A class is a collection of sets in which the sets are members of the class. So, may I ask what are the differences between a class and a set (or more precisely a set of sets) by their nature? What motivates mathematicians to give it a separate name?
I know what a set is. I have no idea what a class is.   As best as I can make out, every set is also a class, but a class can be "larger" than any set (a so-called "proper class").  This obviously makes no sense whatsoever, since sets are of unlimited size. It's not like you get up to 25 elements and go "oh, hey, sets are limited to a maximum of 20 elements, so we'll have to call this 25-element thing something else".  Does anybody know what the actual difference between a set and a class is?  Several people have pointed out that you can write paradoxes which involve sets. I get that. But where do classes come into the picture?
If each suit is represented then we will have two cards from one suit and one card each from the remaining suits. So I am counting the ways this can happen like so -  ${52 \choose 1}$ ways of selecting the first card.  ${12 \choose 1}$ ways of selecting a card from the same suit as the first.  ${13 \choose 1}$ ways of selecting a card for each of the 3 remaining suits.  So I have -  $${52 \choose 1}{12 \choose 1}{13 \choose 1}{13 \choose 1}{13 \choose 1}$$  Then I have the sample space as $52 P 5$ as we are counting with regard to order on the numerator so we need to do the same for the denominator.  So I have $$\frac{{52 \choose 1}{12 \choose 1}{13 \choose 1}{13 \choose 1}{13 \choose 1}}{52P5}$$  Does that look correct? Notation for intervals I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed . I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall someone calling the notation with $[a,b]$ and $]a,b[$ as French notation.   What are the origins of the two notations?  Is the name French notation correct? Are they used frequently in France? Or were they perhaps prevalent in French mathematical community at some point? (In this  Bourbaki is mentioned in connection with the notation $]a,b[$.)      Since several answerers have already mentioned that they have never seen $\langle a,b \rangle$ to be used for closed intervals, I have tried to look for some occurrences for this. The best I can come up with is the article on , where these notations are called Czech notation and French notation. Using $(a,b)$ and $[a,b]$ is called English notation in that article. (I am from Central Europe, too, so it is perhaps not that surprising that I have seen this notation in lectures.) I also tried to google for  or . Surprisingly, this lead me to a  where this notation is used for open interval. Why is Excel truncating my 16-digit numbers? If I type in a 16 digit number (format: number, no decimals) it changes the number on me. Example: 1234567812345678 changes the view to 1234567812345670.    If I type it in as a general format it changes the numbers above so it displays 1.23457E+15 but if you click on the cell, the display shows the last digit as a 0 instead of an 8 once again.   I opened the file on a different computer and same issue now with it.  I have changed the auto correction and auto formatting all to no avail.  Help! Childrens fantasy, blind monk leading boy who can control the elements Prove that the product of four consecutive positive integers plus one is a perfect square I need to prove the following, but I am not able to do it. This is not homework, nor something related to research, but rather something that came up in preparation for an exam.     If $n = 1 + m$, where $m$ is the product of four consecutive positive   integers, prove that $n$ is a perfect square.   Now since $m = p(p+1)(p+2)(p+3)$;     $p = 0, n = 1$ - Perfect Square      $p = 1, n = 25$ - Perfect Square      $p = 2, n = 121$ - Perfect Square   Is there any way to prove the above without induction? My approach was to expand $m = p(p+1)(p+2)(p+3)$ into a 4th degree equation, and then try proving that $n = m + 1$ is a perfect square, but I wasn't able to do it. Any idea if it is possible? Evaluate the limit $\lim\limits_{n \to \infty} \frac{1}{1+n^2} +\frac{2}{2+n^2}+ \ldots +\frac{n}{n+n^2}$ Evaluate the limit   $$\lim_{n \to \infty} \dfrac{1}{1+n^2} +\dfrac{2}{2+n^2}+ \ldots+\dfrac{n}{n+n^2}$$   My approach :   If I divide numerator and denominator by $n^2$ I get :   $$\lim_{ n \to \infty} \dfrac{\frac{1}{n^2}}{\frac{1}{n^2} +1} +\dfrac{\frac{2}{n^2}}{\frac{2}{n^2} +1} + \ldots+ \dfrac{\frac{1}{n}}{\frac{1}{n} + 1}=0$$   but the answer is $\dfrac{1}{2}$ please suggest how to solve this.. thanks. How to uncompress separated tgz files? I am receiving this error message when I'm trying to uncompress a .tgz archive:  $ tar -zxvf OVM_EL5U3_X86_EBIZ12.1.1_DB_VIS_PVM.tgz.1 gzip: stdin: not in gzip format tar: Child returned status 1 tar: Error is not recoverable: exiting now   As you can see, the file name ends with .tgz.1.  When the file name is .tgz.0 it will be uncompressed with no error by the tar -zxvf command. Square root of Positive Definite Matrix Iterating each character in a string using Python Matrix Group induction proof and order of elements question garbled xml output When should I use "shall" versus "will"? Which is the correct use of these two words, and in which context should one be used rather than the other? I'm just wondering where these values are being set and what they default to? Mine is currently 18446744073692774399. I didn't set it anywhere that I can see.  $ cat /proc/sys/kernel/shmmax  18446744073692774399  $ sysctl kernel.shmmax kernel.shmmax = 18446744073692774399 Execute function when post is published I try to use this function i have created when a post is published for the first time.  function a_new_post($post){   $post_id = $post-&gt;ID;    if ( !get_post_meta( $post_id, 'firstpublish', $single = true ) ) {       // ...run code once       update_post_meta( $post_id, 'firstpublish', true );   } } add_action( 'draft_to_published', 'a_new_post' );   I can't see anything wrong with it, but when i try to create some sample posts i check the database and the field "firstpublished" has not been created.  Does anyone see anything wrong? Given $a,b,c,d \in \mathbb{Z}$, there is $x,y$ such that  $$(a^2 + b^2)(c^2 + d^2) = x^2 + y^2$$  One can show this by considering the complex number $a + bi$ and $c+ di$, using complex properties to deduce that $x = ac - bd, y = ad + bc$ is a solution.   However, given that either $a^2, b^2$ are distinct or $c^2,d^2$ are distinct, then how can one can find nonzero solutions $x,y$?  Sorry, I am assuming $a,b,c,d$ are nonzero. Does the hom-tensor adjunction hold for $O_X$ modules also? With sheaf hom and sheaf tensor product, the statement would consist of a natural transformation $Hom_O (M \otimes_O N, K)\cong_{nat} Hom_O(M, Hom_O(N, K))$, $O = O_X$ is the structure sheaf and $M,N,K$ are sheaves of $O_X$ modules.  If true I think I can check this by hand, by describing sheaves in terms of compatible germs, defining this adjunction on the germs and checking that they glue together. It is easier in the quasi-coherent sheaves on a scheme case because then one can just work in an affine cover and the distinguished base.  I am little bit concerned that one might need a finite presentation hypothesis somewhere, in order for $Hom$ to localize well. ($Hom_{R[S^{-1}]}(M[S^{-1}], N[S^{-1}]) \cong Hom_R(M,N)[S^{-1}]$ needs $M$ finitely presented.) Difference between "I'm home" vs "I'm at home" Prove that if $P$ is a $p$-Sylow subgroup of $G$ and $N \triangleleft\&gt; G$ then:    $ PN/N $ is a $p$-Sylow subgroup of $G/N$ $P \cap N $ is a $p$-Sylow subgroup of $N$ If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$. How to get the full executable name of a running process in Linux