Logical equivalence with tautology
Why is it sensical for a proposition with a false antecedent to validate to true?
Graphics file extensions and their order of inclusion when not specified When using 's \includegraphics you can specify the file extension or not. For generic use and perhaps portability, extensions are commonly dropped. The types of "approved" inclusions and their order of precedence depends on the driver/compiler being used. What are these driver-defaults?  Related: For positive real numbers $a,b,c$ with $abc = 8$ prove that   $$ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} + \frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} + \frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}. $$   Can we prove this by Cauchy-Schwarz or Jensen's inequality? If not how? Global inbox message not updated after comment edit At the moment, I have the latest notification in the global inbox, content of which does not correspond either to the comment itself or the copy of it on "recent responses" page. This seems like the comment was edited and the updated version was not copied to the global inbox.    ETA: OK, now this looks more coherent. List of functions not integrable in elementary terms When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means).  However I must admit that the examples I give mostly come from "folklore" or guesswork.  Can anyone point me to a list (not a complete list of course!) of fairly simple elementary functions whose antiderivatives are not elementary?  I'm thinking of things like $\exp(x^2)$ which is the standard example, $\sin(\exp(-x))$ perhaps, things like this, not hugely complicated formulae. Do I make money in the stock market from other people losing money? Can black holes actually merge? If time stops at the event horizon, can we ever detect two black holes merging? In other words, if you are a short distance away, would you encounter a spherically symmetric gravitational field, or a dipole field? I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &amp;= i \\\\\ \frac1{\sqrt{-1}} &amp;= \frac1i \\\\  \frac{\sqrt1}{\sqrt{-1}} &amp;= \frac1i \\\\  \sqrt{\frac1{-1}} &amp;= \frac1i \\\\  \sqrt{\frac{-1}1} &amp;= \frac1i \\\\  \sqrt{-1} &amp;= \frac1i \\\\  i &amp;= \frac1i \\\\  i^2 &amp;= 1 \\\\  -1 &amp;= 1 \quad !!? \end{align} How do I install the Nvidia drivers? How can I make sure that only low and medium security inmates are in a specific area? Is it possible to reverse the overlay you can set? It seems like you are only able to clear areas for low OR medium security prisoners, but sharing is not possible. The thing I want here is to make sure that max security prisoners are not in certain areas. Is it possible to have global color swatches or color styles in Photoshop? I'm looking for a way to use one color in multiple different layers, and then be able to change that color in all the layers in one single place.  Lets say that I have four layers using #ffffff. Now I want to know what my design looks like with #ffff00 instead. How can I do this without manually changing all four layers?  I can add that adjustment layer on hue/saturation is not really what I'm looking for. What I need is something like "change all #ffffff to #ffff00." Experimental observation of matter/antimatter in the universe Ordinary matter and antimatter have the same physical properties when it comes to, for example, spectroscopy. Hydrogen and antihydrogen atoms produce the same spectroscopy when excited, and adsorb the same frequencies. The charge does not make a difference in the potential (regardless if it's generated by a proton or an antiproton) nor in how the positron behaves in this potential (being its mass equal to the mass of an electron)  How can astronomy evaluate if a far galaxy is made of matter or antimatter, given that from the spectroscopy point of view, they behave in the same way? In other words, how do we know that an asymmetry exists between matter and antimatter in the universe? My settlement was attacked by some Raiders and my defenses took care of them along with some of my NPC's. It seems like the Raiders didn't make it through to any of the buildings. What happens when they do?  Can you lose your items if you stored them in a house?  Can the NPC's die?  Do the enemies destroy your buildings? Probability Number A greater Number B Given $a \in \{1,2,...,250\}$ and $b\in \{0,1,...,1000\}$  What is the probability that $a &gt; b$?  I know there already is  question with a very similar question, however it's over an interval of $\mathbb{R}$ so I hope there is another way opposed to the Integral used in the formal solution.  How does one go about calculating said probability? How to set a random value for multiple shape keys? My question is related to my previous asked one:   I have about 300 shapekeys and want to set the random value for all of them . How to do it using scripting? he became firm friend or friends? scrlttr2: Position of the recipient's address Why don't we treat rubbish the same as spam? Reading Excel files from C# Is there a free or open source library to read Excel files (.xls) directly from a C# program?   It does not need to be too fancy, just to select a worksheet and read the data as strings. So far, I've been using Export to Unicode text function of Excel, and parsing the resulting (tab-delimited) file, but I'd like to eliminate the manual step. How do I left-align entries in a matrix with \begin{matrix}? I'd like to left align the -1 entry in the following matrix so that the 1s line up. Is there a way to get matrices to ignore signs in that way?  \begin{matrix}     1 &amp; 1  \\     1 &amp; -1 \\ \end{matrix}   Using \flushright{-1} doesn't seem to work. Proving a graph has no Hamiltonian cycle