It may be the case that this is extremely trivial to prove, but I just cannot see it. I want to prove this:  $|a\alpha-b\beta|\leq|a-b|+|\alpha-\beta|$  for $0\leq a,\alpha,b,\beta\leq1$  I think it is also the case that  $|a\alpha-b\beta|\leq|a-b + \alpha-\beta|$  I actually THINK these might hold because I cannot provide any counter example.  Any kind of help would be extremely appreciated.
Let a,b,c,d be real or complex numbers with absolute value less than $1$. Is the following true:  $$|ab-cd| \leq |a-c| +|b-d|?$$
What is the distance between the centers? Two unit circles overlap in such a way that exactly half the area of each circle is within the overlap. How far apart are the centers of the two circles? The title says it all.  If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?  So my son posed me an interesting question today: since light travels as fast as anything can go, what if I shined light when moving in a car?  How should I answer his question? I want to know where the Metro applications (Mail, People, Messages, ...) executable file path is.  Also, how can I run Mail application when my PC starts? can't send email to addresses at my own domain I just started blender and one of the main things you have to do is to render to make an animation, but I can't seem to do that. Whenever I click Render and then render image or render animation, at the top of the screen it says 'No camera found in Scene "Scene". I don't know why this is happening. Prove that any complex matrix can be expressed in term of Hermitian and anti-Hermitian matrices How can we prove that any complex matrix can be expressed in term of hermitian and anti-hermitian matrices of the following form $A = H_1 + iH_2$ Where, $H_1$ is hermitian and $H_2$ is an anti-hermitian matrix Changing SRID of existing data in PostGIS? While importing my shapefile data to PostGIS, I did not select the proper Projection.  How do I now change the SRID of the data, without transforming the Coordinates? timestamp, modification time, and created time of a file Proving that the Intersection of any Collection of Compact Sets is Compact I was trying to do this problem this way:  Let $\mathcal{B}=\{B_i\}$ be a collection of compact sets. By Heine-Borel, each of the $B_i$'s are closed and bounded. We already know that the intersection of a collection of closed sets is once again closed, so $\cap \mathcal{B}$ is closed.  If I can prove $\cap\mathcal{B}$ is bounded as well, I can use Heine-Borel to say it is compact. So I was not sure how to show it was bounded, but a friend showed me his way of doing it. He did it the following way:  Let $b,a\in\mathbb{R}$ be an upper and lower bound for $B_1$, respectively. Since $\cap\mathcal{B}\subset B_1, a$ is a lower bound for $\cap \mathcal{B}$ and $b$ is an upper bound for $\cap\mathcal{B}$. Therefore, $\cap\mathcal{B}$ is bounded and then by Heine-Borel, it is compact.  My question is on the paragraph right above. I don't understand why what he did means that $a$ and $b$ are lower and upper bounds for $\cap\mathcal{B}.$ Can anyone explain why this is? Thanks. How to make applications open on the correct monitor when using multiple monitors? I'm using Windows 8.1, which has much better multiple monitor support than Windows 7 (sort of). However when I'm in a full-screen app on one monitor, and I go to the Start menu on the other and open a desktop app, it opens on the opposite monitor because of it being on that one last time the app was used.  This suspends the full-screen app and switches to the desktop view on that monitor. I then have to drag the desktop app to the other monitor which I didn't have my mouse pointer near in the first place, then switch back to the full-screen app on that monitor by going to the upper-right corner.  This is so incredibly irritating, every time I plug in my laptop to my second monitor and try to use it, this happens. Not to mention whenever I have Netflix or any kind of video in the second monitor and it just randomly stops playing when I'm trying to work on the other screen.  Most likely there is going to have to be a third-party application to install to do this, right? I would really prefer if it was free, as this was less annoying in Windows 7 when the application would only open on top of the other that was on that monitor. This is one of the maybe two annoying things I don't like about Windows 8/8.1. Everything else is fantastic, to be honest. How can I use a variable in the replacement side of the Perl substitution operator? not working in Phonegap app on Android 2.3.3 Let $G$ be a finite group and $N$ a normal subgroup of $G$ of index $4$. Prove that $G$ has a subgroup of index $2$ (hence normal) and if $G/H$ is not cyclic then $G$ is equal to the union of three proper subgroups.   I understand that $G/H$ is isomorphic to $\mathbb{Z}_4$ or $\mathbb{Z}_2\times \mathbb{Z}_2$. Let $K^*$ be a subgroup of $G/H$ of order $2$. Then $K^*\cong K/H$ for some $K$ such that $H$ is normal in $K$ and hence $[G:K]=[G/H:K/H]=4/2=2$.  If $G/H$ is not cyclic then $G/H\cong \mathbb{Z}_2\times \mathbb{Z}_2$. I don't know how to prove $G$ is the union of three subgroups. Could it be true that as $G/H$ is the union of three proper subgroups $K_1^*,K_2^*,K_3^*$ then the corresponding groups $K_1,K_2,K_3$ have union equal to $G$? Would anyone give me a hint to prove this?  Thank you. If I run history, I can see my latest executed commands.  But if I do tail -f $HISTFILE or tail -f ~/.bash_history, they do not get listed.  Does the file get locked, is there a temporary location or something similar? How does rolling two d20 and taking the higher affect the average outcome? D&amp;D 5E has an "advantage" concept where instead of rolling 1d20, you roll 2d20 and take the higher. Likewise, disadvantage means rolling 2d20 and taking the lower.  How does this affect the expected average outcome of the roll? Show that $\text{C}/c \cong (c/(\text{C}^{\text{op}}))^{\text{op}}$ I'm self studying from Category Theory in Context by Emily Riehl and have encountered the following question regarding slice categories:     Let $\text{C}$ be a category and $c$ be an object in $\text{C}$. Show that $\text{C}/c \cong (c/(\text{C}^{\text{op}}))^{\text{op}}$    The exercise appears in an chapter that introduces the notion of opposite categories and duality proofs.     My intuition of why the statement is true comes from the following observation:  Let $x,y$ be objects in $\mathbf{C}$ (or equivalently $\mathbf{C}^{\text{op}}$). Let $g^{\text{op}} : c \to y$ and $f^{\text{op}} : c \to x$ be objects in $c/(\mathbf{C}^{\text{op}})$. By the definition of slice category, there exists a morphism $h^{\text{op}} : y \to x$ in $\mathbf{C}^{\text{op}}$ such that the diagram below commutes:  $\hskip2in$   Now the 'corresponding diagram' in the opposite category $(c/(\mathbf{C}^{\text{op}}))^{\text{op}}$ would be:  $\hskip2in$    which, as I understand it, would also be commutative. Now, to me, this indicates:   that the objects in $(c/(\mathbf{C}^{\text{op}}))^{\text{op}}$ are the same as the objects in $\mathbf{C}/c$, namely, a morphism in $\mathbf{C}$ from an object $z \in \mathbf{C}$ to $c$. the commutativity of the seconds diagram describes morphisms $h$ in both $\mathbf{C}/c$ and $(c/(\mathbf{C}^{\text{op}}))^{\text{op}}$ if you reverse the arrows again you return to the same diagram. This makes me think you could construct a map between morphisms by first making a map from $ \mathbf{C}/c \to (\mathbf{C}/c)^{\text{op}} ( \cong c/(\mathbf{C}^{\text{op}}) )$ by $h \mapsto h^{op}$     Now, what I am really looking for is a way to structure a proof. I feel like I have some intuition but am very unsure of how to write it down in a concise way. I am new to category theory so I'm still very much learning how to construct these types of proofs using duality; I feel like what I have written is maybe overkill but I'm trying to be as explicit as possible to make sure I understand everything.  In particular, I don't know how I would get a concrete correspondence between morphisms in the two categories that respects associativity of composition etc.  Any help or pointers would be appreciated! Prove that if $B$ is the set of rationals in $[0,1]$ with a finite subcover, then: $1 \leq \sum_{k=1}^n m^*(I_k)$ I have the following homework problem:  Let $B$ be the set of rational numbers in the interval $[0,1]$, and let $\{I_k\}_{k=1}^n$ be a finite collection of open intervals that covers $B$. Prove that $\sum_{k=1}^n m^*(I_k) \geq 1$.    This is my approach, First we have that $B = \mathbb{Q}\cap [0,1]$. Now, $B \subset \bigcup_{k=1}^nI_k$, where $I_k$ are open intervals.In particular $0 \in B \wedge 1 \in B$.Now, since $B \subset \bigcup_{k=1}^nI_k$ then there exist positive numbers, $\epsilon_0, \epsilon_1$ such that: $$(0-\epsilon_0,0+\epsilon_0)\subset \bigcup_{k=1}^nI_k$$ $$(1-\epsilon_0,1+\epsilon_0)\subset \bigcup_{k=1}^nI_k.$$   Clearly, $-e_0 \notin [0,1]$ and $e_1 \notin [0,1]$.I am not sure, if so far I am in the right track. Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$. Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$.   I can find all the infinite subgroups of the form $n\mathbb{Z}\times m\mathbb{Z}$, where $n,m$ run over $\mathbb{Z}$. But I don't know how to write out all the finite subgroups in a uniform expression.  Any suggestions? Thanks in advance. What's the difference between absolute path &amp; relative path when using any web server or Tomcat? Why does ice form on bridges even if the temperature is above freezing? So with this "arctic blast" continuing, I've noticed that for my area, the temperature drops below freezing just long enough to cause freezing rain, but then the sun comes out and the temperature rises immediately. However, on bridges, ice continues to form.   How can ice form even if the temperature is above freezing?