SEO : multiple pages with same content
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Showing that the class of all sets of a particular cardinality is not a set. How to show that the class of all sets of a particular cardinality ,say $h$ is not a set.   My argument:  I assume that I've shown the following lemma.  Lemma: If $X$ is an infinite set of cardinality $h$ , $a\ne x$ for every $x\in X$ then $\{a\}\cup X$ is of cardinality $h$ too i.e. adding new element to an infinite set doesn't change cardinality.  Let $h$ be an infinite cardinal number. Let $T_h=\{x:\text{x is a set of cardinality h}\}$. Suppose $T_h$ is a set.   Let $a$ be any set and consider any $t\in T_h$. Let $R=t\cup \{a\}$. Using the above lemma, $R\in T_h$.   So, $a\in R\subset \bigcup T_h$. As $T_h$ is a set then $\bigcup T_h$ is also a set.  As $a$ was arbitrary set then we have, $a\in\bigcup T_h$ for every set $a$. Hence, the set $\bigcup T_h$ contains all sets which meas that the collection of all sets is a set, contradiction.    If $h$ is non-zero finite cardinal number. Define $T_h$ the same way as above. Let $a$ be any set, Let $A_1=a, A_{n+1}=A_n\times a$. Consider $A=\{A_1,A_2,...,A_h\}$. So, $A\in T_k$. Hence, $a=A_1\in \bigcup T_h=K$. So K is a set containing all sets $a$ since $a$ was arbitary,contradiction.  My main question is, Is there any flaws in my argument? If yes, point it out please.   If the argument is valid, Is there any standard proof to show the same fact? Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $. How does one prove the following limit? $$   \lim_{n \to \infty}   \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}} = 3. $$ Compare nullable types in Linq to Sql I have a Category entity which has a Nullable ParentId field. When the method below is executing and the categoryId is null, the result seems null however there are categories which has null ParentId value.  What is the problem in here, what am I missing?  public IEnumerable&lt;ICategory&gt; GetSubCategories(long? categoryId) {     var subCategories = this.Repository.Categories.Where(c =&gt; c.ParentId == categoryId)         .ToList().Cast&lt;ICategory&gt;();      return subCategories; }   By the way, when I change the condition to (c.ParentId == null),  result seems normal. How did Amix Group get this helicarrier? In Deadpool,       A big fight scene takes place in a battered helicarrier at an Amix Group scrap yard.     What is the history of this particular helicarrier? And why in particular was it sent to the Amix Group scrap yard? I'm using LyX and a custom class from  which is based on the  class.  I'm at another school and my advisor wants me to include titles of the readers on the signature page as well as a date line for each signator.  I've looked through some documentation on the thesis class and poked around in the .cls file in use but didn't come across anything obvious to change.  Perhaps this section would help though?  \newcommand*{\th@degreeyear}{\number\the\year} \newcommand*{\th@signature}[1]{ \begin{center}     \normalsize     \vspace*{16mm}  % works well?     \vrule width 80mm height 0.2mm\\     #1 \end{center} } \newcommand*{\th@dateline}{ \vspace*{9mm} \begin{flushright}     Date~{\vrule width 35mm height 0.2mm} \end{flushright}   How do I include this? I upgraded from 16.04 to 18.04. When I press Ctrl+Alt+F3 I get the terminal which prompts for the username. When I type my username and press Enter, it doesn't wait for my password (its as if I pressed Enter without entering anything). Even if I enter something, the text shown in plaintext and then it keeps on saying my password is wrong. After few times, the screen is cleared and it prompts for my username again and the same loop goes on. How do I run an executable file with a double click? I'm trying to run an executable (testdisk) but double click doesn't work. The only way is to run:  sudo ./testdisk   from terminal. But I'd like to open executables with double click in future. How can I do this? Why is it considered so dangerous to travel in South Africa? As, the above link shows, other countries have similar murder rates but they are not in the radar of international travelers.  For example, as seen in the list, Puerto Rico has similar statistics. But, personally, I never heard of Puerto Rico to be a dangerous place to travel.  Why is it considered so dangerous to travel in South Africa? Is that due to the incompetent law enforcement agencies? CSS transition shorthand with multiple properties? Ready-to-Use Database models example Where can I find ready-to-use database models ?  I don't need a database with data in it, but only schemas (UML diagrams). Perhaps something like the , but much more complex and real world. In the following sentence, I want to know if I've understand the correct meaning or not:     This version holds that the wave function does not so much describe the state of the physical system as the state of our knowledge of the system.    Can we say that it means the wave function describes the state of our knowledge  of the system more than it describes the state of the physical system  ?  It is worth noting that in the Cambridge Dictionary, we read:  not so much sth as sth ​ If you say that something is not so much one thing as something else, you mean it is more the second thing: They're not so much lovers as friends. I don't feel angry so much as sad. Group of even order contains an element of order 2 I am working on the following problem from group theory:     If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd.   That is, for some integer $k$, there are $2k+1$ elements $a$ such that $a \in G,\;\; a*a = e$, where $e$ is the identity element of $G$. Autofill feature isn't iterating numbers properly when using the AVERAGE() formula in Microsoft Excel Is there way to move one vertex and make another move with it proportionally? I'll try to explain in this gif. I am doing this manually in 2 steps - rotate, scale. And I select all vertexes. Is there instrument rotate them in one step or can I move one and make other move too? How to avoid interference in wireless communication? I am working on wireless communication system. We are using around 10 pairs of transmitter and receivers. We are using atmega16 microcontroller for encoding and decoding by USART ports.  Now we are able to transmit the data and receive the same in the receiver end, but there is a major problem, when we are finding the 2 transmitter data coming at the same time. The receiver is unable to get it due to interference.  Suppose one transmitter sends "SENDA" at the meanwhile another transmitter sends "GETTS", at that time the receiver can not receive proper data. As all the transmitters and receivers are working in same frequency, so this interference is occurring. How can I resolve this issue? What's the maximum hand luggage volume you can take with you to match all (or important) airline regulations considering only international connections?  In case my question is not clear, I'm looking for the one biggest carry-on bag that is accepted by all (or important) airlines on international connections. Accepted dimensions tend to be somewhat different on different airlines, meaning that a specific bag might be allowed on one airline, but not on another.  So, my question is what is the biggest bag that meets all (or important) airline requirements? Let $(X, Y)$ be distributed as a multivariate normal with parameters  $$ \mu = \begin{bmatrix} \mu_X \\ \mu_Y \end{bmatrix} \qquad \Sigma = \begin{bmatrix} \sigma_X^2 &amp; \sigma_{XY} \\ \sigma_{XY} &amp; \sigma_Y^2 \end{bmatrix}. $$   I would like to calculate $E(X | y_1 &lt; Y &lt; y_2)$, where $y_1$ and $y_2$ are constants.  From , I have managed to work out that $$ E(X | Y &gt; y_1) = \mu_X + \frac{\sigma_{XY}}{\sigma_Y} \left[\frac{\phi\left(\frac{y_1 - \mu_y}{\sigma_Y}\right)}{1 - \Phi\left(\frac{y_1 - \mu_y}{\sigma_Y}\right)}\right]\\ E(X | Y &lt; y_2) = \mu_X - \frac{\sigma_{XY}}{\sigma_Y} \left[\frac{\phi\left(\frac{y_2 - \mu_y}{\sigma_Y}\right)}{\Phi\left(\frac{y_2 - \mu_y}{\sigma_Y}\right)}\right] ,$$ where $\phi(\cdot)$ and $\Phi(\cdot)$ are, respectively, the p.d.f. and the c.d.f. of the Normal distribution. I could not figure out how to calculate $E(X | y_1 &lt; Y &lt; y_2)$, though.  I have searched for an answer in similar posts from the Stack Exchange network, but I couldn't get a clue from them that would solve this issue. For the record, here are some of them: $(\Bbb Z\oplus \Bbb Z)[X] \not\cong \Bbb Z[X]\oplus \Bbb Z[X]\,$? Well i tried to prove it but failed. They are both commutative and have the same set of units. So i can't see a property of one ring which the other ring does not possess. Thank you for any help. access controller variable in javascript Is there any way to access controller variable set in a Method in Javascript. In the example above the value is set in constructor. But when I set it in a method it is showing as NULL in Javascript  In controller:  public class abc {     public string varA { get;set; }     public void fn(){         varA='hello';     } }   In VFPage:  function hello(){     var bool = '{!varA}';     alert(bool); }   The alert statement in the VF Page(i.e bool) is displayed as null. Can you please help how to access a variable set from a method instead of constructor? I'm looking for a time and memory efficient solution to calculate a  in C. I need to avoid dividing because I'm on a PIC 16 which has no dedicated division unit.  At the moment, I just store all values in a  and simply store and update the sum each time a new value arrives. This is really efficient, but unfortunately uses most of my available memory...