So I did this but I'm not sure if it is an actual proof because it is based on a hypothesis, and also I'm not sure if I understand the Continuum Hypothesis correctly.   Because $\boldsymbol{I} \subseteq \boldsymbol{R}$, l can have either the same or less cardinality than $\boldsymbol{R}$. By the Continuum Hypothesis, l can not have a lower cardinality than $\boldsymbol{R}$, so it must have the same cardinality.   Is this correct, and/or is there a more rigorous method?  Also, can I use the notation $\operatorname{card} A &lt; \operatorname{card} B$?
I need to prove that $\mathbb{R}\backslash\mathbb{Q} \sim \mathbb{R} $  Using Cantor-Bernstein, need to show an injection from $\mathbb{R}\backslash\mathbb{Q}$ to $\mathbb{R}$ and from $\mathbb{R}$ to $\mathbb{R}\backslash\mathbb{Q}$.    $\mathbb{R}\backslash\mathbb{Q}$ is a subset of $\mathbb{R}$ so only need to show injection from $\mathbb{R}$ to $\mathbb{R}\backslash\mathbb{Q}$ to complete the proof.    Possible injection: $f:\mathbb{R}\to \mathbb{R}\backslash\mathbb{Q}$ defined as $f(x) = \pi x$ if $x$ is not a multiple of $\pi$; otherwise $f(x) = \sqrt{2} x$.    Not sure if $f$ actually is an injection...
Given an infinite set of cardinality $X$, is there a chain in $\mathcal P(X)$ of the same cardinalty of $\mathcal P(X)$? $P(\mathbb N)$ = power set of $\mathbb N$. $A \subset P(\mathbb N)$ is a chain if $a,b \in A \implies$ either $a \subseteq b$ or $ b \subseteq a$ That is, we have something like this: $$\cdots a \subseteq b \subseteq c \subseteq\cdots$$ where $a,b,c \in A$ are distinct. We can show easy enough that there is an uncountable chain - this is done by noting $\mathbb N\sim\mathbb Q$ then using Dedekind cuts in $\mathbb Q$ to define $\mathbb R$ we see that a family of (nearly arbitrary) cuts satisfy the condition. For instance the family $L_r=\{q \in \mathbb Q : q &lt; r \}$ for $r &gt; 0$ gives us the sets we need and obviously we can pick others. I tried doing this for $ \mathbb R$ and don't seem to be getting anywhere.  To be more specific, does there exist a chain in $P(\mathbb R)$ with cardinality $2^ \mathbb R$? Given a set of cardinality $X$ (necessarily non-finite), is there a chain in $P(X)$ of the same cardinalty of $P(X)$? Android Seekbar with two thumbs Error: You are missing the following 32-bit libraries, and Steam may not run: libXtst.so.6 [...] I'm aware that many people are having problems on launching Steam (downloaded from Steam not the repos) on Ubuntu 16.04 64-bit with this error message:  svetlana@svetlana-desktop:~$ steam Running Steam on ubuntu 16.04 64-bit STEAM_RUNTIME is enabled automatically Error: You are missing the following 32-bit libraries, and Steam may not run: libXtst.so.6 libXrandr.so.2 libgobject-2.0.so.0 libglib-2.0.so.0 libgtk-x11-2.0.so.0 libpulse.so.0 libgdk_pixbuf-2.0.so.0 Installing breakpad exception handler for appid(steam)/version(1463778979) Installing breakpad exception handler for appid(steam)/version(1463778979) Installing breakpad exception handler for appid(steam)/version(1463778979) Installing breakpad exception handler for appid(steam)/version(1463778979) [2016-05-22 16:33:15] Startup - updater built May 20 2016 18:57:58 [2016-05-22 16:33:15] Opted in to client beta 'publicbeta' via beta file You are in the 'publicbeta' client beta. Looks like steam didn't shutdown cleanly, scheduling immediate update check [2016-05-22 16:33:15] Checking for update on startup [2016-05-22 16:33:15] Checking for available updates... [2016-05-22 16:33:15] Download skipped by HTTP 304 Not Modified [2016-05-22 16:33:15] Nothing to do [2016-05-22 16:33:15] Verifying installation... [2016-05-22 16:33:15] Performing checksum verification of executable files [2016-05-22 16:33:15] Verification complete [2016-05-22 16:33:16] Shutdown   My question has anyone found a solution yet? "Whether" Vs. "If" Why is DML not allowed in Constructor? Irreducibility of an infinite sequence of polynomials When I apply a glossy shader to the material of my text, the text becomes just black. Why?    I have lamp and everything, but the problem persists. The text is red before I add the glossy shader.  I'm using Cycles render.   Thanks! I see a lot of questions comparing full frame (e.g EF for Canon, FX for Nikon) and crop lenses (e.g. EF-S for Canon, DX for Nikon) but I can't find the answer to what would be the differences between the photos taken with say a 50mm f/1.4 EF and EF-S lenses both used with   same APS-C sensor sized camera same ISO same f/1.4 f-stop same focal length same shutter speed   What would be the resulting differences in the photos? Which one would   be brighter than the other due the EF-S lens concentrating the light more? no that's wrong. Both put the same amount of light from the same FoV on the sensor. have greater depth-of-field? neither less problems towards the edges? generally the full frame more lens flare? generally the full frame lens unless there is, or you provide more shading anything else? not really, unless you start down the road of which one is typically of better build quality.   Edit: Added the answers from what seems to be the general consensus from the multiple answers and comments.  Results: My desire to buy a fast medium-high quality lens compatible with both full frame and crop camera's doesn't have to be hampered by the APS-C sensor size of my current camera.  This means I'm willing to part with more money to get a higher performance/quality lens as the performance/quality will be useful now and when I upgrade my camera.  Though reading reviews of some full frame lenses with short focal lengths, look out, as some are really only made to perform well on crop cameras even though they fit on a full frame camera and will give you a full frame picture, the edge quality is very bad. QGIS locked into fullscreen mode on Ubuntu 12.04 No crash in this case, but I made the mistake of toggling to fullscreen mode on my netbook running Ubuntu 12.04. It's now impossible to escape , and it seems to have disabled the popout unity launcher as well. Steps I've already taken:   Alt-Tab to move to a different program, then back - QGIS remains in fullscreen Alt-V, Ctrl-F and F11 get no response I've unistalled QGIS, deleted the /.qgis/ profile data folder, and reinstalled. Still launches fullscreen   To add to the weirdness, it now launches - still in fullscreen - into a different workspace than the active one.  I realize this has the sound of a Unity problem, but it's just this program that has a malfunctioning UI.  Thanks to all the crew; this in no way changes my opinion of QGIS as an essential tool. I've a set of regions within a city. I'd like to split each region into five subregions and then get the centroid of each one of them.  I know how to make a vector grid, but the problem I have is that the generated points are not taking into account the regions, they're only related to the general map.  I have this:    And I want to split each region in five equal size areas. Something like this:    I'd like to build a visualization similar to this one: . In order to accomplish that, I need to split each region, because the original ones are pretty big. Convert XML/HTML Entities into Unicode String in Python When to use icons vs. icons with text vs. just text links? bash regex with quotes? Setting-up SSH-tunnel for VPN? I have a laptop running Ubuntu.  I also rent a VPS running Ubuntu.  I've bought a paid VPN-service, which I use on my laptop.  I was wondering how I can set-up a SSH-tunnel or something similar, from my laptop to the VPS I'm renting, so I then can run VPN (from my laptop) "through" it (to the VPS), to the VPN-server.  The reason I wish to do so, is that a place I often work blocks my VPN-service, but not my SSHing to my VPS...  While I have been using Linux for a while, I'm a novice when it comes to setting-up virtual network-devices, advanced routing, and so on... so please spoonfeed me (using small words and colorful pictures ;-)  I have been googling and looking at various answers on the StackExchange-network, but not seems to be exactly what I'm after.  SSH-tunneling seemed promising, but since I don't have/know an exact domain and port for my VPN, I couldn't figure out how to use it...  I guess I may need to set-up some virtual network-device on my laptop (which ends on the VPS), and connect VPN through it - but how to do this, sadly goes over my head. Is kinetic energy a relative quantity? Will it make inconsistent equations when applying it to the conservation of energy equations? If the velocity is a relative quantity, will it make inconsistent equations when applying it to the conservation of energy equations?  For example:  In the train moving at $V$ relative to ground, there is an object moving at $v$ relative to the frame in the same direction the frame moves. Observer on the ground calculates the object kinetic energy as $\frac{1}{2}m(v+V)^2$. However, another observer on the frame calculates the energy as $\frac{1}{2}mv^2$. When each of these equations is plugged into the conservation of energy, they will result in 2 distinct results (I think). Curve Bevel function, scale? Is there a way to scale up the bevel object on one end of a curve, a little bit over each point?  I notice the vertex "weight" slider will tweak how much influence each vertex has on the others, is there something like this that can scale the bevel object up a little bit?  Preferably with the option to constrain X,Y,Z independently Is it possible to install an iPhone app in MacBook Pro? What is $\int_{-\pi}^\pi \cos(nx)\cos(mx)\,dx$? How much money do you need to have on hand when entering USA? My friend, a Chilean citizen, is entering the US via waiver program to spend 3 weeks in USA in a couple of weeks. She already got her ESTA approval, the plane ticket, all hotels are already booked (but paid by me, not her).  How much money would she need to bring with her when entering US? Do they still check this? She would of course have credit cards with her. How can I find the cheapest prepaid SIM card for a given country? Next month I am traveling to country X, where I would need Y minutes of calls, Z text messages and most importantly Q megabytes of data. How can I find the cheapest/most reliable SIM card for my needs?