So i have the following fitch problem where I have to cite rules of inference. I have only figured out a number of them listed below. I need help with how this problem uses certain rule of inference as it is confusing to me and I only understand some  $\begin{array}{r|l:l} 1&amp; s \to r\vee p \\[-1ex] 2 &amp; p\to(\neg r\to q)\\[-1ex] 3 &amp; \vert\quad s \\[-1ex] 4 &amp; \vert\quad r\vee p\\[-1ex] 5 &amp; \vert\quad\vert\quad r\\[-1ex] 6 &amp; \vert\quad\vert\quad q\vee r \\[-1ex] 7 &amp; \vert\quad r\to q\vee r\\[-1ex] 8 &amp; \vert\quad\vert\quad \neg r\\[-1ex] 9 &amp; \vert\quad\vert\quad p\\[-1ex] 10 &amp; \vert\quad\vert\quad \neg r\to q\\[-1ex] 11 &amp; \vert\quad\vert\quad q\\[-1ex] 12 &amp; \vert\quad\vert\quad q\vee r\\[-1ex] 13 &amp; \vert\quad \neg r\to q\vee r\\[-1ex] 14 &amp; \vert\quad r\vee\neg r &amp; \star\\[-1ex] 15 &amp; \vert\quad q\vee r\\[-1ex] 16 &amp; s\to q\vee r \end{array}$  1:  none   2: none  3:  none  4: -> Elimination, 3 and 1  5:none  6: V Introduction, 5  7: -> Introduction, 5 and 6   8: none  9: ??  10: -> Elimination, 9 and 2  11: ??  12: v Introduction , 11  13: ??  14: ??  15: ??  16: V Elimination , cite all steps
Okay I have the following fitch proof almost figured out, from here I know I have to move forth in conlcuding but I don't know what I am missing  (2 premises)  S → (R ∨ P)  P → (¬R → Q)   my work below  new scope assume S   Rule: > elim  R ∨ P  New scope assume R  Q∨R     Rule: V intro  End scope – (new scope assume P)  ¬R → Q         Rule: > Elim  my confusion comes where I go from here  Q∨R            rule Taut Con ??  Goal:├ S → (Q ∨ R)
Evaluating $\int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\frac{G\ln 2}{2}$ In order to compute, in an elementary way,  $\displaystyle \int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$  (see  )  i need to show, in a simple way, that:  $\displaystyle \int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\dfrac{G\ln 2}{2}$  $G$ being the Catalan constant.  The reason of my interest for this question is, if i am right, that this formula permits to find out a relation between integrals:  $\displaystyle \int_0^1 \dfrac{\ln(1+x)\ln(1+x^2)}{1+x^2}dx$  $\displaystyle \int_0^1 \dfrac{\ln(1+x)^2}{1+x^2}dx$  $\displaystyle \int_0^{\tfrac{\pi}{4}}\big(\ln(\cos x)\big)^2 \dfrac{}{}dx$  and some constants. How are "needs to be washed," "needs washing," and the regional variant "needs washed" to be distinguished"? Can someone remind me of the name of the author, the name of the hero and the name of the series if any? Adventure Sci-Fi books, short by today's epic standards ( = about same length as Heinlein's shorter youth books), probably published late 50s?, early 60s? by Bantam? or Del Rey? I think at least 30, possibly more than 40 books in the series. Not comic books. The author is not a well-known writer. At least, I don't remember seeing the name on any other books, ever.   Subject: a man who can move and respond to threats extremely fast learns about a secret potion that allows a person to change bodies with someone else and thus control them. A powerful organization does not want the secret to be known, because they use the potion to control politicians on many planets. The organization's goons are people who had surgeries to remove their emotions, and function like organic computers. They can communicate over interstellar distances and follow and murder anyone who learns the secret. I remember the following detail about the potion: there are two nearly identical molecules that have to be assembled in a certain sequence. Two of the components had to be switched around and that determined who would be dominant and who would be controlled during the mind switch / body swap.   Every book finds the hero on another planet with new adventures, and usually taking another deep-sleep-passage to another world, fleeing his pursuers. The background story about the secret potion is dished out little by little in each book.  One book in the series starts like this: hero is on very hot planet, diving for valuable marine life, catches a deadly fungus disease and is rescued last moment in exchange for his catch...  One of the themes that returns often is that the hero is an excellent knife fighter.  Of course he's a womanizer. In the second (or close to 2nd) book in the series, he's involved with a woman who rules a planet.  To travel to another world, he usually takes the cheapest dead-sleep passage, and there's always a 10-25% risk you won't wake up from that. How does the tester screwdriver work? How does the tester screwdriver work? If I put the tester screwdriver inside the "hot wire" of an electrical socket, it lits up if I press my finger against the metal cap on top of the screwdriver. This happens also if I stand on a surface of isolating material, such as wood. I read elsewhere that this happens due to stray capacitance formed by the "hot wire", the human body, and the ground. One has  $$ Z = R + \dfrac{1}{j \omega C}  $$  for the impedance, so if C  is high enough, the impedance should be close to r , the "effective resistance" of the formed circuit. Here I get lost; why r  is small enough to cause a current in the range of mA  even if I stand on an isolating surface?  So really what I am asking is how one may represent the system hot wire - screwdriver - human body - wooden floor - building - ground as an electrical circuit, and which parts of the physical system contribute to the resistance, capacitance (and inductance?) and in which proportion, even very approximately. There are no compact minimal surfaces This is one of the exercises of 'Do Carmo' (Section 3.5, 12)     How do you prove that there are no compact (i.e., bounded and closed in $\mathbb{R}^3$) minimal surfaces?   Thanks! Why didn't the Ministry use Veritaserum to question criminals? The truth potion (Veritaserum) is one of the most powerful potions that forces the drinker to speak truth. Then why didn't Ministry used it in any criminal cases instead of long discussions and proceedings? I know there are antidotes to the potion (Dumbledore tells Harry in the 6th book , "Slughorn might have been carrying antidote to veritaserum ever since I asked him about the memory"), but why don't keep criminals in custody until the effect of the antidote wears out and then make them drink veritaserum. Won't that result in perfect judgement? How To Reduce Noise In Cycles Render I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's equations. These equations are invariant under many transformations (Lorentz transformation, rotations, spatial and temporal translations, etc. By the way is there a way, maybe from group theory, to find all the possible transformations that leaves the equation(s) invariant?) which imply as many conserved quantities thanks to Noether's theorem. In wikipedia I can see an equation that seems to give all the conserved quantities () but it involves the Lagrangian and I'm not sure whether the formula is valid for all systems whose Lagrangian is possible to obtain. What happens 'internally' on a unstable/overclocked/undervolted CPU? When you undervolt or overlock an CPU, when it passes a certain limit it starts to make the system unstable, reboot, etc. But these are the effects of it.  What I'm curious about is what happens on the CPU. What happens? It gives wrong answers to instructions? Bits on cache flip? How can I return NULL from a generic method in C#? I have seen a few  here on Meta&nbsp;Stack&nbsp;Exchange that cover parts of this, but they are often too local, and none of them seem to cover all the relevant scenarios.  I will break it up into multiple parts in order to make this more specific:     When should I vote on a question:   What are the qualifications for a question to be deserving of being up-voted? When shouldn't I vote up on a question? What are the things that would cause a question to need to be down-voted?   How should I go about down-voting the question (do I wait a little while to give the OP a chance to change it, or do I down-vote it immediately)?  When shouldn't I vote down on a question?  When should I vote on an answer:   What are the qualifications for an answer to be deserving of being up-voted? When shouldn't I vote up on an answer? What are the thing that would cause an answer to need to be down-voted?   How should I go about down-voting the answer (do I wait a little while to give the OP a chance to change it, or do I down-vote it immediately)?  When shouldn't I vote down on an answer?  When should I vote on a comment:   While votes on comments do not affect reputation points, it might be useful to new users to know when to up-vote them.    It seems that a full, comprehensive list of these would be invaluable to new users looking to figure out how the voting system works, but this list seems to be missing from the meta site. Issue with regex backreference in Python I'm trying to change the cpu frequency on my laptop (running Linux), and not having any success. Here are some details:    # uname -a Linux yoga 3.12.21-gentoo-r1 #4 SMP Thu Jul 10 17:32:31 HKT 2014 x86_64 Intel(R) Core(TM) i5-3317U CPU @ 1.70GHz GenuineIntel GNU/Linux  # cpufreq-info cpufrequtils 008: cpufreq-info (C) Dominik Brodowski 2004-2009 Report errors and bugs to cpufreq@vger.kernel.org, please. analyzing CPU 0:   driver: intel_pstate   CPUs which run at the same hardware frequency: 0   CPUs which need to have their frequency coordinated by software: 0   maximum transition latency: 0.97 ms.   hardware limits: 800 MHz - 2.60 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 800 MHz and 2.60 GHz.                   The governor "powersave" may decide which speed to use                   within this range.   current CPU frequency is 2.42 GHz (asserted by call to hardware). (similar information for cpus 1, 2 and 3)  # cat /sys/devices/system/cpu/cpu0/cpufreq/scaling_available_governors performance powersave   I initially had the userspace governor built into the kernel, but then I also tried building it as a module (with the same results); it was loaded while running the above commands (and I couldn't find any system messages when loading it):  # lsmod Module                  Size  Used by cpufreq_userspace       1525  0 (some other modules)   And here are the commands I tried for changing the frequency:  # cpufreq-set -f 800MHz Error setting new values. Common errors: - Do you have proper administration rights? (super-user?) - Is the governor you requested available and modprobed? - Trying to set an invalid policy? - Trying to set a specific frequency, but userspace governor is not available,    for example because of hardware which cannot be set to a specific frequency    or because the userspace governor isn't loaded?  # cpufreq-set -g userspace   Error setting new values. Common errors: - Do you have proper administration rights? (super-user?) - Is the governor you requested available and modprobed? - Trying to set an invalid policy? - Trying to set a specific frequency, but userspace governor is not available,    for example because of hardware which cannot be set to a specific frequency    or because the userspace governor isn't loaded?   Any ideas? Sort ArrayList of custom Objects by property As a part of this script, I need to be able to check if the first argument given matches the first word of file. If it does, exit with an error message; if it doesn't, append the arguments to the file. I understand how to write the if statement, but not how to use grep within a script. I understand that grep will look something like this  grep ^$1 schemas.txt   I feel like this should be much easier than I am making it.  I'm getting an error "too many arguments" on the if statement. I got rid of the space between grep -q and then got an error binary operator expected.   if [ grep -q ^$1 schemas.txt ] then         echo "Schema already exists. Please try again"         exit 1 else         echo "$@" &gt;&gt; schemas.txt fi A random variable with neither probability density function nor probability mass function... is this example wrong? This is from Shreve's Stochastic Calculus for Finance II, Appendix A.3. Consider infinitely many independent coin tosses with outcomes $\omega_n$, $n = 1, 2, \dots$ such that $\omega_n$ can be either $H$ = heads or $T$ = tails. Define \begin{equation} Y_n\left(\omega\right) = \begin{cases}1 &amp; \text{ if }\omega_n = H \\ 0&amp; \text{ if } \omega_n = T\text{.} \end{cases} \end{equation}  Define $\displaystyle Y = \sum\limits_{n=1}^{\infty}\dfrac{2Y_n}{3^{n}}$. Shreve shows that $Y$ can only take values in the Cantor set, and since that particular set has measure zero, its Lebesgue measure is zero as well, thus it does not have a probability density function (since integrating such a density function would give a value of $0$). This part I understand, as well as my professor.  Now Shreve claims that $Y$ does not have a probability mass function.      If it did, then for some number $x \in C$ [the Cantor set] we would have $\mathbb{P}\left(Y=x\right) &gt; 0$. But $x$ has a unique base-three expansion    \begin{equation} x = \sum\limits_{n=1}^{\infty}\dfrac{x_n}{3^{n}}\text{,} \end{equation}   where each $x_n$ is either 0, 1, or 2 unless $x$ is of the form $\dfrac{k}{3^{n}}$ for some positive integers $k$ and $n$. In the latter case, $x$ has two base-three expansions... In either case, there at most two choices of $w \in \Omega_{\infty}$ [that is, the infinite coin-toss space] for which $Y(\omega) = x$. In other words, the set $\{\omega \in \Omega; Y(\omega) = x\}$ has either one or two elements. The probability of a set with one element is zero, and the probability of a set with two elements is $0 + 0 = 0$. Hence $\mathbb{P}\left\{Y = x\right\} = 0$.   My professor and I were able to follow this until the statement "The probability of a set with one element is zero." Is this a true statement? How so?   I thought it would be perhaps the fact that the Lebesgue measure of $[a, a] = \{a\}$ is zero, but I don't see any reason why Shreve would only use the Lebesgue measure, since it is not a general probability measure. From what I understand, the  states that there is a fundamental natural limit to how accurately we can measure velocity and momentum at the same time. It's not a limit on equipment but just a natural phenomenon.  However, isn't this just an observational limit? There is a definite velocity and momentum, we just don't know it. As in, we can only know so much about the universe, but the universe still has definite characteristics.  Considering this, how do a wide range of quantum mechanical phenomena work? For example, quantum tunneling - its based on the fact that the position of the object is indefinite. But the position is definite, we just don't know it definitely. Or the famous light slot experiment? The creation of more light slots due to uncertainty of the photon's positions?  What I am basically asking is why is a limit on the observer, affecting the phenomenon he is observing? Isn't that equivalent to saying  because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation! I would like my sections to start on a new page if they do not fit on the current page. In other words:  if a section fits on the current page print it, if it does not fit start printing it on the next page.  I would like this to be done automatically.  \usepackage{titlesec} seems to be the solution, but I can not get it to do the job.   Should it be done with titlesec? how ? Is there a better approach? I have a script which calls two commands:  long_running_command | print_progress   The long_running_command prints progress but I'm unhappy with it. I'm using print_progress to make it nicer (namely, I print the progress in a single line).  The problem: Connection a pipe to stdout also activates a 4K buffer, so the nice print program gets nothing ... nothing ... nothing ... a whole lot ... :)  How can I disable the 4K buffer for the long_running_command (no, I do not have the source)? Can grep output only specified groupings that match? Say I have a file:  # file: 'test.txt' foobar bash 1 bash foobar happy foobar   I only want to know what words appear after "foobar", so I can use this regex:  "foobar \(\w\+\)"   The parenthesis indicate that I have a special interest in the word right after foobar.  But when I do a grep "foobar \(\w\+\)" test.txt, I get the entire lines that match the entire regex, rather than just "the word after foobar":  foobar bash 1 foobar happy   I would much prefer that the output of that command looked like this:  bash happy   Is there a way to tell grep to only output the items that match the grouping (or a specific grouping) in a regular expression?