to clarify the difference between this question and the supposed duplicate, absolutely nowhere do I mention the proof being attempted in the other question. It merely serves as an example related to this question.  Now, the way I view proof by contradiction is like this:  We have two logical statements $a$ and $b$ and we claim that "if $a$ then $b$".  We assume $b$ is false (given $a$) and we attempt to find some kind of contradiction with something we assume is true at the beginning (generally it tends to be proving $a$ is false).  Now, proof by contradiction can "fail" by resulting in no contradiction (i.e. that $a$ ends up being true when $b$ is false). However, this seems like a proof that the original statement is false. What I'm asking is more like how one detects or proves that when $b$ is false the truthfulness of $a$ is unprovable. Basically, that $a$ and $b$ are incapable of proving each other.  A good example of this is the parallel postulate of geometry. From what I heard (paraphrasing) "a failed proof by contradiction showed that there existed consistent systems where it was false and that it was unprovable from the other postulates". I'm not quite sure how that worked. Could someone please explain (in general) how one proves a proof by contradiction fails this brutally? I'm not asking specifically how it was done for the parallel postulate, but just how one goes about doing it in general?
What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by contradiction, and the other proves the contrapositive, the proofs look almost exactly the same.  For example, say I want to prove: $P \implies Q$ When I want to prove by contradiction, I would say assume this is not true. Assume $Q$ is not true, and $P$ is true. Blabla, but this implies $P$ is not true, which is a contradiction.  When I want to prove the contrapositive, I say. Assume $Q$ is not true. Blabla, this implies $P$ is not true.  The only difference in the proof is that I assume $P$ is true in the beginning, when I want to prove by contradiction. But this feels almost redundant, as in the end I always get that this is not true. The only other way that I could get a contradiction is by proving that $Q$ is true. But this would be the exact same things as a direct proof.   Can somebody enlighten me a little bit here ? For example: Are there proofs that can be proven by contradiction but not proven by proving the contrapositve?
How to recover files from dead windows using dos? bounding a $C^1$ function that upholds $f'(t)\leq 1-f(t)^2$ I'm unable to uninstall MATLAB;  sandeep@sandeep-VPCEH28FN:~$ sudo rm /var/tmp/lm_TMW* [sudo] password for sandeep:  rm: cannot remove '/var/tmp/lm_TMW*': No such file or directory sandeep@sandeep-VPCEH28FN:~$ sudo apt-get purge matlab E: Could not get lock /var/lib/dpkg/lock - open (11: Resource temporarily unavailable) E: Unable to lock the administration directory (/var/lib/dpkg/), is another process using it? sandeep@sandeep-VPCEH28FN:~$ sudo apt-get purge matlab-support E: Could not get lock /var/lib/dpkg/lock - open (11: Resource temporarily unavailable) E: Unable to lock the administration directory (/var/lib/dpkg/), is another process using it? sandeep@sandeep-VPCEH28FN:~$ Pressing Enter should not refresh the page Consider the following two versions of a question:1   Version 1: What is the difference in meaning between "[to be] not invited" and similar negation forms?  Version 2: What is the difference in meaning among "[to be] not invited" and similar negated forms?    Personally, I think Version 1 might subtly suggest a Star Topology (see below) with the target sentence "[to be] not invited"at the center.  And Version 2 might subtly suggest a more free-form comparison like the Mesh or Fully Connected, with the target sentence simply "one among many":   Between approx equal "Star" and among approx equal "Fully Connected"?  Don't get me wrong: I do think this is quite subtle, not always applicable, and I don't think anyone would intuit such a "network topology" meaning without further explanation.   So the main question is, are both grammatically correct?  If yes, is there a semantic difference to choose one over another, perhaps as I suggested?  Or does the plurality aspect of between and among trump other semantic and stylistic issues?    1. In my question , another user provided a very helpful edit. However, the change of the word between to among prompted this question. What could cause exposure problems after my camera got wet? I had an unfortunate accident where my DSLR camera (and me) briefly ended up in a river. The camera wasn't fully submerged but it did get very wet. It still works, sort of, although the electrics are doing some random things. I'm hoping that's just a bit of dampness doing that, that will fix itself when it dries out fully.  More concerning is that the documents exposure metering seems to be all wrong when using the viewfinder. Pictures are extremely overexposed and nearly white. But if I use live view, then they are fine.  Why could cause this? I assume the light is routed a different way when using the viewfinder and maybe it is going through something with water/condensation on it? Which is the most portable of sed, awk, perl and sh? What do the numbers in a man page mean? What is a word for someone without allegiance? How do I know whether my bike can use disk brakes? RAM not clearing, alway in standby I'm having a problem with my ram on windows 7. The ram is clearing or programs are using and dumping it faster than it can clear. I have 8gb of ram I'm only using 4gb and the rest is in Standby. At best time I can get 100mb of free but its mostly at 15mb or less. I suspect a memory leak but I have no way to figure out what program it is. I have not installed any new hardware recently and newly installed programs have bin closed. Does any one have an idea of what is going on and how to fix it?   I'm sorry this is so little information but that is all I have as well.  Edit: I tried RamMap and found out the my "Mapped File" has 3,291,784k ram in standby how do I figure out what program and how do I limit it's ability to set ram in standby?  Edit: I keep getting a problem when I will have 0 free ram and my computer will stop working properly. It will become outrageously slow where it will take 3-4 minutes to start task manager so I can close some programmes. When I free up some ram by closing some programmes every thing runs normally. It does not matter which programs I closes only that I free up some ram. I know a lot of you say it's supposed to be like that and that's how windows works. But its obviously a problem and my computer has never worked that way neither has any of my other computers. I would like to fix this please help me. Are there tips and tricks for people with fear of flying that can makes them have a more pleasant experience when flying? I know drugs can do magic, and . I'm ideally after official advice and recommendations, but other tips and tricks would be nice as well. Parent chat user still links to discuss.area51 When viewing a user who has their parent user on chat set to area51, it still links to discuss.area51, instead of the new URL, area51.meta. See:      So can we update these links? Find all files that are NOT of a specific type/extension in folder? How to find and replace using sed on records starting with Specific number How to setup Windows from an external hard drive? Is it possible to install Windows from an external hard drive?  I have Windows XP installed on my laptop, and I want to remove it and install Windows 7.  My Windows 7 installation is in an .iso format on an external USB hard drive.  Is it possible to install it or will I need to put in a DVD before I start installing? Generating random points uniformly on a disk I have to randomly generate 1000 points over a unit disk such that are uniformly distributed on this disk. Now, for that, I select a radius $r$ and angular orientation $\alpha$ such that the radius $r$ is a uniformly distributed variate from $r \in [0,1]$ while $\alpha$ is a uniformly distributed variate from $\alpha \in [0, 2\pi]$ using the following code r &lt;- runif(1000, min=0, max=1)  alpha &lt;- runif(1000, min=0, max=2*pi) x &lt;- r*cos(alpha) y &lt;- r*sin(alpha) plot(x,y, pch=19, col=rgb(0,0,0,0.05), asp=1)  Then I look at my sample space and it looks like this:  This obviously doesn't look like a sample with uniform distribution over the disk. Hence, I guessed that the problem might be occurring as a result of a lack of independence between the variables $r$ and $\alpha$ in contingency to how they've been linked computationally. To take care of that I wrote a new code. rm(list=ls()) r &lt;- runif(32, min=0, max=1) df_res &lt;- data.frame(matrix(c(-Inf, Inf), byrow = T, nrow = 1)) for (i in 1:32) {   for (j in 1:32) {     alpha &lt;- runif(32, min=0, max=2*pi)     r &lt;- runif(32, min=0, max=1)     df &lt;- data.frame(matrix(c(r[i],alpha[j]), byrow = T, nrow = 1))     df_res &lt;- rbind(df_res,df)   } } df_res &lt;- subset(df_res, df_res$X1 != -Inf) x&lt;- df_res$X1 *cos(df_res$X2) y &lt;- df_res$X1 *sin(df_res$X2) plot(x,y, pch=19, col=rgb(0,0,0,0.05), asp=1)  And, yet again the sample looks non-uniformly distributed over the disk  I'm starting to suspect that there is a deeper mathematical problem going on in the vicinity. Could someone help me write code that would create a sample space uniformly distributed over the disk, or explain the mathematical fallacy if any in my reasoning? Are there popular but poorly paved roads on Big Island which require an SUV? My grandparents will drive only on roads recommended in printed travel guides; they'll ask the rental car company if Waipio Valley Road and the road to the top of Mauna Kea are sanctioned.       A rented 4WD vehicle is allowed by law to travel ANY public roadway in Hawaii. AND there are some places that require a 4WD vehicle.      [...]      If you travel down into  or to the top of , you are legally allowed to do so, but your rental company will expect you to pay for any damage incurred whether you selected the insurance or not, as the insurance is null and void when you violate the contract.        I do not believe a 4x4 rental is necessary for 99% of visitors. For the other 1%, "necessary" is too strong a word!      But, being in that "1%" category, there ARE worthy places to go on the BI that require 4x4 (the kind you can rent from a normal rental outlet, not the kind locals may drive). I won't publicize those places because I don't want such places to become overused. [I bolded.] Just don't believe it when you read here that there is absolutely no place at all worth visiting that would require 4x4.    I doubt the Tripadvisor comments overhead, as none of them discussed potholes:          Please see the bolded phrase. Where on the BI requires an SUV? In the 10 k tools,  is described as if I had approved it:    However, the review item itself (correctly) displays that I rejected this edit:    Going by   , this should be fixed, but it isn’t.  This could be solved by replacing approved by to reviewed by in the list or making an explicit list, who approved and rejected an edit (which is probably more complicated). I set up a micro instance (Ubuntu, 32 bit) and assigned an elastic IP (23.23.245.104)  Public DNS: ec2-23-23-245-104.compute-1.amazonaws.com  I successfully ssh'ed to the machine and installed nginx.  Then I edited the server_name attribute in nginx's config file (/etc/nginx/sites-available/default)  server_name ec2-23-23-245-104.compute-1.amazonaws.com;   Restarted nginx and hit  but got a "could not connect" error.  What did I forget to do?  Thanks