I think that the question is contained in the title. Suppose we begin from something that is false for example $1=0$. Is it possible using only $\Rightarrow$ (and of course $\lnot ,\wedge,\lor$)  to prove any possible statement? It has been ages since i studied logic at the university so i would prefer a simple explanation (if possible)
Possible Duplicates:             I heard a professor of mathematics explaining that the following conslusions are true.   $\textrm{False}\Rightarrow \textrm{True}$ $\textrm{False}\Rightarrow \textrm{False}$   Could someone please explain to me why both conclusions are true? I am especially interested in the first case. I'd like to hear two explanations if possible: One for mathematicians and one for non-mathematicians. :-)
Returning multiple values from a C++ function Is there a preferred way to return multiple values from a C++ function?  For example, imagine a function that divides two integers and returns both the quotient and the remainder.  One way I commonly see is to use reference parameters:  void divide(int dividend, int divisor, int&amp; quotient, int&amp; remainder);   A variation is to return one value and pass the other through a reference parameter:  int divide(int dividend, int divisor, int&amp; remainder);   Another way would be to declare a struct to contain all of the results and return that:  struct divide_result {     int quotient;     int remainder; };  divide_result divide(int dividend, int divisor);   Is one of these ways generally preferred, or are there other suggestions?  Edit: In the real-world code, there may be more than two results.  They may also be of different types. I am trying to add a timed delay in a C++ program, and was wondering if anyone has any suggestions on what I can try or information I can look at?  I wish I had more details on how I am implementing this timed delay, but until I have more information on how to add a timed delay I am not sure on how I should even attempt to implement this. Real world use cases of bitwise operators Are "white" and "black" ever capitalized when referring to people? Code in beamer presentation Good day! So, I need to present my disertation code to my thesis advisors, but I'm quite unsure how to actually do it in a proper way. I found  showing some ideas.  Is there a way of making the slides 48-67 in beamer without using tikz? Any ideas to implement the whole ideas the author propose?  Thanks in advance. How to to resize an ext4 partition from the command line? What's the easiest way to resize an ext4 partition (or any type partition depending on the method) from the command line (potentially with the fewest commands, but also the easiest to understand)?  Using a tool like Gparted is obviously easy in a GUI, but what about in the command line? I guess text-based GUIs can count for the answer too since it's technically still in the command line. It just needs to be easy.  By partition I mean a simple partition on a single disk of a personal computer (e.g. on a laptop). For example, I want to resize /dev/sda4. There's no RAIDs, there's not more than one disk drive, there's not anything complicated here. Just a simple partition on a single disk (/dev/sdaX on /dev/sda). Can one force-feed an opponent a potion in combat? What does the helpful flag mean? I don't see "Helpful" in the list of things I can do when I press the flag link.   In my profile page I saw "6 helpful flags."  How do I earn these flags?  May I make them on other peoples' posts or comments?   This is what I am talking about: Open a window on a remote X display (why "Cannot open display")? Why does water pouring from a glass sometimes travel down the side of the glass? If you have a glass of water, say, three quarters full and you pour it at an angle of say, $45^{\circ}$ with respect to the the table, the water comes out of the glass and goes directly down towards the floor.  However, when the glass is more full, or even three quarters full and the 'angle of pouring' is far less with respect the table, when the water comes out of the glass, rather than going straight down it kind of stays stuck to the glass and travels down the outside of it.  Why does this happen? (I'm sure many of you have observed it yourself by accident and ended up making a mess). Prove that inequality $\sum _{cyc}\frac{a^2+b^2}{a+b}\le \frac{3\left(a^2+b^2+c^2\right)}{a+b+c}$ Let $a&gt;0$,$b&gt;0$ and $c&gt;0$. Prove that:    $$\dfrac{a^2+b^2}{a+b}+\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}\le \dfrac{3\left(a^2+b^2+c^2\right)}{a+b+c}.$$     $$\Leftrightarrow \frac{(a^2+b^2)(a+b+c)}{a+b}+\frac{(b^2+c^2)(a+b+c)}{b+c}+\frac{(c^2+a^2)(a+b+c)}{c+a}\leq 3(a^2+b^2+c^2)$$  $\Leftrightarrow 2(a^2+b^2+c^2)+\frac{c(a^2+b^2)}{a+b}+\frac{a(b^2+c^2)}{b+c}+\frac{b(a^2+c^2)}{a+c}\leq 3(a^2+b^2+c^2)$  $\Leftrightarrow \frac{c(a^2+b^2)}{a+b}+\frac{a(b^2+c^2)}{b+c}+\frac{b(a^2+c^2)}{a+c}\leq a^2+b^2+c^2$  $\Leftrightarrow \frac{c(a+b)^2-2abc}{a+b}+\frac{a(b+c)^2-2abc}{b+c}+\frac{b(a+c)^2-2abc}{a+c}\leq a^2+b^2+c^2$  $\Leftrightarrow 2(ab+bc+ac)\leq a^2+b^2+c^2+2abc\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c}\right)$  I can't continue. Help me Why is 248x248 the maximum bi dimensional array size I can declare? How do I log in to my personal Windows 10 using a FIDO2 key? I'd like to user a FIDO2 key to log into Windows 10. There are various guides explaining how to add a key to a Microsoft web account, or to Windows 10 sign in options, but neither result in showing up as an option when I try logging into Windows.  How can I configure windows to use my FIDO2 key to log in? How do I set the desktop background on Windows from a script? On X Windows I had a cool 'silent-alarm" reminder script that would change my root window (background) color to solid red, just for a few seconds a few moments before changing it back. Is there a way to do this for Windows XP?  I'm thinking some kind of scheduled task that uses cscript to set registry keys (HKEY_CURRENT_USER\Control Panel\Desktop) . However my attempts don't seem to have any effect. What do I have to signal to read those registry entries and re-draw the desktop? Perform feature normalization before or within model validation? A common good practice in Machine Learning is to do feature normalization or data standardization of the predictor variables, that's it, center the data substracting the mean and normalize it dividing by the variance (or standard deviation too). For self containment and to my understanding we do this to achieve two main things:   Avoid extra small model weights for the purpose of numerical stability. Ensure quick convergence of optimization algorithms like e.g. Conjugate Gradient so that the large magnitude of one predictor dimension w.r.t. the others doesn't lead to slow convergence.   We usually split the data into training, validation and testing sets. In the literature we usually see that to do feature normalization they take the mean and variance (or standard deviation) over the whole set of predictor variables. The big flaw I see here is that if you do that, you are in fact introducing future information into the training predictor variables namely the future information contained in the mean and variance.  Therefore, I do feature normalization over the training data and save the mean and variance. Then I apply feature normalization to the predictor variables of the validation and test data sets using the training mean and variances. Are there any fundamental flaws with this? can anyone recommend a better alternative? How to do a generalized linear model with multiple dependent variables in R? I have six dependent variables (count data) and several independent variables, I see that in a MMR the script goes like this:  my.model &lt;- lm(cbind(DV1,DV2,DV3,DV4,DV5,DV6) ~ IV1 + IV2 + ... + IVn)   But, since my data are counts, I want to use a generalized linear model and I tried this:  my.model &lt;- glm(cbind(DV1,DV2,DV3,DV4,DV5,DV6) ~ IV1 + IV2 + ... + IVn, family="poisson")   And appears this error message:  Error in glm.fit(x = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  :    (subscript) logical subscript too long`   Can anyone explain me this error message or a way to solve my problem? What are the guidelines for reviewing? About reviewing The review queues contain posts that possibly need further action from the community, like improvement, closure or deletion. By presenting those posts for peer-review, the system aims for higher quality content. Some user actions, such as suggesting an edit, casting a flag, close vote or reopen vote, or posting for the first time, can trigger the inclusion of a post on a review queue. There are also posts included on the queues algorithmically, such as the contents of Late Answers and Low Quality Posts. General guidelines  Always read the full post you are reviewing. Don't rush. Take the necessary time to read the post carefully. If you don't care about a post, just click Skip or Not sure. When you need more context, open the post link to see the question and all answers. Most posts coming up for review can be improved. Use the Edit option, and edit thoroughly. If you're unsure how to review a post (perhaps it's outside your areas of expertise), skip it. Someone who understands it better will review it later.  Guidelines for specific review queues         (Stack Overflow only) Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.   What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve this and showing that $|G|$ order element in $G$ exists is enough. How is hotness score calculated at larger sites? To start with, I am aware that hot questions from larger sites are somehow "penalized" and :     This is designed to ensure that a single site doesn't dominate the list, and that users see a good sampling of questions that are likely to interest them throughout the network...   I would like to understand how is hotness score really calculated at these sites. The reason why I ask is that per my observations, this seems to be quite different than is documented in the . Specifically, I would like to understand the difference in the impact of answer(s) score and aging factor.  Regarding impact of answers score, my observations of hot questions from smaller sites appear to be in agreement with known formula: (MIN(AnswerCount, 10) * QScore) / 5 + AnswerScore. For example, a question with low score (+1, +2) can get into hot list if it has got answer(s) with sufficiently high score.  At larger sites, this doesn't seem to be the case. Say, it looks like Stack Overflow questions just don't enter hot list until they reach certain score (+6,+7), no matter how many answers are there and how high is their score.  Does system simply totally ignore answers until question reaches certain score?  Another thing I noticed is that questions seem to age away from the hot list 2-3 times faster than ones from smaller sites (having comparable question and answers scores). Granted, this is harder to track and I am not yet sure if I figured that correctly. Do larger sites questions age away faster than those from smaller sites?    For those interested, here is a (partial) list of example questions I studied before asking this: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .  Although above examples are only from Stack Overflow, my question isn't (I only used it as an example of a large site in my studies). I would like to understand if hot questions from other larger sites are handled in a similar way. Taking notes, I was going to write this:     therefore p′ is a shortest path.   But the topic is APSP which stands for "all pairs shortest paths" so I decided to just write SP instead and...  Is it written with a like this     therefore p′ is a SP   or with an like this (because in my head I would "read" the S)     therefore p′ is an SP