Prove that if H ∪ K is a subgroup of G... Suppose G is a group, with subgroups H and K.   Prove that if H ∪ K is a subgroup of G implies that H ⊆ K or K ⊆ H.   I'm not really sure how to start this, I can prove that H ∩ K is a subgroup but I don't know how to approach the union. Any help is appreciated.
Union of subgroups is a subgroup if and only if one subgroup is a subset of the other Let $H$ and $K$ denote two subgroups of a group $G$.  Prove that the union $H \cup K$ is a subgroup of $G$ if and only if $H \subset K$ or $K \subset H$.
self referential struct definition? In my project requirement to access table data from Mapd Database on GeoServer  Mapd(GPU Based database) Database documentation link:   But in GeoServer provided some extension to connect and access particular database  GeoServer Extension link:    How can I connect MapD databse with geoserver is there any plugin. Let $L$ and $R$ be $n \times n$ matrices. I am trying to solve the following minimization problem  $$ L^{k+1} := \arg \min_{L} \lambda \|L\|_{\ast} + \frac{1}{2\mu}\|L-R^{k}\|_{F}^{2} $$ $$ \mbox{s.t.}\ L \succeq 0 $$  where $\|\cdot\|_{\ast}$ and $\|\cdot\|_{F}$ denote the nuclear and Frobenius norms, respectively. Also, $k$ expresses the $k$th iteration when solving an optimization problem successively.  I don't know how to solve this problem. Would you tell me the way? Issue with particles in blender donut tutorial Is using std::deque or std::priority_queue thread-safe? Can I downgrade to Mavericks after installing Yosemite (beta/developer preview)? I have installed the developer preview for Yosemite.  Huge mistake. Everything in my workspace is broken.  I don't have any sort of backup image. I was wondering if I could re-download Mavericks and run the installer - would that effectively downgrade me back to Mavericks? How does gravity escape a black hole? Prove that there exists such a set of $n$ positive integers Prove that for any positive integers $m$ and $n$ , there exists a set of $n$ consecutive positive integers each of which is divisible by a number of the form $d^m$, where $d$ is some positive integer not equal to $1$.  I don't know how to approach this question. Currently, when rep is changed due to the removal of another user account (to undo the rep gain/loss that happened when they up-/down-voted/accepted/were downvoted by you), the associated message is simply "User was removed". I am not the only one who found that message rather cryptic (I had to go search meta when I saw it, and I found my answer in a duplicate, so that makes me at least the 3rd person to not understand it).  I think it's a good idea to add a link to  or some more descriptive text to that.  Edit: e.g.     Another user was removed; reputation changes linked to the user's activity were reverted.   (I can see how it's hard to fully explain the event succinctly, that's why a "what is this?" link to meta or an FAQ question/answer might be better.) When I add the picture, the word and pdf is the full page and the Latex moves to the right of the page. How can I solve this problem?  this is my work. this is my work and pdf link: https://we.tl/t-cgDSwDuu8H  I couldn't stop the pictures from sliding right. What should I do to make the image look like the original PDF file? If I reduce the scale of the picture is centered but this time it becomes very small.  Is a situation caused by page margins?  \documentclass [12pt, a4paper]{article} \usepackage{color} %\usepackage[turkish]{babel} \usepackage{graphicx} \usepackage{tabularx} \usepackage{multirow} \usepackage{graphicx} %\usepackage[pass,showframe]{geometry} %\usepackage[T1]{fontenc}  \begin{document}  \definecolor{renkbir}{RGB}{0,32,96}  \begin{tabular}{ m{8,5cm} m{6cm} } \Large{\textbf{MsdasdasdN }} \large{\textbf{\textcolor{renkbir}{Firma Adı }}} &amp;\small {Uçak Mühendisi \newline T.C No: asd \newline asd \newline asd \newline asd} \end{tabular} \ %Uçak mühendisi Tc No kısımını sağa yaslamak istedim fakat yapamadım.  \begin{center} \large{\textbf{\textcolor{renkbir}{KİŞİSEL BİLGİLER}}} \end{center}  \begin{center} \begin{tabular}{ l l r } Ad &amp; Mehm &amp; \multirow{9}{*}{\includegraphics[scale=0.15]{2.pdf}} \ Soyad &amp; asdasd \ Doğum Tarihi &amp; sanane (23 Yaşında) \ Adres &amp; asd \ Telefon &amp; asd \ Mail &amp; asd \ Sigara Kullanımı &amp; Hayır \ Ehliyet &amp; B \ Askerlik &amp; Tecilli (31.12.2020) \ \end{tabular} \end{center}  %\vspace{1cm} Burası boşluk bırakma komutu.  \begin{center} \begin{tabular}{ m{4cm} m{6cm} m{5cm} } \multicolumn{3}{c}{\large{\textbf{\textcolor{renkbir}{EĞİTİMLER}}}} \ \hline \multirow{2}{}{Üniversite(YL)} &amp; asd \newline Makine Mühendisliği \newline Özel Öğrenci / Türkçe \newline Durumu : Devam ediyor &amp; \multirow{2}{}{Eylül 2018 / } \ \hline \multirow{2}{}{Üniversite} &amp; asd \newline Uçak ve Uzay Bilimleri Fakültesi \newline Uçak asd \newline Örgün Öğretim / İngilizce \newline Durumu : Mezun &amp; \multirow{1}{}{Ekim 2013 / Haziran 2018} \ \hline \multirow{1}{}{Lise} &amp; asd \newline Anadolu Lisesi (100 / 90,02) &amp; \multirow{1}{}{Ekim 2013 / Haziran 2018} \ \end{tabular} \end{center}  \begin{center} \begin{tabular}{ m{3cm} m{4cm} m{4cm} m{4cm} } \multicolumn{4}{c}{\large{\textbf{\textcolor{renkbir}{YABANCI DİLLER}}}} \ \hline \textbf{İngilizce} &amp;\textbf{Okuma:} İyi &amp;\textbf{Yazma:} İyi &amp;\textbf{Konuşma:} Orta \ \end{tabular} \end{center}  \begin{center} \begin{tabular}{ m{5cm} m{5cm} m{5cm} } \multicolumn{3}{c}{\large{\textbf{\textcolor{renkbir}{BİLGİSAYAR BİLGİLERİ}}}} \ \hline \multicolumn{3}{l}{Donanım - Ofis Programları} % donann ofis koyu yazmak istedim yapamadım. \ Ms Excel &amp; Çok İyi &amp; +5 Yıl \ MS Word &amp; Çok İyi &amp; +5 Yıl \ Ms Powerpoint &amp; Çok İyi &amp; +5 Yıl \ \hline \multicolumn{3}{l}{Yazılım Geliştirme} % donann ofis koyu yazmak istedim yapamadım. \ Office &amp; Orta &amp; 2-3 Yıl \ Matlab &amp; Orta &amp; 2-3 Yıl \ Phyton &amp; Orta &amp; 0-1 Yıl \ Fortran &amp; Başlangıç &amp; 0-1 Yıl \ C++ &amp; Başlangıç &amp; 0-1 Yıl \ Ansys &amp; Başlangıç &amp; 0-1 Yıl \ Miktex &amp; Başlangıç &amp; 0-1 Yıl \ \hline \multicolumn{3}{l}{Bilgisayar Destekli Tasarım} % donann ofis koyu yazmak istedim yapamadım. \ SolidWorks &amp; Orta &amp; 2-3 Yıl \ AutoCAD &amp; Orta &amp; 0-1 Yıl \ \end{tabular} \end{center}  \begin{figure} \includegraphics[width=\linewidth]{2.pdf} \end{figure}  \begin{figure} \includegraphics[scale=1]{2.pdf} \end{figure}  \end{document}   Latex SS is here:  PDF SS is here: Should I remove Adult Dependent from my 2019 tax return so they will receive a COVID Stimulus Check? On my 2019 Tax Return I claimed my 28 year old girlfriend as a dependent. This makes her ineligible to receive a COVID Stimulus Check. My return was already accepted and I have already received my refund. Should I amend my return, remove her as a dependent? Would she then be eligible to receive the stimulus check? We add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers).  Why this trick isn't performed with division by zero? Is there a formal spelling for the English letter names? Passing functions to setTimeout in a loop: always the last value? What can $E=mc^2$ do? When you tap a row in a UITableView, the row is highlighted and selected. Is it possible to disable this so tapping a row does nothing? "viruses" or "virii"? Below you find the design for our business cards. The triangles are yellow, as you can see outside of the canvas. However, inside the canvas, only when I set the Color Mode to CMYK, they appear pink. When I switch the Color Mode back to RGB, they appear yellow as expected. However, when I print them, in both cases they turn out pink...   I would just like to know what is going on here and how I can change them back to the right colors?  CMYK Colors: This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$ let $0&lt;a\le b\le c\le d$, and such $abcd=1$,show that   $$a+b^2+c^3+d^4\ge \dfrac{1}{a}+\dfrac{1}{b^2}+\dfrac{1}{c^3}+\dfrac{1}{d^4}$$   it seems harder than I read this book a few years ago which was part of a series, but I only found the first book and now I've forgotten the name.  The first book featured 3 children/teens from different places that end up on a quest to search for magic gems/stones together and the story takes place in a medieval kind fantasy world. One is a girl, while the other 2 are boys. One of the boys actually finds a stone/gem while he is on a school field trip (I think the gem/stone is red in colour), and when he picks it up, he gets transported to the fantasy land but ends up in some sort of military training ground/camp somewhere in a desert type area.  He isn't familiar with the fantasy world. The other 2 characters each start out on their own on a quest to collect some magical gems/stones which once collected, can be used to fight an evil ruler (I guess).  But halfway through the book, the girl and boy meet up and continue the quest together while the third boy(transported one) doesn't know that the stone he picked up is actually one of the magical stones others are looking for.  A few other details I remember:   The stone the boy finds is red in colour I think  The cover of the book had a border depicting the stones  It's part of a series or trilogy    Some books with similar storylines, but not the one I'm looking for:     Deltora Quest The Seven Towers Sword of Heaven series Westeria series Diadem series