I need to the detriment the cardinality of a group and prove why.  The group:  A = {The group of all real numbers in the range between 0 and 1, where the digits can only be ones and zeroes, and after each zero comes a one}   At first I thought this is א, and was planning to prove it using the good old technique of picking one where there is zero, and zero where there is one.  But then I came up with the following function:  f:A ---&gt; Z f(x) = {  if the first digit after the point is zero, conversion of the binary digits after the point in x (excluding the first zero) into a decimal number   if the first digit after the point is one, conversion of the binary digits after the point in x into a decimal number, multiplied by minus one }   This function seems to be reversible, and since |Z| is א0, |A| should be א0 as well. on the other hand, if proving using the method I intended to use, the result is that |A| is א.  Which one is it, and why?
I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers.  First, take the numbers between zero and one, exclusive.  The following sequence of real numbers is suggested so that we have bijection.    0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.02, ... , 0.09, 0.10, 0.11, ... , 0.99, 0.001, 0.002, ... , 0.999, 0.0001, etc.  Obviously, this includes repeats, but this set is countable.  Therefore, the set of all numbers between zero and one is a subset of the above countable set, and is thus countable.  Then we simply extend this to all real numbers and all the whole numbers themselves, and since the real numbers, as demonstrated above, between any two whole numbers is countable, the real numbers are the union of countably many countable sets, and thus the real numbers are countable.    Please help me with this.  I understand the diagonalization argument by Cantor, but I am curious specifically about this proof which I thought of and its strengths and flaws.  Thanks.
What is a complete list of the usual assumptions for linear regression? How to show that $\log (\frac{2a}{1-a^2}+\frac{2b}{1-b^2}+\frac{2c}{1-c^2})= \log\frac{2a}{1-a^2}+ \log \frac{2b}{1-b^2}+ \log \frac{2c}{1-c^2}$ If $\log (a +b +c) =\log a+\log b+\log c$ then show that $$\log \left(\frac{2a}{1-a^2}+\frac{2b}{1-b^2}+\frac{2c}{1-c^2}\right)= \log\frac{2a}{1-a^2}+ \log \frac{2b}{1-b^2}+ \log \frac{2c}{1-c^2}$$  Trial: If put $a=\frac{2a}{1-a^2}$ and the similar we are done. Can we do this in this way? Otherwise how to do. Please help. What should main() return in C and C++? What is the correct (most efficient) way to define the main() function in C and C++ — int main() or void main() — and why? And how about the arguments? If int main() then return 1 or return 0?  There are numerous duplicates of this question, including:       — For C++, with a very good answer indeed.     Related: My chooser (file-picker dialog) goes under the dock so I can't see buttons. What can I do? This is a  about DNS/Hostnames resolution to IPs/Ports   Example 1  I'm running a web server on port 80 and another on port 87. I would like to use DNS so that www.example.com goes to port 87. How can I accomplish this using DNS only?  Example 2  I'm running a service on my server on a non-standard port. How can I get clients to connect to this non-standard port automatically? Can I use DNS? Is there some application specific support where DNS could indicate the IP and Port?   Example 3  Do some application protocols specifically support hostname awareness, and allow special actions to be taken based on this information? Are there other questions on Server Fault that cover some of these?  Commandeering: This question was originally asking about running IIS and Apache on the same server, but the same concepts can be applied to any server software receiving connections from clients. The Answers below describe the technical problems and solutions of using DNS and application protocol support to assign a port number for a client to connect. Making some tests in Windows XP for customizing a user interface in QGIS 1.8 (see ), I have unchecked, among other options, the Settings Menu. This means that, after restarting QGIS, there's no way of recovering the older settings because there is not already a Setting Menu. So, I cannot reload any ini file.  For your information, as a precaution, previously I have saved in a folder both the original ini file, with all the options, and the customized ini file with only some options (of course, not included here the necessary Settings Menu).  Before reinstalling QGIS (version 1,8), I would like to try other smarter options if there's any. Where and what does QGIS look at where this applications starts? Finding my house in minecraft I wandered too far to go look for sheep and now I'm on a completely different island.  How do I find my way back to my house? If the plural of ‘man’ is ‘men,’ shouldn’t the plural of ‘German’ be ‘Germen’? What makes these two words so different that 'man' is changed to 'men', but 'German' is changed to 'Germans'? Can I start and stop a tile floor job? I am tiling my bathroom floor and don't soon have a full day to dedicate to the job. Is it ok to lay down a few tiles each day and come back to the job when I have time? I would only lay mortar for the tiles I have time to lay each day.    My concern is that the mortar wouldn't properly "join" to the previously laid sections of tile and cause an inferior result, but I'm not really basing this concern on any facts. That is what I'm hoping to learn with this question. When do you use POST and when do you use GET? What WordPress API function lists active/inactive plugins? I'm getting "jumpy" loadings in rollapply PCA in R. Can I fix it? Two figures side by side I'm making a map where I want players to be able to choose there kits/classes. The problem I'm having with it though is giving players already enchanted Tools/Armour/Weapons.  I'm not sure if you can use the /give [player] [item] [amount] because you'll have to use the whole {} thing and I'm not sure how to make enchanted Tools/Armour/Weapons using that.  Can anyone help? Reason for the current trend to use «she» as the gender-neutral pronoun? There are some questions on gender-neutral pronouns both  and on .  User Christine Letts :     In academia, there is currently a   movement toward using the feminine   pronoun at all times.   I wonder why that is. I came across several examples on papers I read, but the only one I can remember at the moment is a book: Seth Godin's . While some might not be comfortable labeling it as part of academia, it suits my point perfectly. Every time he refers to a person, he uses the feminine pronoun.  User Senseful  the following, potentially identifying affirmative action as the precursor for this trend.     I remember reading somewhere that it   was recommended to use the opposite of   what most people stereotype the   profession as. So, for example, when   talking about a chiropractor, you   would use "her", and when talking   about a secretary, you would use   "his".   So, where do you think this trend comes from? Can I travel to Canada with valid visa on old/expired passport? I recently renewed my passport. I also have an expired passport which has my Canada visa (visitor's) valid for another three years. I have already used my Canadian visa which is multiple entry. I am planning to visit Canada this summer.  Since my valid Canadian visa is in my old/expired passport, do I need to renew/transfer my Canada visa or can I simply travel with valid visa in old passport? How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$? I have an entry-level Nikon DSLR, and I cant decide if I should upgrade my camera body or upgrade from a  AF-S DX Nikkor 18-200mm to whatever else might be recommended? I use Microsoft Outlook 2013 for work and Google Calendar for everything else. I'd like to find an easy way to sync my work calendar to my personal calendar. Ideally, this would be a software application that sits in the background and automates the syncing.  Requirements   Syncing should be one-way only: Outlook to Google. Must support repeating calendar items. Must allow me to specify which calendar in Google to sync to. I will set up a separate calendar for work items. No intermediate server. Export is direct.   Nice to Haves   Automated. Syncs in real time or regularly (say, every fifteen minutes or so). Do spice containers need to be hermetic?