I've tried using direct proof and contrapositive proof to prove this. I'm stuck. Is anyone able to help solve or give me some hints?
Let $m$ and $n$ be two integers. Prove that if $m^2 + n^2$ is divisible by $4$, then both $m$ and $n$ are even numbers.   I think I have to use the contrapositive to solve this. So I assume $\neg P\implies Q$ and I have to derive $\neg Q$? I know its easier to show odd numbers rather than even... but im still unsure how to do this practice problem.
How do the radian have a unit? The radian is defined as the ratio of the circumference and the radius. Both are measured in meters. So there should not be a unit for that. But we use 'rad' as the unit of the radian value.   The coefficient of static/kinetic friction also the same, it is a ratio of both forces. Therefore it doesn't have a unit.  So, is there a special reason to have a unit for the radian value? primes whose difference is a multiple of $n$ Calculating the longest distance within polygon in QGIS I want to calculate the "diameter" of polygons in QGIS.  The "diameter" defined as the distance between the two most distant points of the polygon.  I couldn't find a fitting solution in the field calculator. Maybe you have some ideas? Remove line containing certain string and the following line How can I query the name of the current SQL Server database instance? It is a bit of a "chicken or egg" kind of query, but can someone dreamup a query that can return the name of the current database instance in which the query executes? Believe me when I say I understand the paradox: why do you need to know the name of the database instance if you're already connected to execute the query? Auditing in a multi-database environment.  I've looked at all the @@ globals in Books Online. "SELECT @@servername" comes close, but I want the name of the database instance rather than the server. Multi-Question Questions: How Best to Handle Them? Sometimes I come across a question on SO that is actually a number of questions all in one. If these &quot;sub-questions&quot; are closely related, it's not a problem. But if they are only weakly related (or not at all) it gets ugly. I'm now (ironically or hypocritically) going to ask a few questions of my own for the community to respond to:  Do you consider a question made up of a number of weakly-related sub-questions &quot;inappropriate&quot;? Why or why not? How many is too many sub-questions? What do you think is the best way to deal with such questions?  As an example,  I had some discussion about with another user. I will add a more complete version of my response in an answer below, but in short I felt that this question should be closed and many of the questions asked separately. Proving that without the axiom of choice there is a set with an accumulation point that isn'tthe limit of a convergeant sequence I've been reading the simple parts of Thomas Jech's book on the Axiom of Choice and came across the following proof on page 141. The proof assumes a mathematical model without the axiom of choice.     Lemma. $X$ is D-finite if and only if $X$ does not have a countable subset.       Theorem. There is a set $S$ of real numbers and a real number $a \not \in S$ such that a is in the closure of $S$ but there is no sequence $(x_n)^{\infty}_{n=0} \subset S$ such that $(x_n)^{\infty}_{n=0} \longrightarrow a$.       Proof. Let $D$ be a D-finite infinite set of reals. We claim that the set $D$ must have an accumulation point. To prove this claim suppose that all $d \in D$ are not accumulation points, then let $\{I_n:n=0,1,...\}$ be a fixed enumeration of open intervals with rational endpoints. Assign the least $n$ such that $I_n \cap D={d}$ to each $d \in D$. This makes $D$ countable and by the above lemma gives us our contradiction. So $D$ has an accumulation point $a$. Let $S=D \setminus\{a\}$ and since $S$ is D-finite, every convergent sequence in $S$ is eventually constant.   Could somebody explain this proof? Why is the last sentence true/ Why does this mean every convergent sequence is eventually constant?  I think the following theorem is important     Theorem. There is a model of ZF which has an infinite set of real numbers without a countable subset Wrong hyperref with repeated counter I'm having an interesting problem with hyperref.  Apparently, when a counter is reset by \setcounter{mycounter}{0} then hyperref always links to the first possible instance of the counter and not the one where the respective label sits.   Check this MWE out:   % !TEX TS-program = xelatexmk \documentclass{article} \usepackage[paperwidth=6in,paperheight=3in]{geometry} \usepackage{hyperref} \hypersetup{colorlinks=true, allcolors=blue, } \newcounter{mycounter}\setcounter{mycounter}{0} \newcommand\countme{\refstepcounter{mycounter}\themycounter}  \begin{document}  First line \countme{} has some interesting text.  The second line \countme \label{onelabel} has \textbf{more} interesting text.  As we saw at line \ref{onelabel} ...  \newpage  A new page brings new numbers \setcounter{mycounter}{0}   A very different first line \countme{} has other interesting text.  The second line \countme \label{otherlabel} has \textbf{different} interesting text.  As we saw at line \ref{otherlabel} ... \end{document}   As you can see in the screenshot below, the pop up preview of the \ref on the second page doesn't refer to the second line with \label{otherlabel} but to the one on the first page (also has the tooltip "Go to page 1" -- in TeXShop).   Am I not understanding something fundamental about hyperref?      The context for all this is an environment that resets my counter to 0 at begin.  And all subsequent \refs to labels in a second or third iteration of the environment point to the very first instance of that environment, not the correct one where the label sits. Word for seeing both sides of an argument From the TV Show Krypton, Dru-Zod is the son of Seg-El and Lyta-Zod. In S02E05 'A better yesterday', Lyta-Zod's throat gets slit by Jax-Ur leading to Lyta's death. How is Dru-Zod still alive? Shouldn't he be wiped from existence? Questions can be protected (i.e. marked "highly active").   What does it mean for a question to be protected? Why are some questions protected? Who can protect and unprotect questions? Who can answer a protected question? How are protected questions displayed? When should I protect or unprotect a question? When does the Community user protect a question? I have a PC with an OEM install of Win 8 updated to 8.1.  It has 3 physical hard drives in it.  The drive where Windows is installed has a LOT of partitions on it already.    Can I install Ubuntu on a different physical hard drive than Windows, and be able to dual-boot?  I am able to boot to a live USB of Ubuntu 14.04, so I think I'm about ready to proceed.  But if I can make my installation easier by using a different physical drive, I'd like to do that. How do I remove all loose edges? I have cut a hole in a plane, but I am left with some loose edges.  Is there an easy way to select and delete them all?  See this screenshot:    See this edges siting out with no face? What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors? I get the physical significance of vector addition &amp; subtraction. But I don't understand what do dot &amp; cross products mean?   More specifically,    Why is it that dot product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\cos\theta$? Why is it that cross product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\sin\theta$, times a unit vector determined from the right-hand rule?   To me, both these formulae seem to be arbitrarily defined (although, I know that it definitely wouldn't be the case).   If the cross product could be defined arbitrarily, why can't we define division of vectors? What's wrong with that? Why can't vectors be divided? Enumerate or list all variables in a program of [your favorite language here] A friend asked me last week how to enumerate or list all variables within a program/function/etc. for the purposes of debugging (essentially getting a snapshot of everything so you can see what variables are set to, or if they are set at all). I looked around a bit and found a relatively good way for Python:   #!/usr/bin/python                                                                                                                                                                                                                            foo1 = "Hello world" foo2 = "bar" foo3 = {"1":"a",         "2":"b"} foo4 = "1+1"  for name in dir():     myvalue = eval(name)     print name, "is", type(name), "and is equal to ", myvalue   which will output something like:   __builtins__ is &lt;type 'str'&gt; and is equal to  &lt;module '__builtin__' (built-in)&gt; __doc__ is &lt;type 'str'&gt; and is equal to  None __file__ is &lt;type 'str'&gt; and is equal to  ./foo.py __name__ is &lt;type 'str'&gt; and is equal to  __main__ foo1 is &lt;type 'str'&gt; and is equal to  Hello world foo2 is &lt;type 'str'&gt; and is equal to  bar foo3 is &lt;type 'str'&gt; and is equal to  {'1': 'a', '2': 'b'} foo4 is &lt;type 'str'&gt; and is equal to  1+1   I have so far found a partial way in PHP (courtesy of ) but it only lists all variables and their types, not the contents:   &lt;?php // create a few variables $bar = 'foo'; $foo ='bar'; // create a new array object $arrayObj = new ArrayObject(get_defined_vars()); // loop over the array object and echo variables and values for($iterator = $arrayObj-&gt;getIterator(); $iterator-&gt;valid(); $iterator-&gt;next())         {         echo $iterator-&gt;key() . ' =&gt; ' . $iterator-&gt;current() . '&lt;br /&gt;';         } ?&gt;   So I put it to you: how do you list all variables and their contents in your favorite language?    Edit by : I propose this question follows the spirit of a little "". If you do not agree, just edit and remove the tag and the link. Cron jobs in Drupal 7 I have a PHP file that does some task in consistent intervals.   I have placed that PHP file inside a custom module folder, but I found that Drupal functions like drupal_set_message() are not available inside the PHP file.   How do I include Drupal functions inside this PHP file? How do I set up this php file to be run as a cron job? I have been there first time.   The reason why I think this might not be natural is that the expression "first time" seems to describe "past". You do not use the present perfect with an adverb indicating "past", in my understanding.  However, I feel the phrase below is not that funny.     This is the first time that I have been there.   I am wondering about the relationship between "the first time" and "I have been there". "That" could behave like a relative adverb. If that is the case, the antecedent of "that"—"the first time"—is shared with the following clause: "I have been there". Then, is it OK to say "I have been there first time"?  By the way, Google searches for "" and " in the UK domain show just 33 hits for the first phrase and 10 hits for the second. I want to do a search on StackOverflow for a few terms, but also by user, since I know which user referenced the item I'm looking for. Discrete Probability: Four dice are thrown, what's the probability that... Four dice are thrown, what's the probability that:  a) None of them fall higher than three?  b) None of them fall higher than four?  c) That four is the highest number thrown?  So for a, I wanna think about the denominator first. There are 6 possible outcomes for each dice, and we have four dice. So we're technically picking r out of n objects, aka picking 4 possible values out of 6. So the denominator/sample space should be $6^4$ right?  Now this is where I'm getting tripped up. Order technically shouldn't count, because we only care about the quadruplets that don't have a value higher than 3. I might be wrong, maybe I am, but could someone explain a bit more? My professor stated that typically in the sample space/denominator, we want order to count.   So I'm trying to think of the numerator now, so we want to find the probability that no values appear higher than 3, so some events that can occur are: $(1, 1, 2, 3), (1, 2, 2, 3), (1, 1, 1, 1)$ etc. Again, I can't see why order should count here, because if we only are concerned about what appears rather than how they appear, then we can say $(1, 1, 2, 3)$ is equal to $(1, 2, 1, 3)$? So if order does not count, and we have replacement, then does this lead the case where we use $$\binom{n-1+r}{r}$$to find out the probability?   I'm pretty sure if I can do a), I could prob do b), but if someone could maybe lead me in the right direction for c) I would appreciate that too! Linux - Help, I'm running out of inodes!