Suppose that $S_1$ , $S_2$ are disjoint countable sets of T .Then their union is countable   ATTEMPT  Let  $S_1$ = ${x_1 ,x_2,...}$  $S_2$ = ${y_1,y_2,...}$  I am thinking of making pairs by doing $S_1 \times S_2$ ,but i donot know what to do furthure .Need hints  Thanks
This is a question from my proofs course review list that I have had trouble understanding.   I understand the concept of disjoint sets.  I'm not sure what they mean by countable. How would one prove a set is countable and furthermore, that the union of two countable sets is countable.  Please help me get started with the question by understanding this concept.
How can I make iconv replace the input file with the converted output? Triangular Numbers Modulo $k$ - Hit All Values? Why does C++ not let baseclasses implement a derived class' inherited interface? Opening the menu and selecting an item from the "Favorite Sites" menu allows us to see a feed for a specific site. In the top right corner of this page is a drop-down which allows us to select the "sort order" for the posts. The default is "Active" with other options like "Newest", "Hot", etc.  The problem is that when I select "Newest" or any other option. This preference is not remembered when I change between a different "Favorite Site".  So I am requesting that the selection is persisted when changing between site feeds. I have no preference whether this is implemented per site (to allow a different default for each site) or just one single preferred default that works on all sites (last option selected). I just want it to remember that I have selected "Newest" when I am flicking though the sites. Observe the following results for searching the text "dictionary" on Google:    Notice that there is a custom search box at the bottom of the listing for dictionary.reference.com. Is there any documentation floating around that explains the criterion by which Google decides to display such a search box? You are here, but where's here? I find the .youarehere a styling for the main navigation links causes the text to be a little difficult to read on the also-blue header background. Could we give it a little bit of background colour or something else that would provide more contrast?  For example:    which is the product of the following CSS*:  .nav ul li {     margin: -2px 10px 0 0;     padding: 5px; }  .nav ul li.youarehere {     background-color: rgba(0, 80, 112, 0.25);     border-radius: 5px; }  .nav ul li.youarehere a {     color: #A1E9FF; }   *For bonus points, also reduce the line height for the codeblocks! How to make a bootable ISO image from a bootable DVD? I want to make a .iso image of a bootable OS DVD, to use in VirutalBox.    Nautilus can make a .iso, but only of data files, it seems. The .iso it created was not bootable.  What is available to make a bootable .iso from a bootable DVD? grep strange behaviour with single letter words I am removing stop words from a text, roughly using this   I have the following  $ cat file file types extensions  $ cat stopwords i file types   grep -vwFf stopwords file  I am expecting the result: extensions  but I get the ( I think incorrect)  file extensions   It is as if the word file has been skipped in the stopwords file. Now here's the cool bit: if I modify the stopwords file, by changing the single word/letter i on the first line, to any other ascii letter apart from f, i, l, e, then the same grep command gives me a different and correct result of extensions.  What is going on here and how do I fix it?  I'm using grep (BSD grep) 2.5.1-FreeBSD on a Mac OSX GNU bash, version 4.4.12(1) Is there an English word meaning "the use of uncommon words"? Is there an English word meaning "the use of uncommon words" or similar? Finding $\lim_{n\to\infty}\binom{2n}{n}$ I found a limit problem which is $\lim_{n\to\infty}{2n \choose n}$ as we know that $\sum_{i=0}^{n}{n \choose i}^2={2n \choose n}$ now we could transform the given problem as $\lim_{n\to\infty} \sum_{i=0}^{n}{n \choose i}^2$ and as $\sum_{i=0}^{n}{n \choose i}^2$ is a polynomial in $n$, therefore, $\lim_{n\to\infty}\sum_{i=0}^{n}{n \choose i}^2=\infty$ Please tell me if my solution is correct or not,and if it is incorrect then please provide me a solution.  Thanks How can I disable a tab inside a TabControl? Is there a way to disable a tab in a ? In addition to whatever files I may choose to exclude, what files does the system exclude by rule or classification?    It might make sense not to backup Trash or Caches, but I'm looking for a way to tell what files are skipped "behind the scenes." Knowing if the system maintains an exclusion list or exclusion pattern would be very helpful in managing larger groups of macs. What is the best way to parse html in C#? I'm looking for a library/method to parse an html file with more html specific features than generic xml parsing libraries. How to reset Canon 70D WiFi? Why shouldn't we allow NULLs? Note: this post is different to , which just cites part from somewhere else, without any further analysis and thinking. Furthermore, none of the answers to that post answers my question - the situations where &quot;will (going to) be doing&quot; is more appropriate.   I will be participating in IELTS test next Tuesday afternoon.  In an online class, the teacher says the following at the beginning of the lesson.  I will be using NotePod (pointed out by red rectangle) to write relevant points we talk about today.   I guess &quot;will be using&quot; is more appropriate than &quot;will use&quot; there, right?  A  explains  This &quot;tense&quot; is used for reporting a scheduled activity with special focus on its timing— as if one is standing in the future and experiencing the flow of the time. The focus is on the temporary, ongoing, or repetitive nature of the activity.   I understand the grammar rules of those verb tenses and the difference between this group  I am going to drive home tomorrow.   I am going to be driving home tomorrow.  The former is just convey my activity in the near future while the latter probably makes an implication that &quot;don't disturb me as I am driving in that time&quot; However, in the following situations, I don't see the difference between 2 verb tenses.  Welcome back to this class. Today's lesson is for the IELTS students who are going to be taking the IELTS test. it's for everybody to be honest with you. this lesson is useful for anybody who needs to do some writing.   who are going to take the IELTS test ... $$\int_0^{2\pi} \frac{1}{2+\sin(x)} \ dx$$  $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$  $$z=\alpha(x)=e^{ix}$$   $$\sin(x)=\frac{z^2-1}{2zi}$$ $$2+\sin(x)=\frac{4zi+z^2-1}{2zi}$$ $$\frac{1}{2+\sin(x)}=\frac{2zi}{4zi+z^2-1}$$    $$\int_\alpha \frac{2zi}{4zi+z^2-1} \ \ \frac{1}{zi}$$    $$4zi+z^2-1=0$$ $$z_1=\frac{-4i+2\sqrt{3}i}{2}$$ $$z_2=\frac{-4i-2\sqrt{3}i}{2}$$    $$\rvert z_1 \rvert &lt;1$$ $$\rvert z_2 \rvert &gt;1$$   Using Residue theorem:   $$\int_\alpha \frac{2}{4zi+z^2-1}=2 \pi i \ \lim_{z\rightarrow-2i+\sqrt{3}i} \ \frac{2}{z+2i+\sqrt{3}i}=\frac{2 \pi}{\sqrt{3}}$$    Is it correct?  Thanks! Calculate $\lim_{n\rightarrow +\infty}\binom{2n} n$ If $A$ and $B$ are positive definite, then is $B^{-1} - A^{-1}$ positive semidefinite? How to "freeze" a cloth animation at X frame? How would one freeze a simulated body at frame X and use it as a static mesh?  Following a , I've been able to do a cloth animation to my model, now I'd like to have that made static, without the use of the timeline.   The thing is, I'm modelling something that's not going to be an animation, but a static object (something like a statue/figurine). So I don't need the animation, only the shape at a particular frame that I find convenient. I'd like eventually once this is done, to also modify that deformed grid.   I've tried to follow what was said , but couldn't figure out what chipmasque was saying. I'm quite new to blender (been using it on and off, just enough to forget everything).  I'm using blender 2.63, if it is of any relevance.