True or False. For any set of sentences, T, if T semantically entails (double turnstile symbol) p v q, then either T semantically entails p or T semantically entails q.
In Chiswell's Mathematical Logic, there is the following problem is posed using the language of propositions (LP):     In some systems of logic (mostly constructive systems where 'true' is taken to mean 'provable') there is a rule      if $(\Gamma\vdash(\phi\,\lor\psi))$ is a correct sequent then at least   one of $(\Gamma\vdash\phi)$ and $(\Gamma\vdash\psi)$ is also correct.      By giving a counterexample to a particular instance, show that this is   unacceptable as a rule for LP. [Start by giving counterexamples for   both the sequents $(\vdash p_0)$ and $(\vdash(\neg p_0))$.]   In my , it was answered that one can let $\Gamma = \{p_0\,\lor\,(\neg p_0)\}$.  From there, one can deduce neither $(\Gamma\vdash p_0)$ nor $(\Gamma\vdash (\neg p_0))$.  But this does not proceed by the counterexample route that the author suggests.  Namely, he seems to imply that we should (1) take an interpretation of propositional variables $p_0,\,p_1,\,\ldots$ (setting each one equal to T or F), and (2) write $\phi$ and $\psi$ as functions of these propositional variables, and (3) pick sentences for $\Gamma$ which are all T, all with the conditions that (A) $(\phi\,\lor\psi)$ is evaluated as T, yet (B) both $\phi$ and $\psi$ are evaluated as F.  This, of course, seems impossible.  What, then, am I missing?  It seems that taking a concrete interpretation of the propositional symbols makes the constructivist's rule irrefutable.  (Note: this is supposed to be a very simple problem set, and I am an elementary logician).
Why can't XeTeX handle this dank meme? "Failed to download repository information" error I'm using a  freely downloadable latex class file and the style is set to \bibliographystyle{ieeetr}. But when I cite something with \cite{} I get it inside ().   For example I get it as abc et al. (1) but what I want to get is abc et al. [1].   Please tell me how to get my citations inside [ ] ? I'm trying to install the Jupiter power settings app but, I read it is no longer compatible with the new kernel. Is there a similar program or a way to manage the power settings on 13.04? How can I test the quality of a RNG? On electronics.stackexchange we had  about constructing a True Random Number Generator. Since the method relies on noise, which isn't deterministic, the only way to test the quality of the RNG seems to be empirical. Not being a statistician I suggested to test a long bit sequence for normality, but I have no idea what outcome is acceptable and what isn't. For instance counting single bits I guess we have a thumbs-up for a 499500/500500 distribution, but when is the ratio too much skewed to be acceptable? Same for 2-bit and longer sequences. Of course, if testing for normality is a Bad Idea™, I'd like to hear it, including better alternatives.    edit Diehard was mentioned a few times, but I'm not sure this answers my question. The normality test should give a normal distribution, with better bell curve approximations for longer sequences. Diehard seems also to have tests which either should result in normal or exponential distributions. But my question remains: how do I judge the results? Just by looking at the curve and discern a bell curve in it? To get back at my first example, a 499500/500500 distribution is definitely OK, and a 950000/50000 distribution definitely is a no-no, so where does the switch-over happen? My linear algebra professor proved that      Diagonalizable matrices with complex values are dense in set of $n \times n$ complex matrices.   He defined a metric (I believe) that was somehow related to the usual metric on $\mathbb{R}^{n^2}$.  Then somehow proved that diagonalizable matrices were dense because for any matrix $A$ if $\det(A - \lambda I) = 0$ on an open subset, then $\det(A - \lambda I)$ was the zero polynomial.  I Googled around a bit and found some stuff talking about the Zariski topology, and I am not sure this is what I want. Most of what I have found on this topology is much more general than what he was doing.  Does anyone know any resources that would explain how to go about proving this statement?  Edit: I found out how to prove this the way my professor did.  We define the norm of a matrix by $$ |A| = \max\{|Ax| : |x| = 1 \}.$$ Now we have a distance $d(A, B) &lt; \epsilon$. You can prove that if $(A - B)_{ij} &lt; \epsilon/n$, then $|A - B| &lt; \epsilon$.  Let $p(x)$ be the characteristic polynomial of $A$, an $n \times n$ matrix.  $$p(x) = \prod_1^n (x - x_i) = x^n + (-1)\sigma_1 x^{n - 1} + \cdots + (-1)^n\sigma_n$$ where $\sigma_1, \ldots, \sigma_n$ are the elementary symmetric polynomials of $x_1, \ldots, x_n$ which are the eigenvalues.  The discriminant of the characteristic polynomial is a symmetric polynomial, therefore it can be written in terms of the elementary symmetric polynomials, which in turn can be written in terms of the entries of the matrix.  But since the discriminant is a polynomial, it only has finitely many roots. Therefore for $\epsilon &gt; 0$, by changing the entries of the matrix less than $\epsilon / n$ we can find a new matrix $B$ such that $|B - A| &lt; \epsilon$ and the discriminant is not zero.   The discriminant being not zero means $B$ has distinct eigenvalues, thus has a basis of eigenvectors. Therefore, it is diagonalizable.  Thus the set of diagonalizable matrices is dense in the set of matrices with respect to that metric. The following  (links to pdf) in the  has no provided source. The book  by Ash and Gross seems to use a similar style. Where can I find hints for typesetting publication standard articles/books as above in TeX. I looked around a bit but couldnt find much from else from keyword searches.  Particularly, I am looking at the chapter headings, title page design, and the mirror effect. Sorry for the title, as I dont really know what its called. I have hundreds of figures with filenames containing "#" to include using \includegraphics, I've tried packages like grffile or currfile, but none of them work. Can anyone suggest a real solution so that I don't need to change hundreds of file names? Thanks.  Update: 1. I'm using Windows. 2. When I compile it, it shows "Illegal parameter number in definition" I believe that umask is something that controls , but do not fully understand it.  After running umask 0644 in a , I cannot read the files I create with the command-line text editor nano. I noticed that the permissions of that file are set to 0022 instead of the default 0755.  How does umask work? I thought I could just remove the each digit in the umask from 0777, 7 - 6 = 1 and 7 - 4 = 3, so I expect the permissions to be 0133, but apparently, this is not the case.   What is umask exactly? Explain it to me like I was a "Linux noob" How do I calculate with umask? What are use cases for umask? Why does ice make such peculiar sounds? I've come across a couple of videos where some interesting sounds are produced using ice.  (Click on the images to see the video.)      Here, they drop a block of ice into a deep crevice and as the block falls, you can hear some strange sounds. These sounds remind me of some sound effects in old cartoons and sci-fi movies. The pitch of the sounds start high and end up low. Is this some kind of doppler effect?      Here, there's a man performing 'nordic skating', making the same sounds as he skates on top of a thin ice sheet. Maybe they are caused by the cracking ice. But why does that sound like this?    There are some other videos too, where they skip stones on frozen lakes.    Here the ice makes some high-frequency sounds which correspond to the chirping of birds.  What are these sounds and why do they sound the way they sound? Determining likely relationship based upon autosomal DNA shared cM Given two tests that share some cM of autosomal DNA, how can I easily find the predicted relationships that such shared cM would suggest? For example, a 2,200 cM match suggests it could be either: grandparent, half-sibling, or grandchild. Predict() - Maybe I'm not understanding it Do I have to do something "special" to get the Color -> Color to alpha menu option to work in GIMP? I am trying to convert a background color on a GIF image to transparent by converting the background color to an alpha channel but the Color to Alpha menu option is not selectable.  I've gone through the GIMP documentation and it looks pretty simple :-) I must be missing something rather obvious here! How can I remove Ubuntu from my system and install the Windows bootloader? How to extend USB beyond the recommended maximum cable length of 5m? I'm planning to use USB port for an application(from a device to a PC) but the distance between the device and the PC will be around 15m. The recommended maximum USB cable length is 5m.  How can I buffer the data without loosing any information, safely? Would a USB hub work? Word for "moving up one conceptual level" Clone internal HDD to new SSD I recently installed an SSD into my machine. The machine itself is a Lenovo thinkpad W520, and it previously had an internal HDD. I moved the internal HDD into an expansion bay (replacing the CD-ROM), and put the new SSD into the internal bay.   The problem I'm having is that I have my Ubuntu configuration EXACTLY the way I want it - I originally spent many hours configuring it to get it to the way it is now. I'd rather not do this again. But, I'd also like the boot-up gains I would get from the OS being on the SSD.  So, what I'd like to do is clone my Ubuntu partition onto the SSD. The catch is that the standard HDD is significantly bigger than the SSD. And it has a windows partition that I don't need on the SSD (I never use Windows, so if it boots off of the other hard drive, that's fine). The layout of my hard drives are as follows:  /dev/sda (SSD):     Model: ATA M4-CT256M4SSD2 (scsi)     Disk /dev/sda: 256GB     Sector size (logical/physical): 512B/512B     Partition Table: msdos  Number  Start   End    Size   Type     File system  Flags  1      1049kB  147GB  147GB  primary  ext4         boot   /dev/sdb (HDD): Model: ATA ST9500420AS (scsi) Disk /dev/sdb: 500GB Sector size (logical/physical): 512B/512B Partition Table: msdos  Number  Start   End     Size    Type      File system     Flags  1      1049kB  1259MB  1258MB  primary   ntfs            boot  2      1259MB  269GB   268GB   primary   ntfs  4      269GB   483GB   214GB   extended  5      269GB   416GB   147GB   logical   ext4  7      416GB   475GB   58.9GB  logical   linux-swap(v1)  6      475GB   483GB   8470MB  logical  3      483GB   500GB   16.8GB  primary   ntfs   What I've tried so far:  1) Resizing the partitions /dev/sdb5 and /dev/sda1 to be the same size. 2) Booting into Ubuntu 11.04 (from /dev/sdb5) and running dd if=/dev/sdb5 of=/dev/sda1 (of course this causes problems with booting, so I had to reinstall grub.. I can get it to boot, but then I have problems with initrd not finding some files ... presumably it can't load some partitions I think).  Now, I think those two steps are the wrong approach, because it will clone /dev/sdb5 EXACTLY - including references in fstab which point to the wrong hard drive. I'm not sure exactly how to rectify this. I could install Ubuntu 11.04 onto the SSD, then try and copy over all of my configurations, but I'm concerned that I'm going to lose something, or that I'm going to overwrite something like fstab that points back to the original hard drive.   Note that currently, I can still boot from the HDD, so it's not imperative that I get this figured out right away, but I do want it to be exactly how it is right now, so that I can maintain my current level of productivity (it's a work laptop).  Suggestions on how I might be able to overcome this difficulty?   Thanks in advance! I have 8 individual SQL Server 2008 R2 machines, each hosting 1 database. Each database has an identical table structure and schema, and entirely unique data.  I would like to establish a reporting server (may be 2008 or 2012), that consolidates the rows  from selected tables across the 8 source servers into a single instance of those tables on the reporting server. This is one-way replication (no changes will be made to the reporting server). I will need to replicate changes from the source databases with relatively low latency (say 20-30 seconds).  Furthermore, I'd like to find methods to achieve this with as little impact to the source servers as is practical.  3rd-party agents, triggers, or schema mods to those servers are difficult in my environment.  My questions:   What are promising architectures and technologies for  achieving this goal?  I've looked at SQL Server Merge Replication, but I am concerned about latency. Is this an appropriate technology for this goal? Are there many-to-one architectures for transactional replication?   Should I be looking at 1-to-1 replication into 8 databases on my reporting server, followed by some custom merge function (a 2-step replication)?   Thanks, John I am trying to prove that all vector spaces can be written as the direct sum of two subspaces.  My attempt: If each vector in vector space V can be uniquely written as the sum of two other vectors in V, the two vectors can be categorized into subspaces and thus V can be written as the direct sum of these subspaces. I am not sure how to show that this is true for all cases though. Edit Sorry for the confusion,to clarify I meantGiven a vector space V with a subspace W, prove that there exists a subspace W' such that the direct sum of W and W' is V.I am not sure about my proof method at all. Thank you. $$\sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}= 5$$