I need to prove that the set of all functions $\mathcal{F}:\mathbb{N}\rightarrow \left \{ 0,1 \right \}$ is uncountable.  I'm not too sure at all how to do this. My initial idea was to try and show that $\mathcal{F} \rightarrow \mathbb{N}$ was not a bijection but I'm not clear at all and need some big help.  Thanks!!!
How can I prove that the set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?  Edit:  This answer came to mind.  Is it correct?  This answer just came to mind.  By contradiction suppose the set is $\{f_n\}_{n \in \mathbb{N}}$.  Define the function $f: \mathbb{N} \to \{0,1\}$ by $f(n) \ne f_n(n)$.  Then $f \notin\{f_n\}_{n \in \mathbb{N}}$.
I have a website that sells music artists posters. Each artists has a page containing a small bio, and a list of posters we are selling representing the artist.  Currently our HTML looks like that  &lt;h1&gt;{Artist name}&lt;/h1&gt; bio &lt;h2&gt;Posters&lt;/h2&gt; &lt;ul&gt;   &lt;li class="poster"&gt;...&lt;/li&gt;   ... &lt;/ul&gt;   We are redesigning this page, and we have a new template that we really like. Problem is that in this new version the &lt;h2&gt;Posters&lt;/h2&gt; does not appear anywhere on the page.  As you might expect, a significant part of our traffic comes from people typing something like : "{Artist name} poster" on Google.  I fear that, by removing the word "poster" from the page, we will reduce our chances of getting these people to come to the site, since Google might down rank us.  I see several options here :   Give Google a slightly altered page that contains the H2 tags. But I think this falls into the "Black Hat SEO" category. And I don't want that. Put the heading. Hide it with CSS. Everyone wins, including accessibility tools which can determine the structure of the page using the "invisible" heading. But I do not know if this is considered a good practice. Forget about the heading, and find other ways to tell Google that we really are about posters. (via keywords, but do they really have the same "weight" as an H2?) Adapt the design to our SEO need, and find a way to put that heading nicely.   Are there webmasters out there that have been facing the same issue, how did you resolve it? Is there any difference between "offense" and "offence"? "Offense" vs. "offence", which is more correct? If both are correct, are there any differences in shades of meaning and/or usage? Equivalent of typedef in C# Is there a typedef equivalent in C#, or someway to get some sort of similar behaviour? I've done some googling, but everywhere I look seems to be negative. Currently I have a situation similar to the following:  class GenericClass&lt;T&gt;  {     public event EventHandler&lt;EventData&gt; MyEvent;     public class EventData : EventArgs { /* snip */ }     // ... snip }   Now, it doesn't take a rocket scientist to figure out that this can very quickly lead to a lot of typing (apologies for the horrible pun) when trying to implement a handler for that event. It'd end up being something like this:  GenericClass&lt;int&gt; gcInt = new GenericClass&lt;int&gt;; gcInt.MyEvent += new EventHandler&lt;GenericClass&lt;int&gt;.EventData&gt;(gcInt_MyEvent); // ...  private void gcInt_MyEvent(object sender, GenericClass&lt;int&gt;.EventData e) {     throw new NotImplementedException(); }   Except, in my case, I was already using a complex type, not just an int. It'd be nice if it were possible to simplify this a little...  Edit: ie. perhaps typedefing the EventHandler instead of needing to redefine it to get similar behaviour. I have two questions. First, which command lists files and directories, but lists directories first?  Second question: I want to copy a list of files into a single directory, but make the target directory the first filename in the command. So I paid for a minecraft account and everything but when I try to log in it says that I don't have a username. Is it an email or an email that you've made just for this account? Bypass GeneratedValue in Hibernate (merge data not in db?) Make mouse left-handed and touchpad right-handed I see that I can configure mouse and touchpad all in one panel (Ubuntu 13.10, defaults, Gnome). Which is to some extend. Namely, I want to have the touchpad differently configured then the mouse and ...  When I have an (USB) mouse attached I use it normally left-handed. Therefore I switch the buttons to left-handed. But when I use the touchpad I feel little bit lost with that kind of configuration. Therefore I wanna have it right-handed.  Is there a way to achieve the following: when I plugin a mouse (respectively when it is detected) automatically switch to left-handed and when I unplug it, switch automatically to right-handed? What is the effect where some objects are a single bright color but the rest is black and white? There is an option to filter only one color (R,G,B,Y) in my camera. What I want to know is can this effect be achieved in photoshop? And what is this called? I tried googling monochrome and came up with nothing. My camera calls it part color. Equivalent of "both" when referring to three or more items? What would be the correct word to use when referring to three or more items, in the same manner as the word both?  For example, using two words, with the word both:     "There are several recommendations I   have to further improve the sites —   both to improve their profit, and   decrease their cost."   Using three words, with a blank space in place of the correct word:     "There are several recommendations I   have to further improve the sites —   _ to improve their profit, decrease their cost and improve their   usability."   So, what would be the correct word to use in place of the __? Recently, my answers are showing at the bottom of the page, even though I got good scores on those answers.       How do I change this back? The glorified global variable - becomes a gloried global class. Some say breaking object-oriented design.  Give me scenarios, other than the good old logger where it makes sense to use the singleton. How to change the snapping increment in different units? I would like to know how to change the increments used for snapping transforms.  This would be useful in a number of situations but my problem now is that when I change the units it does not scale the snapping increments to match.  So when holding down ctrl the cube snaps from here:    To here:    I would like it to snap to the 1 foot grid, or possibly any other amount I find useful.  It would also be nice to be able to change the rotation snap as well. Let's say it again, we all make mistakes, and I have just made one.  A brief history: I was doing some stuff on a VPS (Debian) I'm renting, when I noticed some strange behaviour. Using the netstat command I saw an non-authorized connection through SSH. I didn't know what to do, so I decided to close his connection using iptables:  iptables -A INPUT -p tcp --dport ssh -s IP -j DROP   But I am tired, and I wrote  iptables -A INPUT -p tcp --dport ssh -j DROP   and I kicked myself (and everyone else) out...  How do I fix this? How to sort a list of objects based on an attribute of the objects? In a file with lots of lines I want to delete lines that starts with HERE IT IS.  How can I do this using only command-line tools? Showing just color ramp in legend without values Consequences of a master thesis with errors I finished my master thesis 6 years ago, since then I gained a lot of experience on my field and also in academia. I can say that in that time I was a "child" , very inexperienced. I think that also my advisor did not help a lot, and I tried to do  a lot of things by my own, without someone to really review my things. Despite that, I still received a good mention by the master committee on final results.   Some days ago, I just decided to read part of master thesis, and I just realized a big nightmare coming. I identified a lot of typos, some references errors, and some improvements that I should do, but I did not. The worst thing is that I wrote a paper to a conference with my master thesis results (and some errors are there). The conference is not a high impact conference, but still the paper has my name and is available for everyone...   As I am finishing my PhD studies for now, I am concerning a lot that situation. My dream and goal is to pursue a Professor career. So, my questions are: What are (or could be) the consequences of a master thesis and paper with errors (some minor and some major) in academia? Revoke degree after years by errors in master is possible ( I am concerning a lot this situation that I thought on it)? How can I minimize it (I thought that I can put a fixed version of my thesis and paper in the future, for instance on my personal website)? How those errors can impact me in a Professor recruitment process?  Sorry for the long text, I am blaming myself all the time, and really afraid about the possibilities since I know that Academia career is really competitive. How can I prove that the Cartesian product of two countable sets is also countable? Latex create a second keywordstyle using listing doesn't work Any Video formats that render with Alpha Channel? Does anyone know if there are any video formats (besides the HUGE uncompressed avis) that you can render in that maintain the alpha channel? (i.e. for after FX etc.)   I used to use quicktime, but that's no longer an option.  I know you can render in png. or other image formats, but video is significantly smaller, and I often work on machines with less space &amp; resources.