I'm working on a mathematical proof, but the problem I'm having is more to do with logic than math per se.   I'm wondering if these two are equivalent:   If A and B then C   If A and not B, then not C.
For example:     Is (A &or; B &or; C) &and; (D &or; E &or; F) the same as   (A &and; D) &or; (A &and; E) &or; (A &and; F) &or; (B &and; D) &or; (B &and; E) &or; ( B &and; F) &or; (C &and; D) &or; (C &and; E) &or; (C &and; F)?   How would I check equivalence in general?
What does "it" refer to in "it's raining"? I wanted to leave the question title as is so as not to take away from my amusement :).  Anyway,     It's raining.   What is raining? Is it the sky? The clouds? The weather? The rain? What is "it"? Any historical insights on the statement? Transpose Ranges repeated by row I would like to convert several range of columns into rows, and keeping the association to the original col/row source  Sample source sheet  --------------------------------------------------- | Data 1 | 1/1/2018 | 1/2/2018 | 1/3/2018 | etc...365 days | --------------------------------------------------- | name 1  |    25    |   30     |    5     | etc...|  --------------------------------------------------- | name 2  |    2     |          |    4     | etc...| --------------------------------------------------- | etc ... 100 rows   Expected Results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- | name 1  |  1/2/2018   |   30     |    --------------------------------------------------- | name 1  |  1/3/2018   |   5      |    --------------------------------------------------- | name 1  |  etc.. 365 days --------------------------------------------------- | name 2  |  1/1/2018   |   2      |    --------------------------------------------------- | name 2  |  1/2/2018   |          |    --------------------------------------------------- | name 2  |  1/3/2018   |   4      |    --------------------------------------------------- | name 2  |  etc.. 365 days --------------------------------------------------- | etc ... for each change of name pull all the col dates/values   Current results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- |         |  1/2/2018   |   30     |    --------------------------------------------------- |         |  1/3/2018   |   5      |    --------------------------------------------------- |         |  etc.. 365 days                      I have to copy down to bring in "name 1" for each date  --------------------------------------------------- | name 2  |  1/1/2018   |   2      |             now update the formula to bring in the second item for the sequence of dates --------------------------------------------------- |         |  1/2/2018   |          |    --------------------------------------------------- |         |  1/3/2018   |   4      |    --------------------------------------------------- |         |  etc.. 365 days --------------------------------------------------- | etc ...     =Sheet1!A:A I can bring in the name from the source list, but I have to copy it down for the number of dates.  How can I have it match the number of dates?  and be dynamic to update as dates/values are added Edit: trying to use split and rept to count the number of times to repeat, then transpose.. =transpose(rept(split(arrayformula(concatenate(Sheet1!A:A&amp;"1")),"1",True, True),COUNTA(Sheet1!I1:ZZ1)))  =TRANSPOSE(Sheet1!$C$1:$Z$1) I can use transpose to get the dates/values, but it's fixed range.  How can I make it dynamic to import dates as they are added? match first column of file a with paragraphs of file b I have 2 files. The first, fileA looks like  TCONS_00000066  XLOC_000030     -       u       q1:XLOC_000030|TCONS_00000066|0|0.000000|0.000000|0.000000|0.000000|- TCONS_00000130  XLOC_000057     -       u       q1:XLOC_000057|TCONS_00000130|0|0.000000|0.000000|0.000000|0.000000|- TCONS_00000395  XLOC_000206     -       u       q1:XLOC_000204|TCONS_00000393|0|0.000000|0.000000|0.000000|0.000000|-   FileB looks like:  &gt;TCONS_00000001 gene=XLOC_000001 AGATGAGCTGGTGGGGATGCTCTAAGAGAACGAGAGAAGCACAGAGCAGATAAACCACACCCACAGGCAC CACCGTCCTTGTTGGTAATGAAGAAGACGAGACGACGACTTCCCCACTAGGAAACACGACGGAGGCGGAG ATGATCGACGGCGGAGAGAGCTACAGAAACATCGATGCCTCCTGTCCAATCCCCCCATCCCATTCGGTAG TTGGATTGAAGACTACCGAATAAGAGAAGCAGGCAGGCAGACAAACCCTTGAACCAAGGAGTCCTCGCTG AGGAAGCTTTGGATCCACGACGCAGCTATGGCCTCCCCGCCCACCAGGCCGCCAGCCACAACCAGCTGAC TAGGTCGCATGCATCATCAGATTTCAATCTCCCTTCGTTCCCTGTCCCTAATCCAATACCAATAGGGAGC AATCAGCTGCTCCTCGACGGCGAGGGAGATGTCGTCGGCCGCGGGCCAAGACAACGGAGATACCGCTGGG GACTACATCAAGTGGATGTGCGGCGCCGGTGGCCGTGCGGGCGGCGCCATGGCCAACCTCCAGCGCGGCG TTGGCTCCCTCGTCCGTGACATTGGCGACCCCTGCCTCAACCCATCCCCCGTTAAGGGGAGCAAAATGCT CAAACCGGAAAAATGGCACACATGTTTTGATAATGATGGAAAGGTCATAGGTTTCCGTAAAGCCCTAAAA TTCATTGTCTTAGGGGGTGTGGATCCCACTATTCGAGCTGAAGTTTGGGAATTTCTTCTTGGCTGCTATG CCTTGAGTAGTACCTCAGAGTATAGGAGGAAACTAAGAGCTGTTAGAAGGGAAAAATATCAAATTTTAGT TAGACAGTGCCAGAGCATGCACCCAAGCATTGGTACAGGTGAGCTTGCTTACGCTGTTGGATCAAAGCTA   Now, fileA contains selected transcript numbers in the first column and fileB contains sequences of all transcripts. I want to scan fileB for the first column of fileA and print the trailing sequences of matching transcripts along with the transcript number. Where is the preferred place to set the PATH envvar?  ~/.profile or /etc/environment?  What is the case when PATH is set in both places? Is the end result a concatenation of both values set in those two places? Retail boxed Steam activated games: can they be resold? I recently purchased retail boxed versions of   Dead Island (which is awesome) Warhammer 40k: Space Marine (not so much)   I finished both games and I was thinking of reselling them or giving them to a friend to play. However, these are both Steam activated games that had a required Steam registration code I had to enter -- and validate on Steam -- as part of the installation.  If I give these games to a friend or resell them, will they even work? Is the registration code single use and tied to a single person / computer / Steam account? I am moving from Egypt to the Netherlands and I need to take my PC with me. I won't be taking the monitor, just the case, but I'm wondering what the best way would be.  Am considering taking it as carry-on because I'm allowed 8kg and at least then I can handle it myself. Has anyone done this before and can give me tips or warn me that this doesn't work? Would putting it in my normal suitcase as checked baggage be a better idea? vim and system clipboard I can cat a file, copy and paste text with the middle button or shift-insert.  How can I copy text in vim and paste it from the system clipboard? Adjective, if adjective: eg "pointed, if fatherly" They were pointed, if fatherly, remarks that echoed the themes he has stressed in his papacy but ones that resonated all the more in a newly renovated cathedral surrounded by the luxurious shops of Fifth Avenue. —   I'm not sure the meaning of the phrase &quot;if fatherly&quot; in the above sentence. I'm not familiar with the syntax using the structure 'adjective, if adjective'. Isomorphism $H_n(D^n, S^{n - 1};A)$ and $A$ The homomorphism $a \mapsto [[a \cdot f]]$ is supposed to be an isomorphism between $A$ and $H_n(D^n, S^{n - 1};A)$ for any homeomorphism $f: \Delta^n \to D^n$. I have proven that it is well defined and injective, but why is it surjective?  The group $H_n(D^n, S^{n - 1};A)$ is generated by elements of the form $a \cdot \sigma + b \cdot \tau + c \cdot \partial(\kappa)$ where $\sigma: \Delta^n \to D^n$, $\tau: \Delta^n \to S^{n - 1}$ and $\kappa: \Delta^{n + 1} \to D^{n + 1}$ and I do not see how an element of this form represents the same equivalence class of $[[m \cdot f]]$ for some $m \in A$. We can write $\sigma$ as $f f^{-1} \sigma$ to get $f$ in there, but I don't see that working out. Any hints? Is there an easy way to log all commands executed, including command line arguments? I am trying to find how to log a specific instantiation of rrdtool to see whether the path it is receiving is incorrect.  I know I could wrap the executable in a shell script that would log the parameters, but I was wondering if there was a more kernel-specific way to monitor for that, perhaps a filesystem callback that sees when a particular /proc/pid/exe matches a given binary? I have CSV data loaded into a multidimensional array. In this way each "row" is a record and each "column" contains the same type of data.  I am using the function below to load my CSV file.  function f_parse_csv($file, $longest, $delimiter) {   $mdarray = array();   $file    = fopen($file, "r");   while ($line = fgetcsv($file, $longest, $delimiter))   {     array_push($mdarray, $line);   }   fclose($file);   return $mdarray; }   I need to be able to specify a column to sort so that it rearranges the rows.  One of the columns contains date information in the format of Y-m-d H:i:s and I would like to be able to sort with the most recent date being the first row. I have read this topic:   There, the issue is handling two different generations of PHP (PHP 4 vs PHP 5), but what I need is to make all but one of my server accounts work under 5.3 and just a single domain under 5.2.  I cannot make this through Apache models, and the answer from  does not apply since in my case I can't use a different port - this is a regular website  on the same server. What is a NumberFormatException and how can I fix it? I’m looking for an adjective/ noun that describes a person who loves life, traveling, great food, always doing things which gets adrenaline kicking and is a always ready for attempting things for pleasure only How do I align an image to centre? How do I align an image to centre?  In a beamer presentation I have a 2-3 items on a slide followed by an image. The image is not wide enough to cover whole slide, so it leaves around 30% space from right. How do I adjust it to leave enough space on both sides?  \begin{frame} \frametitle{Outlook} \begin{itemize} \item apps \item apps \item apps \includegraphics[scale=0.3]{P2P}     \end{itemize} \end{frame} Keyboard shortcut to focus on (idle) terminal I would like to know, if there is possible to use keyboard shortcut to focus on first idle (non working) terminal window in unity (13.10)? Something like Ctr+Alt+T, but before opening the new instance of terminal, searching for any idle instance? How do I disable the guest session in Ubuntu 11.10 or higher? I don't want people to be able to use my computer without using a password to log in! How do you open a command line? How do you access the command line or run option to put the commands into the system? What is EV, when used as an absolute measurement? As a non-native speaker, I feel that it is ridiculous that can and can't could sound so similar in American accent.  Just now, I was just listening to a video in which the speaker with an American accent says     Fat cells can't reproduce themselves.   The speaker dragged the word "can't" longer for emphasis, but perhaps as a result, what I heard at first was     Fat cells can reproduce themselves.   Of course from the context I later figured out what he was really saying, but I felt extremely surprised by how these two words which literally mean the opposite could sound so similar in American accent. How is that not affecting American people's daily conversations? As a non-native speaker, is there a way for me to further clarify them when my listeners are confused? (Like when you say "one-eight" to clarify that you were trying to say "eighteen" as opposed to "eighty".)