The Axiom of Choice, and the statement that whenever there is a surjection $A \to B$ there is an injection $B \to A$
Is the following equivalent to the axiom of choice?
Background: I'm giving a presentation to colleagues at work on hypothesis testing, and understand most of it fine but there's one aspect that I'm tying myself up in knots trying to understand as well as explain it to others.  This is what I think I know (please correct if wrong!)   Statistics that would be normal if variance was known, follow a $t$-distribution if the variance is unknown CLT (Central Limit Theorem): The sampling distribution of the sample mean is approximately normal for sufficiently large $n$ (could be $30$, could be up to $300$ for highly skewed distributions) The $t$-distribution can be considered Normal for degrees of freedom $&gt; 30$   You use the $z$-test if:   Population normal and variance known (for any sample size) Population normal, variance unknown and $n&gt;30$ (due to CLT) Population binomial, $np&gt;10$, $nq&gt;10$   You use the $t$-test if:   Population normal, variance unknown and $n&lt;30$ No knowledge about population or variance and $n&lt;30$, but sample data looks normal / passes tests etc so population can be assumed normal   So I'm left with:    For samples $&gt;30$ and $&lt;\approx 300$(?), no knowledge about population and variance known / unknown.   So my questions are:   At what sample size can you assume (where no knowledge about population distribution or variance) that the sampling distribution of the mean is normal (i.e. CLT has kicked in) when the sampling distribution looks non-normal? I know that some distributions need $n&gt;300$, but some resources seem to say use the $z$-test whenever $n&gt;30$... For the cases I'm unsure about, I presume I look at the data for normality. Now, if the sample data does looks normal do I use the $z$-test (since assume population normal, and since $n&gt;30$)? What about where the sample data for cases I'm uncertain about don't look normal? Are there any circumstances where you'd still use a $t$-test or $z$-test or do you always look to transform / use non-parametric tests? I know that, due to CLT, at some value of $n$ the sampling distribution of the mean will approximate to normal but the sample data won't tell me what that value of $n$ is; the sample data could be non-normal whilst the sample mean follows a normal / $t$.  Are there cases where you'd be transforming / using a non-parametric test when in fact the sampling distribution of the mean was normal / $t$ but you couldn't tell? I just read a description of CVE-2020-1971, which indicates that a patch was released today for openSSL, the patched version is 1.1.1i -- my version of Ubuntu 20.04.1 LTS is running openSSL version 1.1.1f.  From last March. Software Updater says that my system is up to date.  I am hoping that this openSSL update is released by Canonical by tomorrow... can anyone confirm they are working on this? What is accountsd and why does it want to use the "login" keychain? On every bootup it's the same:  /dev/sda1: clean, 908443/38690816 files, 44176803/154733312 blocks   Is it some kind of option Ubuntu uses to ensure filesystem consistency or is there something wrong with my HDD? fsck takes up to 30s while booting and so about triples the time needed otherwise.  Full output (partly in German):  Begin: Loading essential drivers ... done. Begin: Running /scripts/init-premount ... done. Begin: Mounting root file system ... Begin: Running /scripts/local-top ... done. Begin: Running /scripts/local-premount ... done. Begin: Running /scripts/local-bottom ... done. done. Begin: Running /scripts/init-bottom ... done. fsck von util-linux 2.20.1 /dev/sda1: sauber, 908443/38690816 Dateien, 44176803/154733312 Blöcke udevd[623]: unknown key 'SYSFS{idVendor}' in /lib/udev/rules.d/45-libticables.rules:6  udevd[623]: invalid rule '/lib/udev/rules.d/45-libticables.rules:6'   * Starting mDNS/DNS-SD daemon                                                 [ OK ]  * Starting Reload cups, upon starting avahi-daemon to make sure remote queues are populated                                                                   [ OK ]  * Starting configure network device security                                  [ OK ]  * Starting bluetooth daemon                                                   [ OK ]  ####* Starting all other stuff Is there a word that describes a parent whose child has died?   Along the lines of "orphan", "widow", and "widower", is there a single word for a parent who has lost a child (of any age)? Inner join using an array column Having trouble indexing and executing a query in O (log n) time.  The query includes an INNER JOIN, an ORDER BY, and an equality operation. If I understand the laws of databases correctly, a query can be indexed and executed in O (log n) time (or thereabouts) if a non-equality operator is not used on more than one column. In this case, I believe the INNER JOIN does count as an equality operator and the non-equality operator would be the ORDER BY portion of the query. This table has upwards of 10,000,000 rows and needs to handle several reads and writes per second.  Using PostgreSQL. This is what the table looks like. As you can see, the column 'Names' is a list property and it is the column that the INNER JOIN goes against:  Age Names                       Date 34  ['carla', 'john', 'sam']    3/13/2011 26  ['json', 'cindy', 'joel']   3/13/2011 72  ['beth', 'amber', 'susie']  3/13/2011 14  ['john', 'jim', 'debie']    3/13/2011   This is the query that we are trying to do:  SELECT * FROM the_table WHERE Age==26 AND Names=='john' ORDER BY Date   My background is from using App Engine's Big Table, so I've used equality operators here to indicate that 'john' should be one of the names in the Names column. This would be an acceptable query in GAE's big table, it would execute in O (log N) time as all Big Table queries are reqyured to do. I am assuming there is a way to do this in PostgreSQL as well since PostgreSQL accepts list data types as columns.  Is this possible to do in PostgreSQL?  If so, how should the index be set up (we can't figure out how to set up an index that takes into account the three properties)? Unexpected output of xargs I have been looking at the plugin API a bit more in depth recently and I was wondering what real differences there were between action and filter hooks. They both are events that receive data as a parameter and they seem to both be able to do the same things.  Obviously I see that actions are called when actions take place and filters are called when data is manipulated, but it seems to just be a semantic naming difference.  Besides semantics and what they are used for, what real differences are there between them? My question might appear the same as the question , but actually I did read all the answers and none of them helped me with the details I need.  I am new to cryptography and my problem is with two time pad attacks on OTP. The problem I had in my course was that I have 10 ciphertexts encrypted with the same key $K$. I am then given another ciphertext that I should decrypt.  I know that XOR-ing two ciphers gives me the XOR of their original messages.  My question is what is the correct thing to do after that?  I tried to take 3 ciphertexts $C_1, C_2$ and $C_3$.  Then get $S_1 = C_1 \oplus C_2  \oplus $' ', also get $S_2 = C_1 \oplus C_3 \oplus$ ' '.  After that I compared all corresponding characters in $S_1$ and $S_2$,  and if $S_1[i] = S_2[i]$ then I calculate $S_1[i] \oplus C_2[i]$ to get $K[i]$.  I tried this on paper before coding and it worked, but I might be missing something.  Is this the right approach? Why does it work? Increase close vote weight for gold tag badge holders The use of "real" in the following cases Possible Duplicate:             If you listen real close...      Can you swing by real quick...   Sentences like the above two are what I often hear in daily life. If I didn't hear them in the real world, I would probably be more ready to say "If you listen really close..." and "Can you swing by really quick...".  Is real being used as an adjective in this scenario? If so, is the usage of adj.+ adj. a common practice in English language? Can someone decide to be hit? Uk tourist visa - duration of stay My mother is given a UK tourist visa for 2 years with a duration of stay as 180 days. It is a c type multiple entry visa. Is this 180 days is the total stay she can take or is it duration of stay per visit.  Thanks in advance for your reply. Redirect the output using `sudo` I have a small question regarding using sudo with output redirect &gt;. To enable IP forwrding, someone can use the command:  echo 1 &gt; /proc/sys/net/ipv4/ip_forward   Executing this command will give permission denied as it requires root privileges. However, executing the same command with sudo gives also permission denied error! It seems that output redirect &gt; does not inherit the permissions of the preceding command echo. Is this right?  As a workaround I do:  echo 1 | sudo tee /proc/sys/net/ipv4/ip_forward   Is this the best way to do it? Am I missing something?  Please, note that this is an example and it applies to all commands that use output redirect. (a.k.a. "god rays" and many other names) can be created in Blender Internal by checking the Halo checkbox in the lamp properties box.  How can I create a similar effect using Cycles? Why does DMM show AC and DC current when source is a battery? (How to calculate power in this case?) The circuit is an altered Joule Thief of my design that is driven by an AA or AAA battery. I’m observing the current at the battery anode with my DMM in series. (Extech EX330) This circuit is extremely efficient (In relation to other Joule Thief I've build.) and does some weird stuff - like, it needs a pushbutton sometimes.    Measurements: It is drawing 9.21 mA DC and 6.48 mA AC in series with the battery. It shows 1.364 VDC and 0.010 VAC in parallel with the battery. Output on the LED is 2.572 VDC and 0.011 VAC with no measurable frequency. (with my DMM at least) All measurements are taken with the same configuration and voltage.    But how is it drawing AC and DC and how do I calculate power consumption? --> Could it be because the emitter pin of the NPN is directly connected to ground? --> Or is the circuit just "switching on and off"?     Download schematic:  (I tried to simulate the circuit in LTspice. Thanks for making me aware of this amazing program, Sredni Vashtar!)     What the pushbutton is used for: The pushbutton is used to "kickstart" the oscillation when it wont start by itself. For example: I have to manually "oscillate" using this button to start the circuit if the capacitor is too big and battery voltage is low. Pushing the button releases the energy in the transformer into the capacitor. If it reaches LED forward voltage the circuit keeps oscillating on it's own. It does work without manual intervention when then capacitor in parallel with the LED is not too big, like &lt;5000 µF, and it's a fresh battery.    . Thank you for your answers! I'm learning so much! :D Is it possible to simulate key press events programmatically in JavaScript? Copy maintenance plans to a different server Is there a way to copy maintenance plans from one SQL Server 2012 to another? Both servers are not accessible via the same copy of Management Studio at the same time (on different networks). Is $\mathbb{Z} \left[ \frac{1+i \sqrt{7}}{2} \right]$ euclidean? Will cases with front USB port support USB 3.0?