recall that a countable set $S$ implies that there exists a bijection $\mathbb{N}\to S$. Now, I consider (0,1). I want to prove by contradiction that $(0,1)$ is not countable.  First, I assume the contrary that there exists a bijection $f$, and I can find an element in $S$, but not in the range of $f$. But I can't find such element. How can you construct such $f$? Smoothness constraint of wave function Is there anything in the physics that enforces the wave function to be $C^2$? Are weak solutions to the Schroedinger equation physical? I am reading the beginning chapters of Griffiths and he doesn't mention anything. Planet where humans fight for access to algae Set out on another planet where a human spaceship lands for a research and they observer a huge mass moving below them. On closer look this mass appears to be humans, and the density of the population increases towards the sea. The reason for this is algae,the only nutrition found on the planet. On a beach they discover tall, huge humans (developed due to the natural selection), controlling the best access to the algae. Eventually the researchers die, because humans eat them. Conjunction Puzzle: Is this clause dependent or independent? Third grade teacher here. I plan to teach students to distinguish between simple, compound and complex sentences — but only if I can demonstrate a clear and meaningful difference between the latter two!  Complex or Compound? Each sentence in the following pairs is similar to its partner.   All dogs are mammals, so huskies are mammals. All dogs are mammals, therefore huskies are mammals. I come bearing soup, for Kate is sick. I come bearing soup, because Kate is sick.   Are the sentences above complex or compound?   First, we need to decide whether each italicized clause is independent or dependent. To me, they all look dependent. If that is the case, there's not much need to teach about compound sentences. Is there a meaningful difference between the sentences in each pair that I am missing? (Are the FANBOYS as special as they'd like to think?) Prove that $ n^3 + 5n$ is divisible by 6 for all $n\in \textbf{N}$ Prove that $ n^3 + 5n  $ is divisible by 6 for all $ n \in \textbf{N} $.  I provide my proof below. I have been around on Twitter for awhile and I regularly find people using "Please to retweet this" or "Please to help me", etc. Is this proper English? I do not think prefixing the infinitive form with a 'to' is necessary here at all. Distance from the surface as an input node for use with volumetric materials says the following:     Saving an OS can be achieved by running sudo grub-set-default if   DEFAULT=saved is set in /etc/default/grub. It may also be saved if   GRUB_SAVEDEFAULT=true is also set in /etc/default/grub. In this case,   the default OS remains until a new OS is manually selected    from the   GRUB 2 menu or the grub-set-default command is executed.   I put the lines DEFAULT=saved AND GRUB_SAVEDEFAULT=true in /etc/default/grub, and ran sudo grub-set-default.  Here is the output:  $ sudo grub-set-default entry not specified. Usage: grub-set-default [OPTION] entry Set the default boot entry for GRUB.    -h, --help              print this message and exit   -v, --version           print the version information and exit   --boot-directory=DIR    expect GRUB images under the directory DIR/grub                           instead of the /boot/grub directory  ENTRY is a number or a menu item title.  Report bugs to &lt;bug-grub@gnu.org&gt;.   Am I not following the documentation correctly?  What's the correct way to do this? Increasing notebook magnification pushes output onto a new line Difference between int[] array and int array[] Can we interchange phrases " What's the date today?" and "what day is it today?" Splitting pieces of a mesh into a new object I've just made a set of armor by using a male body mesh model I downloaded and selecting certain faces from the model and duplicating them to create viable armor pieces.   Since the pieces are just duplicated from the same mesh male model, then it's still part of the same object.   How do I separate the armor piece from the model to make the pieces separate? How to make grass appear on top of ground not in middle? I would like to use pgfplots to create a figure where I plot curves that have a recursive formula. I tried a very simple case that uses the declare function command to see whether this is possible, and I can't compile (TeX capacity exceeded). Is it even possible to do this?  \documentclass{standalone}  \usepackage{pgfplots}  \begin{document}  \begin{tikzpicture}[   declare function={test(\p)=ifthenelse(\p&gt;1, test(\p-1), 0.5);}, ]     \begin{axis}[]     % use TeX as calculator:     \addplot [mark=none] {x};     \addplot [mark=none] {test(0)};     \end{axis} \end{tikzpicture}  \end{document} Is the latest version of Ubuntu 100% FREE How should I structure my URLs for both SEO and localization? Change bounding box of math glyphs in LuaTeX I'm trying to use the Garamond-Math font in LuaTeX (despite the documentation saying I should use it with XeTeX only :-). It's working more or less OK with LuaTeX, except for some unfortunate glyph combinations, mostly at the end of inline math, such as “where $T$ is the period”.  I used \fbox to draw the bounding box of the glyph and noticed that the bounding boxes are different in LuaTeX:        and in XeTeX:        In XeTeX the glyph fits almost entirely in the bounding box, but in LuaTeX it overshoots far too much.  I'm not a font guy, so I have no idea why this happens. I thought the bounding box was a property of the font, so in my imagination the result should be the same. Thus the first question is: what actually happens that the bounding boxes are different?  The second question, in the title, is: can I change the bounding box of specific glyphs (using LuaTeX) so that the output looks more like XeTeX's?  I tried manually adding some spacing after the glyphs, but this looks hideous and will go down the drain if I happen to change the font or engine, so I'd really like not to do that. I also tried using the italic correction \/ but it seems to have no effect in math.  Here's the code to produce the pictures above:  \documentclass{standalone} \usepackage{unicode-math} \setmathfont{Garamond-Math.otf} \begin{document} \fboxrule=1sp \fboxsep=0pt In Lua\TeX{} this \fbox{$T$} looks horrible, but the \fbox{$X$} looks OK. \Huge\fbox{$T$}\quad. \end{document}   I'm using TL 2018:  This is LuaTeX, Version 1.07.0 (TeX Live 2018)  (format=lualatex 2019.3.29)  13 APR 2019 12:43  restricted system commands enabled. **test.tex (./test.tex LaTeX2e &lt;2018-12-01&gt; Lua module: luaotfload-main 2019-02-14 2.96 luaotfload entry point