Logic: Proving that ~(P->Q) and P^~Q are equivalent Informally, I realize that ~(P-&gt;Q) is the same as P^~Q, since the only way that the original conditional P-&gt;Q would be false is if we had both P^~Q.  I think that given ~(P-&gt;Q), the next step would be to assume ~(P^~Q) and try to obtain P^~Q via proof by contradiction, but after this step, I am not sure how to proceed.   The system I am using is Fitch. Unfortunately, I cannot use AnaCon or TautCon, nor can I use truth tables.
How to solve ~(P → Q) : P & ~Q by natural deduction? I'm trying to solve the following by natural deduction:  ~(P → Q) : P &amp; ~Q  It's a trivial problem if identities are used, as can be seen by the following:   {1}      1.   ~(P → Q)                    Prem. {1}      2.   ~(~P ∨ Q)                   1 Mat. Impl. {1}      3.   ~~P & ~Q                    2 De Morgan's {1}      4.   P & ~Q                      3 DNE   Starting from step 2 of the previous proof, it's also fairly straightforward to reach a conclusion using natural deduction; i.e. starting from ~(~P ∨ Q), which is an identity of ~(P → Q):  ~(~P ∨ Q) :  P &amp; ~Q   {1}      1.   ~(~P ∨ Q)                   Prem. {2}      2.   ~P                          Assum. {2}      3.   ~P ∨ Q                      2 ∨I {1,2}    4.   ~(~P ∨ Q) & (~P ∨ Q)        1,3 &I {1}      5.   P                           2,4 RAA {6}      6.   Q                           Assum. {6}      7.   ~P ∨ Q                      6 ∨I {1,6}    8.   ~(~P ∨ Q) & (~P ∨ Q)        1,7 &I {1}      9.   ~Q                          6,8 RAA {1}      10.  P & ~Q                      5,9 &I   I also found that it's fairly easy to conclude ~Q:   {1}      1.   ~(P → Q)                    Prem. {2}      2.   Q                           Assum. {3}      3.   P                           Assum. {2,3}    4.   P & Q                       2,3 &I {2,3}    5.   Q                           4 &E {2}      6.   P → Q                       3,5 CP {1,2}    7.   ~(P → Q) & (P → Q)          1,6 &I {1}      8.   ~Q                          2,7 RAA ... {1,2}    12.  P & ~Q   In spite of how easy it looks, I can't figure out how to do it. I believe it can be done in 12 steps. Thanks!
Determine wether the differential equation is stable or not for the following conditions:  $x'''(0) = x''(0) = x'(0) = x(0) =0$  $\frac{d^3x(t)}{dt^3}+\frac{d^2x(t)}{dt^2}+4\frac{dx(t)}{dt}+4x(t) = e^t(H(t)-H(t-3))$  Now, I got that the solution to this differential equation reduces x(t) by use of Laplace Transforms, however, how do I determine whether or not it is stable?  $x(t) = 0.1e^{-t} + 0.1e^t - 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})*H(t-3)$  Now I first thought I would look at it graphically and I used Matlab software to get a plot as shown below:    And from the plot above it clearly appears to diverge (though this is not a proof).  Anyone have an idea as to an elegant proof to show the solution is unstable?  Anyone help would be greatly appreciated! How to disable Ctrl+Shift+U? Ctrl+Shift+U is a combined shortcut in Eclipse. But when I press it, it shows up a u, just like input with an underline. I guess this shortcut has been declared by Ubuntu, so I can't use it. I used to solve this problem by typing Caps Lock first. Is there any better method? Our Epson dot matrix printer is not printing on the following DOS command.:  Type text.txt&gt;prn   But it does print perfectly on:  open TEXT.txt&gt;File&gt;Print    What is the issue here? Why does my procedure run slowly until I update statistics? dev/sda1: clean, 552599/6111232 files, 7119295/24414464 blocks   After I turn on my laptop this message appears. The system never boots, the message just stays there. "physical block size is 2048 bytes, but Linux says it is 512" when formatting USB - How to create a bootable USB without this error Why commutator subgroup is normal to G? If $G$ is a group and $G'$ is generated by $\{xyx^{-1}y^{-1}|x,y\in G\}$, then $G'\trianglelefteq G$ and  $G/G'$ is Abelian.  At first, I thought this is easy because I thought $g[x,y]g^{-1}=[gxg^{-1},gyg^{-1}]$. But it was a mistake. I could not prove that it is also subgroup of G and $G/G'$ is abelian. I need help. Who wrote about an energy free space elevator? Knockout js loop up to a value What's the best way to communicate between view controllers? The parmesan cheese in pasta recipe is quite expensive.....do we have cheaper alternative??  Are there are cheddar cheese which can replace parmesan cheese? Someone found this quiver with surprisingly high values, although I'm unable to find a level 30 rare quiver on     Given that the item's internal level (IIL) determines which affixes are possible, I'd like to know whether the IIL affects the minimum/maximum rolls for an attribute, a graph or table would be great.  For example, let's say items are able to get a pure +intellect affix from IIL 20, what's stopping you from rolling 250 at 20?  Is it a bell curve, which has shallower gradients with a higher IIL of the item?  Is it capped, dependant on IIL?  Do the min/max caps step up at certain IILs,though it's the same attribute? For example, IIL 20-40 is capable of 5-120, IIL 41-60 is capable of 20-320, ..., IIL 63 = 40-400 Why do we say there is an overall negative charge on the intracellular side of the plasma membrane at rest, and an overall positive charge on the extracellular side when both potassium and sodium are positively charged ions, and they are in relatively equal amounts on either side? It seems to me that both sides should be positively charged, and I can't get my head around it. Unable to install Springseed 2 on Ubuntu 15.04 Limit of $\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} }$ I have a Dell XPS 12 running Ubuntu 13.10 and it doesn't appear to find it's wireless.  I have tried the suggestions  but they made no difference  peter@xps:/lib/firmware$ ls -l /lib/firmware/iwlwifi-7260-7.ucode  -rw-r--r-- 1 root root 682892 Aug  4 12:50 /lib/firmware/iwlwifi-7260-7.ucode peter@xps:/lib/firmware$ sudo lshw -c network   *-network UNCLAIMED             description: Network controller        product: Wireless 7260        vendor: Intel Corporation        physical id: 0        bus info: pci@0000:06:00.0        version: 63        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress cap_list        configuration: latency=0        resources: memory:f7c00000-f7c01fff  peter@xps:~/backports-3.10-2$ grep IWL7260_UCODE_API_MAX drivers/net/wireless/iwlwifi/iwl-7000.c  #define IWL7260_UCODE_API_MAX   7     .ucode_api_max = IWL7260_UCODE_API_MAX,         \ peter@xps:~/backports-3.10-2$ dmesg | grep iwl peter@xps:~/backports-3.10-2$ iwconfig lo        no wireless extensions.  peter@xps:~$ find / -name *7260-6* peter@xps:~$ find / -name *7260-7* /lib/firmware/iwlwifi-7260-7.ucode   peter@xps:~$ dmesg | grep -i -A 20 intel [ 1318.013278] Intel(R) Wireless WiFi driver for Linux, in-tree:d [ 1318.013281] Copyright(c) 2003-2013 Intel Corporation [ 1318.015154] cfg80211: World regulatory domain updated: [ 1318.015157] cfg80211:   (start_freq - end_freq @ bandwidth), (max_antenna_gain, max_eirp) [ 1318.015159] cfg80211:   (2402000 KHz - 2472000 KHz @ 40000 KHz), (300 mBi, 2000 mBm) [ 1318.015161] cfg80211:   (2457000 KHz - 2482000 KHz @ 40000 KHz), (300 mBi, 2000 mBm) [ 1318.015162] cfg80211:   (2474000 KHz - 2494000 KHz @ 20000 KHz), (300 mBi, 2000 mBm) [ 1318.015164] cfg80211:   (5170000 KHz - 5250000 KHz @ 40000 KHz), (300 mBi, 2000 mBm) [ 1318.015165] cfg80211:   (5735000 KHz - 5835000 KHz @ 40000 KHz), (300 mBi, 2000 mBm)     What I have found is that it appears to be loading an old version of the firmware but I don't know how or why  peter@xps:/lib/firmware$ modinfo iwlwifi | grep 7260 firmware:       iwlwifi-7260-6.ucode peter@xps:/lib/firmware$ ls *7260-6* ls: cannot access *7260-6*: No such file or directory   Any suggestions of how to get this setup very welcome. Does "a couple" always mean two? How to participate in clan war instead of being a spectator I joined a new clan and now that the clan war has started I tried to attack but it says "Spectator can't attack". How do I get out of being a spectator (I haven't used either of my 2 attacks in the war). I went to the clan castle and the clan option and I selected "Yes I'll participate in this war" but nothing happens. How do I tell Maven to use the latest version of a dependency? In Maven, dependencies are usually set up like this:  &lt;dependency&gt;   &lt;groupId&gt;wonderful-inc&lt;/groupId&gt;   &lt;artifactId&gt;dream-library&lt;/artifactId&gt;   &lt;version&gt;1.2.3&lt;/version&gt; &lt;/dependency&gt;   Now, if you are working with libraries that have frequent releases, constantly updating the &lt;version&gt; tag can be somewhat annoying. Is there any way to tell Maven to always use the latest available version (from the repository)? Set the location in iPhone Simulator How can I set the location (as it's picked up in CoreLocation services) in the iPhone Simulator?