So doing an exam question for set theory this was the question:  Let X be a set. Prove there is no injection $f : P(X) → X.$  Would X being an empty set be a counter example because in this case the powerset is a set of a single element $\{\}$ hence it is mapped to a null element.   However would this be the case that $f$ would not be a function, since there isn't $(a,b)\in f$ hence $f$ isn't defined for the whole of the domain hence $f$ is not a function.  So yes I understand that this isn't a proof for this problem but while thinking about the problem I was trying some examples and non-examples and couldn't work out why this failed until I actually put it into words.  Sorry for a convoluted question, I don't really have the correct notation to explain myself.
Cantor's diagonal argument can be used to show that a set $S$ is always smaller than its power set $\wp(S)$.  The proof works by showing that no function $f : S \rightarrow \wp(S)$ can be surjective by constructing the explicit set $D = \{ x \in S | x \notin f(s) \}$ from a function $f$ and showing that no element of $S$ maps to $D$.  This proof works because all bijections are surjections, so if no surjection from $S$ to $\wp(S)$ exists, then there cannot be a bijection between $S$ and $\wp(S)$.  My question is whether it is possible to run a "reverse diagonalization" that works by instead showing that there is no injection from $\wp(S)$ to $S$.  I am curious about this because I have never seen Cantor's theorem proved this way (or, more generally, any diagonal argument structured like this).  Is it possible to take Cantor's diagonal argument and "reverse" it to show that there cannot be an injection from $\wp(S)$ to $S$, rather than showing that there can be no surjection from $S$ to $\wp(S)$?  Thanks!
I have to move a MongoDB Collection (Size: 1TB+) from Sharded MongoDB Cluster to another[5 shards on both clusters]. During this operation, it is assumed that the accompanying program will be offline, so no need to worry about inconsistent data, but I need to minimize the downtime.  I've tried testing it with mongodump and mongorestore, but looks like that is taking ages to finish.  Please share your experiences for such kind of scenarios.  Note: Cluster has only one collection, so I am open at cluster level sync as well. Custom thumbnailers don’t work on Ubuntu 18.10 and 18.04 Recently I had a bunch of my own scripts for the thumbnails in Nautilus: , , office documents and so on.  Seems like either I’m doing something wrong (thumbnail generation behavior changed), or custom thumbnailes don’t work in latest GNOME at all, even my previous scripts, which worked smoothly on previous versions of Ubuntu.  I’ve done some research, tried to run a simple script (see below) and it didn’t work out.  My test thumbnailer in /usr/share/thumbnailers/z.thumbnailer:  [Thumbnailer Entry] Exec=/home/maximal/thumb.sh %s %i %u %o MimeType=image/webp;image/x-webp   Where /home/maximal/thumb.sh is:  #!/bin/bash echo $0 &gt;&gt; /home/maximal/thumb.log echo $1 &gt;&gt; /home/maximal/thumb.log echo $2 &gt;&gt; /home/maximal/thumb.log echo $3 &gt;&gt; /home/maximal/thumb.log echo $4 &gt;&gt; /home/maximal/thumb.log   Then, when I remove thumbnail cache rm -rf ~/.cache/thumbnails and restart Nautilus nautilus -q, my log file doesn’t contain anything and ~/.cache/thumbnails/fail/gnome-thumbnail-factory contains all the failed thumbs (looks like corrupted/empty PNGs):    So, apparently, Nautilus even didn’t try to execute my thumbnailer script. When I try to run thumb command itself, it works perfectly. 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For some weeks my MacBook 6.2 does not switch to the integrated graphics adapter (Intel HD). This is especially crucial when I work on battery (which happens quite often). I guess the runtime could be enhanced when it would take the HD adapter for display.  I checked that it automatically changes the graphics mode. I am running Snow Leopard 10.6.8.  I recently installed 8GiB RAM (not from Apple). Could this probably related to this issue? Actually I do not want to downgrade or use the expensive Apple RAM.  So, how can I manually turn on the internal graphics adapter? I want to prove that: $\ln(x+1)&lt; x$.  My idea is to define:  $f(x) = \ln(x+1) - x$, so:  $f'(x) = \dfrac1{1+x} - 1  = \dfrac{-x}{1+x} &lt; 0, \text{ for }x &gt;0$.  Which leads to $f(x)&lt;f(0)$, so $\ln (x+1)-x&lt;0$.  Is that a valid proof? Any other ideas?  Thanks. Finite Summation of Fractional Factorial Series I've heard that you always want the same size of RAM (2x1GB sticks) but does their placement matter? I have four slots on my motherboard but does it matter if the sticks are not adjacent?  This particular question involves the Deleted comments are persisting in the global inbox I realise that there is other questions out there about the  (First open one I found is ), but this is marked .  I often get the below issue where it will notify me about comments on my  (in this case what appears to be the user re-writing after deleting their comment) that no longer exist.  Also, occasionally, I see comment snippets with text that doesn't match what is on the page when I view it.  Let me know if I should be trying to edit that one to remove the tag indicating that it has been fixed, and add the above as another answer. I'm totally new to Ubuntu/Linux, using Ubuntu Server at the moment. Just trying to  figure out something basic.   How can you tell where you are installing a program. For example I just installed Sphinx search engine by placing the tarball that I downloaded from their site to my:  /home/sphinx   directory. I created the sphinx directory to place that tarball in. Then I ran these commands:  tar xvzf sphinx-0.9.8.1.tar.gz cd sphinx-0.9.8.1/ ./configure --with-mysql-includes=/usr/include/mysql --with-mysql-libs=/usr/lib/mysql   and then these:  make sudo make install   Now I have a lot of files sitting in the directory where I ran these commands. Is this my Spynx installation or did it install somewhere else?  In windows if you run an installer (.exe file) anywhere the program will still install in your C:\Program Files directory. Does something similar apply to linux where all programs are installed in a central place, or can you install programs anywhere on the system.  Questions   I would prefer to keep all my installed programs in one place so what is the right place for this in terms of best practice. 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