Because of distributive properties we know this is true, but I am stuck on showing the proof. I know: Let $x\in A\times (B-C)$. This means $x\in\mathbb{A}$ and $x\in B$ but $x\ not in C$. Assume $x\in A$ then from here im not sure where to go.
This is what I have come up with so far, but I am rather lost at lines 3-4...  $$\begin{align} (a,b)\in A\times (B\setminus C)\iff&amp; a\in A \land b\in (B\setminus C)\\ \iff&amp; a\in A \land (b\in B \land \lnot(b\in C))\\ \iff&amp; (a\in A \land b\in B) \land (a\in A \land b\notin C)\\ \iff&amp; (a\in A \land b\in B)\setminus (a\in A \land b\in C)\\ \iff&amp; (a,b)\in (A\times B)\setminus ((a,b)\in A\times C)\\ \iff&amp; (a,b)\in (A\times B)\setminus (A\times C) \end{align}$$
Maximum Likelihood Estimate with different parameters Vertical line only in one part of a table I need some help on  putting a vertical line in the second part of the table, I don't know how to do it since the tabularx I declared does need to have that only in the second part. The code I used is:  \begin{tabularx}{\linewidth}{|XX|} \hline \multicolumn{2}{|c|}{\textbf{title name}}\\ \hline something: &amp; else \\ something: &amp; else \\ something: &amp; else \\ \hline \multicolumn{2}{|c|}{bla bla blah}\\ \hline title text: &amp; description 1 \\ title text: &amp; description 1 \\ \hline \end{tabularx}   And the render of the table looks like this:    I just need a \vline between title text and description (it I use \vline it works only for the first line but the description text and sometimes the title, are few lines long). "Error: Invalid signature" when configuring dual boot Ubuntu 14.04 and Windows 10 Given a set $\{1,2,3,4\}$, how is the following relation $R$ antisymmetric?  $$R = \{(1, 2), (2, 3), (3, 4)\}$$  Note: Antisymmetric is the idea that if $(a,b)$ is in $R$ and $(b,a)$ is in $R$, then $a = b$. In my textbook  it says the above is antisymmetric which isn't the case as whenever $(a,b)$ is in $R$, $(b,a)$ is not. I'm finishing professional school and over the years have acquired a pile of notes and articles that I'd like to hang onto.  I'd like to add to them and create sort of an archive of article and files that may be useful down the road.  I'd also like to organize this collection of files not only by simple grouping but also with tags.  I feel like that will make searching through them years later much easier.  Suggestions on software that would be good for this?  Just a general file manager, something that uses tags that can be attached to each file? Find the maximum value of $72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx$ Find the maximum value of $72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx  $ for $y\in[0,1].$  I tried to differentiate the given function by using DUIS leibnitz rule but the calculations are messy and I tried to solve directly by integrating it but that also is not working.Can someone please help me in solving this question? creating a diagonal matrix with a variable amount of parameters I've always wondered why the number of protons in the Universe exactly matches the number of electrons. They are such different particles with totally different cross sections. So, first of all, is this true? And, secondly, how could it be that they have been produced in the same amount after the Big-Bang? We know Euler formula:  $$ e^{ix}=\cos(x)+i\sin(x). $$  Does this formula work if we replace real number $x$ with complex number $z$? What is the meaning of modular fractions? What is the meaning of $$ \\\frac{m}{n}\equiv k\pmod l $$ and $$ \\\frac{m}{n} \mod l $$ $m, n, k, l \in\mathbb N$ and $\gcd(m, n)=1$. What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15 Question: Are my proofs below valid?   In both cases we are using: $f:A\to B, g: B\to C$  Notation of your type converted: $(g\circ f)(x)=g(f(x))=xfg$    If $fg$ is injective what can be said about $f$ and $g$?  If $fg$ is injective then $f$ is injective:  Proof:  $(x_1fg = x_2fg \implies x_1=x_2)$  $x_1fg=x_2fg$ is clearly true when $x_1f=x_2f$  Which requires $x_1=x_2$, hence $f$ is injective.  If $f$ were not injective, then $x_1fg\ne x_2fg$  $\blacksquare$    If $fg$ is surjective what can be said about $f$ and $g$?  If $fg$ is surjective then $g$ is surjective:  Proof:  $(\forall c\in C)(\exists a\in A) |( a(fg))=c$  This means for $g$ every element must have had been mapped to. How can this be proved Mathematically?  well $(af)g=c$ is surjective, hence $(af)=(b\in B)$ and $bg=c$  $(\forall c\in g)(\exists b\in B)|(bg=c)$  Hence $g$ is surjective.  $\blacksquare$ No file output after rendering? I usually am able to render successfully to the c:\tmp folder, but when I came into work today, I rendered out the image on my work laptop, and everything seems to work, but I'm not getting an output to the tmp directory.  Does anyone know how to make this work? How can I make a macOS Mojave bootable USB installer in Windows 10? Calculating polygon area within polygons in QGIS? I am new to QGIS - I want to estimate the area of the reef under a polygon grid. My shapefiles are already in meters, however I am having a hard time calculating the area under the grid layer. Attached are the photos. I want to calculate the total area of the bold black polygons (reefs) under each grid square (image 2). How much reef is inside each square, for example, grid1: 25 sq.km, grid 1 sq.km, etc. Each reef polygon has its corresponding area in the layer attribute table.    ADDITIONAL PHOTO: RED IS THE REEF  THE ATTRIBUTES OF THE REEF LAYER Is this correct? I thought it would be but when I entered it into wolfram alpha, I got a different answer.   $$\int (\cos^3x)(\sin^2x)dx  = \int(\cos x)(\cos^2x)(\sin^2x)dx  = \int (\cos x)(1-\sin^2x)(\sin^2x)dx.$$  let $u = \sin x$, $du = \cos xdx$  $$\int(1-u^2)u^2du = \int(u^2-u^4)du  = \frac{u^3}{3} - \frac{u^5}{5} +C$$  Plugging in back $u$, we get $\displaystyle\frac{\sin^3 x}{3} - \frac{\sin^5 x}{5}$ + C Why does $\sum_{k=1}^{\infty}\frac{{\sin(k)}}{k}={\frac{\pi-1}{2}}$? Inspired by  (and far more straightforward, I am guessing), Mathematica tells us that $$\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}$$ converges to $\dfrac{\pi-1}{2}$.   Presumably, this can be derived from the similarity of the Leibniz expansion of $\pi$ $$4\sum_{k=1}^{\infty}\dfrac{(-1)^{k+1}}{2k-1}$$to the expansion of $\sin(x)$ as $$\sum_{k=0}^{\infty}\dfrac{(-1)^k}{(2k+1)!}x^{2k+1},$$ but I can't see how...  Could someone please explain how $\dfrac{\pi-1}{2}$ is arrived at? Changing the resistor to be in front or behind an LED doesn't affect brightness? Consider the sequence $(\phi_n)_{n\in \mathbb{N}}$ of test-functions $\phi_n\in \mathcal{D}(\mathbb{R})$ defined by  $$\phi_n(x) = \dfrac{n}{\sqrt{\pi}}e^{-n^2x^2}.$$  I want to show that this is a delta sequence, in the sense that:  $$\lim_{n\to\infty}(\phi_n,\psi)=(\delta,\psi)=\psi(0), \quad \forall \psi\in \mathcal{D}(\mathbb{R}).$$  In that case we know that we have  $$(\phi_n,\psi)=\dfrac{n}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-n^2x^2}\psi(x)dx.$$  The problem here is to compute this integral. The function $x\mapsto e^{-n^2x^2}\psi(x)$ has no singularities, hence contour integration wouldn't help. Integration by parts or substitution also doesn't seem to be of great help.  The major problem is that there is a function $\psi$ inside the integral about which we don't know really anything apart from the fact that it is $C^\infty$ and with compact support.  How can we compute this integral and in the end show that the sequence $(\phi_n)$ is a delta sequence?  Just to make clear, I'm here doing all of this in the context of the Riemann integral. Noun for "receive"? Receipt?  Receiving?  I'm trying to come up with a name for "the acceptance of a delivery."  I'd like to be able to say      Ten deliveries were made but there are only five [fill_in_the_blank_here] on record.   Any suggestions? Why don't wizards always use multiple wands?