It is a principle and proof from Introduction to Set Theory, Hrbacek and Jech.   In the proof, line 1 and 2, I couldn't understand why $Q(0)$ is true.   $Q(0)$ means that "$P(k)$ holds for all $k&lt;0$".   I understood there are no $k&lt;0$.   And then I couldn't proceed.
I was pondering a bit more about  question regarding being able to &quot;omit&quot; the base case in a proof by strong induction due to vacuous truth. The post states:  Strong induction proves a sequence of statements $P(0), P(1), …$ by proving the implication &quot;If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true.&quot; for every nonnegative integer $n$. There is no need for a separate base case, because the $n=0$ instance of the implication is the base case, vacuously.  However, if we consider $n=0$, we would have that the statement is vacuously true, which I would take to mean that the implication is true regardless of the validity of $P(0)$. However, clearly it's necessary for $P(0)$ to hold for an induction proof to be valid. So I'm confused on how, by omitting the base case, $n=0$ isn't just a tautology, making the implication true regardless of whether $P(0)$ actually holds.
If $A$ has two eigenvalues $\lambda _1, \lambda_2$ and $\dim (E_{\lambda_1})=n-1$, then $A$ is diagonalizable Alice and Bob play the following game with an $n \times n$ matrix, where $n$ is odd. Alice fills in one of the entries of the matrix with a real number, then Bob, then Alice and so forth until the entire matrix is filled. At the end, the determinant of the matrix is taken. If it is nonzero, Alice wins; if it is zero, Bob wins. Determine who wins playing perfect strategy each time.    When $n$ is even it's easy to see why Bob wins every time. and for $n$ equal to $3$ I have brute-forced it. Bob wins. But for $n = 5$ and above I can't see who will win on perfect strategy each time. Any clever approaches to solving this problem? Eclipse Shortcut to Split Long Strings I swear I've seen someone do this, but I can't find it in the various lists of shortcuts.  Given:  String s = "A very long ............................ String";   Is there an Eclipse shortcut to turn it into:  String s = "A very long ............................ "      + "String"; Print  only? How do I print the indicated div (without manually disabling all other content on the page)?  I want to avoid a new preview dialog, so creating a new window with this content is not useful.  The page contains a couple of tables, one of them contains the div I want to print - the table is styled with visual styles for the web, that should not show in print. Injecting variable values into javascript and HAML in RoR Explanation of finite correction factor Proof that the Casimir invariant of a representation commutes with everything stl::multimap - how do i get groups of data? Multimap essentially has groups of data sorted by the key. I want a method by which I could access these individual groups and get their aggregate values. For example, in a std::multimap&lt; string, int &gt; I store   {"Group1", 1},  {"Group1", 2},  {"Group1", 3},   {"Group2", 10},  {"Group2", 11},  {"Group2", 12}   Having stored these values, I should be able to iterate this multimap and get the aggregate values of each "group". Problem is there aren't any functions defined in STL to access MultiMaps in such a way. I could use lower_bound, upper_bound to manually iterate the multimap and total the group's contents, but I am hoping there could be better ways already defined in STL ? Can anyone propose a solution as to how I could get the aggregate values for a group in the above example. QGIS update from 2.14 to 2.16 failed on Ubuntu 14.04 Until yesterday my whole qgis and python-qgis worked fine with version 2.14 under Ubuntu 14.04. After a simple sudo apt-get update and sudo apt-get upgrade qgis failed to update... Several attempts to reinstall it with different versions failed.  Current error message (for version 2.14.3) goes like this:  qgis-providers (2.14.3+dfsg-2~trusty1) is configured ... /usr/lib/qgis/crssync: symbol lookup error: /usr/lib/libgdal.so.20: undefined symbol: GEOSDelaunayTriangulation_r   I also tried to reinstall all gdal libraries - not sure if I got all?!  Maybe it is somehow related to this one:  ?  UPDATE:  Trying to install version 2.16.0 it gives different but similar error:  /usr/lib/qgis/crssync: symbol lookup error: /usr/lib/libqgis_core.so.2.16.0: undefined symbol: GEOSNearestPoints_r   UPDATE Sources:  For version 2.16.0 there are these sources in /etc/apt/sources.list:  deb http://qgis.org/debian trusty main deb-src http://qgis.org/debian trusty main deb http://qgis.org/debian trusty main deb-src http://qgis.org/debian trusty main   which are all the same, right?   When I do apt-cache policy qgis it gives me:  qgis:   Installiert:           1:2.16.0+20trusty   Installationskandidat: 1:2.16.0+20trusty   Versionstabelle:  *** 1:2.16.0+20trusty 0         500 http://qgis.org/debian/ trusty/main amd64 Packages         100 /var/lib/dpkg/status      2.0.1-2build2 0         500 http://de.archive.ubuntu.com/ubuntu/ trusty/universe amd64 Packages Is it correct to change the common structure in these phrasal verbs? I just read a book to learn English. And the topic I read is about the phrasal verbs, but a big doubt has come up to my mind.  Is it correct to change the position of the preposition (putting it with the verb instead of after the direct object)?  As in:  Common structure:   My wife backs me up over my decision to quit my job. We have to blow 50 balloons up for the party.  I need to break these shoes in before we run next week. Please help me out to get this job done.   So, would it be correct to do the following:   My wife backs up me over my decision to quit my job. We have to blow up 50 ballons for the party.  I need to break in these shoes before we run next week. Please help out me to get this job done. Formal power series ring, norm. Let $k$ be a field. Let $R$ be the formal power series ring $k[[x]]$. Define $N$ on $R \setminus \{0\}$ as follows: $N(f)$ is the smallest $n$ of which the coefficient of $x^n$ in $f$ is nonzero.  (a) Prove that $R$ is a Euclidean domain with Euclidean norm $N$.  (b) For $a$, $b$, $a+b$ nonzero elements of $R$, prove that $N(a+b)$ cannot be bounded as a function of $N(a)$ and $N(b)$.  (c) Prove that if $a$ and $b$ are two power series such that $b$ does not divide $a$ (and $b \neq 0$), there are infinitely many pairs $(q, r)$ for which $a = bq + r$ and $N(r) &lt; N(b)$. What evidence is there supporting an alternative parentage for Jon Snow? I've heard speculation that Jon Snow's actual parents may be Rhaegar and Lyanna Stark, but in reading the books I'm not sure what evidence could be used to support this conclusion.  I thought there might have been something in the conversation Davos had before being smuggled into White Harbor, but I couldn't stretch anything that far.  What evidence do proponents of this theory cite?  Does any of it involve stuff in the texts or just sweeping generalizations of the characters' personalities (e.g. "Ned Stark is too honorable to father a bastard"). In SF1, Stop refreshing the embedded vf page on a layout Below is my VF page which is embedded on a layout. In SF1 mobile app, it's getting refreshed whenever we touch on it. How can I avoid refreshing the VF page while touching it on the screen? In Desktop it's working fine. Please provide your inputs. &lt;apex:page standardController=&quot;PremiseObj__c&quot; lightningStylesheets=&quot;true&quot;&gt; &lt;apex:includeLightning /&gt; &lt;apex:slds /&gt; &lt;div id=&quot;premiseServDetails&quot;&gt; &lt;/div&gt; &lt;script&gt; window.onload = function (){     $Lightning.use(&quot;c:PremiseObjUpdateApp&quot;, function() {         $Lightning.createComponent(&quot;c:PremiseObjUpdatecmp&quot;,                                    {                                         recordId : &quot;{!Premiserecord__c.Id}&quot;                                     },                                    &quot;premiseServDetails&quot;,                                    function(cmp,event) {                                        setTimeout(function() {                                            //document.getElementById('loading').style.display = 'none';                                        }, 3000);                                    });     }); } &lt;/script&gt; &lt;/apex:page&gt; Should Emergency Funds be Used for Infrequent, but Likely, Expenses? Say we break down non-monthly costs into the following three categories.   High level of predictability: This would include oil changes that may be needed every 2 to 4 months depending on usage, car insurance (if paid every 6 month), car registration, etc. Medium level of predictability: There are some expense which one can reasonable assume will occur on an infrequent basis.  For example, car repairs, house repairs, or even doctor visits during flu season. Low level of predictability: For example, losing a job, significant medical expense, etc.   Which of these categories are emergency funds meant to cover?  I think it is pretty clear that a high level of predictability costs should not use emergency funds while a low level of predictability should.  But medium level of predictability costs seem different, and it left me wondering what is generally defined as an "emergency".  While you do not know when these costs will occur or how much they will be, you could devise some pretty good rules of thumb on frequency and cost. Say you know your 2007 Volvo should have a repair cost averaging out to $100 per month.  Conceptually you could save $100 each month and when your car breaks, not have to spend any emergency funds on the repair.  Is doing this "double-saving" and not suggested if you already have the 6 month emergency fund saved up?  Note: this is a question on how to save money for expenses.  It is not a question about if you should spend emergency fund money to fix a car when you have no other money set aside. Problem in solving functional equation $f(x^2 + yf(x)) = xf(x+y)$ To find all functions $f$ which is a real function from $\Bbb R \to \Bbb R$ satisfying the relation  $$f(x^2 + yf(x)) = xf(x+y)$$  It can be easily seen that the identity function $i.e.$ $f(x)=x$ and $f(x)=0$ (verified just now) satisfies the above relation!! And putting $y=0$ I have got $f(x^2)=xf(x)$.  Help needed to find other functions satisfying the relation. Is it possible to activate an iPhone 4 without a SIM? How do I delete all empty directories in a directory from the command line? Say I have a directory named foo/. This folder includes subdirectories. How can I delete all the empty directories in one command? Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is:  $$ \nabla_A \operatorname{trace}( ABA^{T}C )  = CAB + C^T AB^T $$  There is a proof . However, I was not able to understand the second line of the proof. it says something like this:  $$ \nabla_A \operatorname{trace}( ABA^T C )= \nabla_\bullet \operatorname{trace}(f(\bullet)A^{T}C) + \nabla_\bullet \operatorname{trace}(f(A) \bullet^T C) $$  Where  $f(A) = AB$  I tried searching the proof or any hint using google, but I couldn't fine anything. Could anyone please help me with this second line. I cannot sleep. I am obsessed with this proof. HDMI/Displayport audio control for OS X Uncaught exception thrown after moving my website to localhost