I wonder how to build a 1-1 correspondence between [0,1] and [0,1). My professor offers an example such that 1 in the first set corresponds to 1/2 in the second set, and 1/2 in the first set corresponds to 1/4 in the second. I don't quite understand it. Does it mean every element in the first set corresponds to its half value in the second set? wouldn't that make some elements left out in the second set? Does it still count as 1-1 correspondence? Does it connect to Schroder-Bernstein theorem?
How to define a bijection between $(0,1)$ and $(0,1]$?   Or any other open and closed intervals?   If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it): I make a function $f : (-1, 2)\rightarrow (-5, 4)$ of the form $f(x)=mx+b$ by \begin{align*} -5 = f(-1) &amp;= m(-1)+b \\ 4 = f(2) &amp;= m(2) + b \end{align*} Solving for $m$ and $b$ I find $m=3\text{ and }b=-2$ so then $f(x)=3x-2.$  Then I show that $f$ is a bijection by showing that it is injective and surjective.
How do I change my timezone to UTC/GMT? Currently when I execute the date command from bash I get the following displayed  Thu May 17 12:21:00 IST 2012  Below is my timezone config based on executing dpkg-reconfigure tzdata     Current default time zone: 'Europe/Dublin'   Local time is now: Thu May 17 12:32:52 IST 2012.   Universal Time is now: Thu May 17 11:32:52 UTC 2012.     Any idea how I change from IST to UTC/GMT? I would like to add texts with halo, in the composer (not in the map). Do you know a way to do it?  I'm using QGIS 2.0.1 How to check the maximum number of allowed connections to an Oracle database? In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance. I have to prove the  by tomorrow and I am really unconfident about what I did:  $$\sum_{n=1}^\infty a_n\text{ converges } \iff \sum_{n=1}^\infty 2^n a_{2^n}\text{ converges}$$  I did the following:  Let $(b_n)$ be a sequence as follow: $b_{2^k+m}:=a_{2^k}$ with $k\in\mathbb N_0$ and $0\leq m&lt;2^k$.  It's $a_{n+1}\leq a_n$ and so $0\leq a_{n+p}\leq a_n$ for all $n,p\in\mathbb N$.  So $\sum\limits_{n=1}^\infty b_n$ converges by the majorizing series $\sum\limits_{n=1}^\infty a_n$. And it's $\sum\limits_{n=0}^\infty b_n=\sum\limits_{n=0}^\infty\sum\limits_{m=0}^{2^n-1}a_{2^{n+1}}=\sum\limits_{n=1}^\infty 2^{n-1}a_{2^n}$ so $\Rightarrow$ is done.  For $\Leftarrow$ consider $c_{2^k+m}:=a_{2^k}$ with $k\in\mathbb N_0$ and $0\leq m&lt;2^k$.  It's $|a_n|\leq c_n$ and $\sum\limits_{n=0}^\infty c_n=\sum\limits_{n=0}^\infty\sum\limits_{m=0}^{2^n-1}a_{2^{n}}=\sum\limits_{n=1}^\infty 2^{n}a_{2^n}$ and so $\sum\limits_{n=1}^\infty a_n$ converges by the majorizing series $\sum\limits_{n=0}^\infty c_n$.  Is this in form and content correct? I have Sanguinare Vampiris. I don't really want to become a vampire. How can I cure myself? How to combine array items with string substitution? Clone a hard disk content to another hard disk I subjected my hard disk (C drive) to a disk scan and found that there was 12 KB of bad sectors. I now consider that disk to be unreliable, and am looking forward to transferring all my data to another file.  The problem is, I want to clone the complete drive (including the apps, OSes, files and everything, verbatim) to another drive. I know there would be some crazy software activation issues, like Windows Not Genuine problem and so on. But I wonder whether is there a tool powerful enough to do the cloning and at the same time maintain the  OS and apps? Why don't C++ compilers define operator== and operator!=? I am a big fan of letting the compiler do as much work for you as possible.  When writing a simple class the compiler can give you the following for 'free':   A default (empty) constructor A copy constructor A destructor An assignment operator (operator=)   But it cannot seem to give you any comparison operators - such as operator== or operator!=.  For example:  class foo { public:     std::string str_;     int n_; };  foo f1;        // Works foo f2(f1);    // Works foo f3; f3 = f2;       // Works  if (f3 == f2)  // Fails { }  if (f3 != f2)  // Fails { }   Is there a good reason for this?  Why would performing a member-by-member comparison be a problem?  Obviously if the class allocates memory then you'd want to be careful, but for a simple class surely the compiler could do this for you? How can I fix an out-of-focus blurred photo in Photoshop? How can I fix this picture?  Somebody else took this picture and well he wasn't able to do it very well, and it's blurred.    I have photoshop For $q, p$ odd primes such that $p \neq q$, there is not primitive root modulo $pq$. What and where are the stack and heap? Programming language books explain that value types are created on the stack, and reference types are created on the heap, without explaining what these two things are. I haven't read a clear explanation of this.  I understand what a stack is. But,    Where and what are they (physically in a real computer's memory)? To what extent are they controlled by the OS or language run-time? What is their scope? What determines the size of each of them? What makes one faster? How do I rename all folders and files to lowercase on Linux? set of almost complex structures on $\mathbb R^4$ as two disjoint spheres The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ gives two disjoint sets.  How do we show that $M_2=S^2 \sqcup S^2$? Moreover, I'm a bit confused by dimensions: how does $\dim_{\mathbb R}M_2=8$ relates with the latter decomposition?  This is exercise 1.2.1 from the book Complex geometry by Huybrechts. Preserving order with LINQ I use LINQ to Objects instructions on an ordered array.  Which operations shouldn't I do to be sure the order of the array is not changed? How do I summon a named mob? In Java Edition, I can name a mob by using NBT tags like this: /summon zombie ~ ~ ~ {CustomName:'{&quot;text&quot;:&quot;TheUndead&quot;}'}  But in Bedrock Edition, I've heard NBT is not accessible from commands. What can I do to summon a mob with a custom name tag using the /summon command? dd vs cat -- is dd still relevant these days? I recently realized we can use cat as much as dd, and it's   I know that dd was useful in dealing with tapes where block size actually mattered in correctness, not just performance. In these days, though, are there situations where dd can do something cat can't? (Here I would regard a less than 20% performance difference irrelevant.)  Concrete examples would be nice! How to modify a text file? I'm using Python, and would like to insert a string into a text file without deleting or copying the file. How can I do that? How can I horizontally center a &lt;div&gt; within another &lt;div&gt; using CSS? &lt;div id=&quot;outer&quot;&gt;   &lt;div id=&quot;inner&quot;&gt;Foo foo&lt;/div&gt; &lt;/div&gt; How is $J^{PC}$ value experimentally determined for new types of particles?