I'm revising for an exam - two questions:  i) What's the cardinality of the set of strict total orders on $R$?  This is a subset of the power set of $R$ (because $|R|=|R.R|$), so I know its cardinality is less than this - but I can't find a way to show its cardinality is equal to this, i.e. an injection from the power set into the set of strict total orders.  ii) What is the cardinality of the set of countable subsets of F, where F is the set of functions from $R$ to $R$?  Cardinality of F is same as cardinality of power set of $R$ (one way given by reasoning above, and the other way by the fact that $|P(R)|=|2^R|=|R^R|$. The set of countable subsets of this I am not sure how to calculate though.
For a finite set $S$ there are $\vert S\vert!$ orderings of its elements.  What is the cardinality of all orderings of $\mathbb{N}$?  What would $$\vert \mathbb{N}\vert!$$ mean?  Is it $\prod_{n\in\mathbb{N}} \aleph_0=\aleph_0^{\aleph_0}=\aleph_1$?   says the "equivalence classes of well-orderings of the natural numbers" has cardinality $\aleph_1$.  But I can't put together in my mind the connection between well-orderings of $\mathbb{N}$ and all orderings of $\mathbb{N}$.  (Are there non well-ordered orderings of $\mathbb{N}$?)  This leads to wondering what the cardinality of all orderings of $\mathbb{R}$ is.  (Actually my thought process went in the other direction, from $\mathbb{R}$ to $\mathbb{N}$.)  I think knowing the answer to the above question will teach how to determine what $$\vert \mathbb{R}\vert !$$ means, if anything.  I will be happy if you assume the continuum hypothesis to be true for the purposes of this question, but any level of completeness is welcome.
A film or TV series about classic monsters getting together Setting constraints causes the location of the constrained objects/bodies not be accessible I set up 2 bodies manually &amp; assigned Copy Location constraint to one with respect to another body. Then I set up the keyframe for the guiding body &amp; then I started to view the change in the location of the constrained body. What I noticed in the 3D viewport was the constrained object moves, but when I look into its real position the values are stagnant. Why is that? I was unable to find my answer in previously asked questions by other people !! shade plane with a color function color(x,y) I have a third party php library which I would like to use in a module. How do I do that in Drupal 8? In drupal 7, I have done things like the following code, but libraries_get_path() doesn't exist in Drupal 8, and I'm not exactly sure how things change, not that we use namespaces (of which I am a beginner).  require(libraries_get_path('twitter-api-php') . '/TwitterAPIExchange.php');   And, where should I place my library? Should I create a new libraries directory and put it in the root, just like modules and themes are now in the root? Minkowski Metric Signature When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(c^{2}dx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}   $$ where $ x^0,x^1,x^2,x^3 $ come from $ x^{\mu} : \mu = 0,1,2,3 $, and $ c $ is the speed of light. The first resource I had access to--I have to do a bit of digging for the exact paper--of course addressed that this invariant takes the tensor form: $$ ds^2 = g_{\mu \nu}dx^{\mu}dx^{\nu} $$ as well. This two things I've seen in all of my texts and online resources regarding resources. The element in question that varies between authors, is the time coordinate, $ x^0 $. When it was first explained to me, it was using the standard Cartesian representation for the spatial portion of the coordinates, and the time coordinate was labeled as $ x^0 = ict $. Squaring gives $ (x^0)^2 = -c^2t^2 $, and applying differential calculus, we get $ (dx^0)^2 = -c^2dt^2 $. Sensible and expected to have the tensor formula spit out the Minkowski Metric. The author then later explicitly states the coordinates are $ x^0 = ict, x^1=x,x^2=y,$ and $x^3 = z$.  My questions is then, why is it most authors on the subject omit the imaginary unit on the time coordinate? For example, .   The only reason I can fathom omission is if the author is using metric signature $ [+,-,-,-] $, where I started off learning the theory with signature $ [-,+,+,+] $ which may be the reason seeing the time coord with no imaginary unit seems dissonant to me. All help appreciated!  Edit: After reading the other answers, my questions is now one of why and how (mathematically) do we obtain the Minkowski Metric Signature. More specifically the one element with a different sign. Why use a z test rather than a t test with proportional data? How do I calculate standard errors for sums of OLS coefficients? Ill implement the some UI but its not working in the visualpage , can u know how implement jquery ui in viusalpage ,  Kindly tell the procedure ,     Im using this url its not working ,,,,, I have just migrated from Windows environment. I have installed Python 3.2 in a separate directory. How can I get the python installation path in Ubuntu shell?  Is there any way I can let the shell know/choose at runtime which python version is to be used for further code execution?  Are there any environment variables and search path kind of things in Ubuntu Linux as well? How do I become a Werewolf? I've heard that one of (maybe the only?) secret races in this game is a werewolf.  How do I become a werewolf?  And by extension, how do I cure lycanthropy, that is, if there even is a cure? In IE 9 you can type in whatever number you want to get the perfect zoom but in Chrome all I see is the + and - in the drop menu.  How do I choose a custom zoom and set it as the default? Is there more to an interface than having the correct methods So lets say I have this interface:  public interface IBox {    public void setSize(int size);    public int getSize();    public int getArea();   //...and so on }   And I have a class that implements it:  public class Rectangle implements IBox {    private int size;    //Methods here }   If I wanted to use the interface IBox, i can't actually create an instance of it, in the way:  public static void main(String args[]) {     Ibox myBox=new Ibox(); }   right? So I'd actually have to do this:  public static void main(String args[]) {     Rectangle myBox=new Rectangle(); }   If that's true, then the only purpose of interfaces is to make sure that the class which implements an interface has got the correct methods in it as described by an interface? Or is there any other use of interfaces? What exactly do quotation marks around the table name do? I thought that the quotation mark (") was simply a type of grouping marker but I'm debugging some NHibernate code and notice that while  SELECT * FROM site WHERE site_id = 3;   Works fine  SELECT * FROM "site" WHERE site_id = 3;   fails with a table or view does not exist error.  What gives? Im new to Illustrator and have found the clipping mask is not so easy compared to Phtoshop. Could any one tell me how i can make this line texture a clipping mask over the circle? For $n \geq 2$, show that $n \nmid 2^{n}-1$ Here is a problem which i have not been able to do for quite sometime.   For $n \geq 2$, show that $n \nmid 2^{n}-1$.   I have thought of proving this in two ways: One by using induction which didn't actually work. Next by Fermat's little theorem we have $2^{p-1} \equiv 1 \ (\text{mod} \ p)$, which actually says that for $p \mid 2^{p}-2$. But i couldn't proceed more than this. Any ideas by which i can actually solve the problem? How do I prove or disprove $X \times (Y \cup Z)= (X \times Y) \cup(X\times Z)$  for all sets $X$, $Y$, and $Z$? I'm lost on the steps here. Is there a difference between DR/evil and DR/unholy? Best Fake Proofs? (A M.SE April Fools Day collection) Is there any difference between "word-for-word translation" and "word-by-word translation" and is the latter actually valid? First off, some data: According to COCA word-for-word has 60 usages, 3 of them are &quot;word-for-word translation&quot;. Word-by-word has 26 usages, none of them are &quot;word-by-word translation&quot; (but some with &quot;transcription&quot;). The definition of word-for-word:  : In exactly the same or, when translated, exactly equivalent words : being in or following the exact words, verbatim : one word at a time, without regard for the sense of the whole  Only the last dictionary contains a definition for word-by-word, too:  : one word at a time  The definitions given by The Free Dictionary are, obviously, identical to each other. Google hits: Word-for-word ~21m Word-for-word translation ~318k Word-by-word ~3.8m Word-by-word translation ~95k According to usages and dictionaries word-by-word is, at least, less popular. And assuming that there may be a lot of usages from non-natives among the Google hits, this could be an indicator for word-by-word being even utterly wrong. In another forum I found the following :  When I translate something &quot;literally,&quot; (wörtlich) it still follows the main rules of the language I'm translating into. What you mean is &quot;word-by-word&quot; (wortwörtlich) to me.  I assume that this was written by a German but I don't know it. However, if this would be true a &quot;word-by-word translation&quot; would be a translation where I keep, for instance, the order of the words, disregarding if it makes sense in the target language. Some examples:  Original: word-by-word Word-by-word translation: Wort bei Wort (That's a terrible translation!) Original: It is critical to know... Word-by-word translation: Es ist kritisch zu wissen... (That's a terrible translation!) Original: Ich glaub, ich spinne. Word-by-word translation: I think I spider. (I guess only Germans understand this.)  A &quot;word-for-word translation&quot;, however, would be an attempt to keep the word-choice as close as possible but following the rules of the target language (e.g. order of words) and also considering if the statement still makes sense in the other language. Here are better translations for the examples above:  Wort für Wort Es ist wichtig zu wissen... I think, I'm going nuts. (Actually, this is not a word-for-word translation but rather a .)  So, my questions again:  As neither Oxford nor Merriam-Webster have any entries for word-by-word in their dictionaries: is word-by-word actually valid? If yes, is there any difference between &quot;word-by-word translation&quot; and &quot;word-for-word translation&quot;? If yes again, what is it specifically? When playing as a spellcaster, is it valid the use of the Ready action to prepare Dispel Magic to counter a spell cast by an opponent? In the same vein, when declaring said action, do I have to specify which spell slot I will use when Readying the Dispel Magic spell? If that is the case, does this tactic nullify an opponent's spell? I get the impression that this is basically turning Dispel Magic into a spell with a casting time of &quot;1 reaction&quot;. Is my reading of the rules accurate?