I have seen this example in a number of course material Show that the hypotheses “It is not sunny this afternoon and it is colder than yesterday”, “We will go swimming only if it is sunny”, “If we do not go swimming, then we will take a canoe trip”, and “If we take a canoe trip, then we will be home by sunset” lead to the conclusion “We will be home by sunset”. and the propositions are identified as p - “It is sunny this afternoon” q- “It is colder than yesterday” r- “We will go swimming” s- “We will take a canoe trip” t- “We will be home by sunset” All of them translate the second premise as r--&gt; p. I am not able to understand why. Doesn't the second premise mean &quot; if it is sunny then we will go swimming&quot; and if yes then it should be p--&gt;r or if the second premise is a biconditional then it should be r&lt;--&gt;p
For some reason, be it some bad habit or something else, I can not understand why the statement "p only if q" would translate into p implies q. For instance, I have the statement "Samir will attend the party only if Kanti will be there." The way I interpret this is, "It is true that Samir will attend the party only if it is true that Kanti will be at the party;" which, in my mind, becomes "If Kanti will be at the party, then Samir will be there."   Can someone convince me of the right way?  EDIT:  I have read them carefully, and probably have done so for over a year. I understand what sufficient conditions and necessary conditions are. I understand the conditional relationship in almost all of its forms, except the form "q only if p." What I do not understand is, why is p the necessary condition and q the sufficient condition. I am not asking, what are the sufficient and necessary conditions, rather, I am asking why.
Deferring admission to a computer science graduate program Inferring father/son relationships from Senior (Sr.) / Junior (Jr.) naming? As in my other questions, I am investigating the spread of the Blackstone family name in early America. I came across the following tidbits and I would like to know if my interpretation is (likely to be) correct.  In the "All Kent County, Delaware Probate Records, 1680-1800" collection on ancestry.com, we have the following items.   In 1792, there was a will executed by Eben Blackiston, Sr. In 1794, there was a will executed by Ebenezer Blackiston, Jr. (and Abraham Redgrave).   My question is simple.  Is this enough evidence to reasonably conclude that there was a father/son combo of Ebenezer Blackistons living in Kent County, Delaware during the 1790s?  This would be an interesting result for me as my current research only has two people by this name in the state at the time, and none of them were part of a Jr./Sr. combination.  If this is not enough evidence, what are some other possibilities I need to consider? Experimental Fractional scaling makes fonts dirty in Ubuntu 17.10 I updated to 17.10 from 17.04.   I activated the fractional scaling by running the following command.   gsettings set org.gnome.mutter experimental-features "['scale-monitor-framebuffer']"   Then, I changed 125% scale. The scale was changed, but the fonts and icons are so dirty.    The launcher icon and font in Google Chrome and other application except the system setting are dirty.  How can I resolve this problem?  For some reason, font in the system setting is not dirty... I want to know if there is another way to solve the equation:  $$(x+1)^y-x^z=1$$  for positive integers $x,y,z$ greater than $1$ without using the Catalan identity. Does exist? By now I only know that $x\mid y$. A floor is paved with rectangular marble blocks,each of length $a$ and breadth $b$.A circular block of diameter $c(c&lt;a,b)$ is thrown on the floor at random.Show that the chance that it falls entirely on one rectangular block is $\frac{(a-c)(b-c)}{ab}$  I thought over this problem,i found total number of cases as area of rectangular marble block,but i cannot find the favorable number of cases.Favorable number of cases cannot be area of rectangular marble block$-$ area of circular block,as the answer suggests.What should be the correct logic,please help. Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$? Finding the derivative of $x\uparrow\uparrow n$ I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general expression for $f^{\prime \prime}$ and I have stumble opon the series in the picture below, I wonder, do anyone have an idea of what sort of expantion it migth be ? I have tried breaking it down into three different sums, but I wasn't very successful!  EDIT  So since this post has gotten some attention and the title has been change I will tell you what I am on about, although I already have a post just for the question in the title from before. I am as I said earlier trying to find a general derivative for the function $f(x)=x^{x^{x^{...^{x}}}}$ where $n$ is an arbitrary number, natural number and greater or equal to 2. I have already made success when I tried to find the first derivative, and I have also found some interesting pattern emerge when the derivatives as well as n incresses. But my question here was if anyone knew what this expation might be or look similar to be, since I have yet to be lernt series etc. I did not know that the question needed such background since it's straight a forward question. How did the AT-ATs get to the surface of Hoth? Stirling's Approximation Proof I was searching for the reason why Stirling's Approximation holds true. I found the website  which apparently shows why this is the case. Does this part of the equation make sense in the proof? If so why?  $$\int_{1}^{n}\ln(x)dx\approx\sum_{k=1}^{n}\ln(k)$$ Why do we need to be root in terminal for shutdown and restart? Implementing a PIVOT query How can I pivot the following SELECT statement:           SELECT sc.CID,                 sc.CodeName     as OverviewText,                 scRAG.CodeName  as RAGStatusText             FROM StatusCode      sc LEFT OUTER JOIN ProjectOverview po               ON sc.CID          = po.ProjectOverviewCID              AND po.ProjectId    = 180 LEFT OUTER JOIN StatusCode      scRAG               ON po.RAGStatusCID = scRAG.CID           WHERE sc.SCID         = 18   Such statement yields the result set that follows:  CID OverviewText    RAGStatusText --- ------------    ------------- 153 Cost            Green 154 Requirements    Yellow 155 Schedule        NULL 156 Technical       NULL 157 Testing         NULL   But, I want it to return one single row with 10 values, as shown bellow:   ----------------------------------------------------------------------------------------------  | Cost | Green | Requirements | Yellow | Schedule | NULL | Technical | NULL | Testing | NULL |  ----------------------------------------------------------------------------------------------   Can I pivot on CID? Recovery of deleted folders in home folder By accident I've removed some folders (they were empty) in my home folder. I've created a new ones but they doesn't have proper icons etc... how can I recreate them step by step? Invisible software updates list in Software Updater Today I received an update notification for three updates but update manager doesn't say what updates are they.    Does anyone have a clue what this is about? How am I to check what is to be updated? $ \frac{1}{a^{2} + b^{2} + 1} + \frac{1}{b^{2}+c^{2}+1} + \frac{1}{c^{2} + a^{2} + 1} \ge 1 $, prove $ab + bc + ca \le 3$ If $a,b,c$ real numbers such that $$ \frac{1}{a^{2} + b^{2} + 1} + \frac{1}{b^{2}+c^{2}+1} + \frac{1}{c^{2} + a^{2} + 1} \ge 1 $$,  prove that $ab + bc + ca \le 3$.     Attempt:  Assume that $a,b,c &gt; 0$.   $$ \frac{1}{a^{2} + b^{2} + 1} + \frac{1}{b^{2}+c^{2}+1} + \frac{1}{c^{2} + a^{2} + 1} \ge 1  $$ $$ \implies  (a^{2} + b^{2} + 1)(b^{2}+c^{2}+1) + (a^{2} + b^{2} + 1)(a^{2}+c^{2}+1) +  (a^{2} + c^{2} + 1)(b^{2}+c^{2}+1) \ge (a^{2} + b^{2} + 1)(b^{2}+c^{2}+1)(c^{2} + a^{2} + 1)$$  $$ \implies  ((ab)^{2} + (ac)^{2} + (bc)^{2} + a^{2} + 2 b^{2} + c^{2} + b^{4} + 1) + ((ab)^{2} + (ac)^{2} + (bc)^{2} + b^{2} + 2 a^{2} + c^{2} + a^{4} + 1) + ((ab)^{2} + (ac)^{2} + (bc)^{2} + a^{2} + 2 c^{2} + b^{2} + c^{4} + 1) \ge (a^{2} + b^{2} + 1)(b^{2}+c^{2}+1)(c^{2} + a^{2} + 1)$$  $$ \implies  3((ab)^{2} + (ac)^{2} + (bc)^{2}) + 4(a^{2} + b^{2} + c^{2}) + (a^{4}+b^{4}+c^{4}) + 3  \ge (a^{2} + b^{2} + 1)(b^{2}+c^{2}+1)(c^{2} + a^{2} + 1)$$  Next, notice  $$(a^{2} + b^{2} + 1)(b^{2}+c^{2}+1)(c^{2} + a^{2} + 1) = ((ab)^{2} + (ac)^{2} + (bc)^{2} + a^{2} + 2 b^{2} + c^{2} + b^{4} + 1)(c^{2} + a^{2} + 1) $$ $$ = (abc)^{2} + a^{2}c^{4} + b^{2}c^{4} + (ac)^{2} + 2(bc)^{2} + c^{4} + b^{4}c^{2} + c^{2} + a^{4} b^{2} + a^{4}c^{2} + (abc)^{2} + a^{4}  + 2(ab)^{2} + (ac)^{2} + b^{4}a^{2} + a^{2} +  (ab)^{2} + (ac)^{2} + (bc)^{2} + a^{2} + 2 b^{2} + c^{2} + b^{4} + 1 $$ $$ = 2(abc)^{2} + 3(ac)^{2} + 3(bc)^{2} + 3(ab)^{2} + c^{4}a^{2} + c^{4} b^{2} + b^{4}a^{2} + b^{4} c^{2} + a^{4}b^{2} + a^{4} c^{2} + a^{4} + b^{4} + c^{4} + 2(a^{2} + b^{2} + c^{2}) + 1  $$  So we have  $$ \frac{1}{a^{2} + b^{2} + 1} + \frac{1}{b^{2}+c^{2}+1} + \frac{1}{c^{2} + a^{2} + 1} \ge 1 $$ $$ \implies  2(a^{2} + b^{2} + c^{2}) +  3  \ge 2(abc)^{2}  + c^{4}a^{2} + c^{4} b^{2} + b^{4}a^{2} + b^{4} c^{2} + a^{4}b^{2} + a^{4} c^{2} + 1 $$ I would like to use the sentence     He was 'over-eager' to implement his plan, leading to failure.   I am looking for a single word which expresses over-eagerness in a negative sense.  TIA mysqli::query(): Couldn't fetch mysqli I just discovered an incredibly annoying "feature" of Windows 7 tonight.  If you click on your optical drive in Explorer, and there is no disc in the drive, Windows 7 is kind enough to open the tray for you.  I'm finding it hard to contain my rage over this idiotic design decision.  WTF?  According to , this behavior was apparently introduced in Vista.  The author posted a workaround, but I don't want a workaround.  I want it to stop doing this.  Like he, I have a case with a door that covers the drive bays.  If it's closed, and I accidentally click on that optical drive, it knocks on that door 10 or 12 times before giving up.  Can this behavior be turned off in Windows 7? Can Git hook scripts be managed along with the repository? On , sorted by votes, the accepted answer is #3 in order of votes.  It appears as number 3 in the list as well. Is this new behavior intended? I have a sprite sheet from a classic video game (if you're curious, Final Fight) that I'd like to be broken up into indivual images to be used in . However, the images aren't on an even grid, and all have a different amount of space between them.  Is there an automated way to export images based on contiguous pixels or something like that?