So this is the thing I've got to prove: $ (A \cap C = B \cap C ) \land ( A \cup C = B \cup C ) \Leftrightarrow A = B $  I can understand this intuitively but the formal proof is taking me some time. So far, I've got this:   $ (A \cap C \subset B \cap C ) \land (B \cap C \subset A \cap C ) \land ( A \cup C \subset B \cup C ) \land ( B \cup C \subset A \cup C ) \Leftrightarrow (A \subset B) \land (B \subset A) $  $ (\forall x) (x \in A \land x \in C \Leftrightarrow x \in B \land x \in C ) \land (\forall x) (x \in A \lor x \in C \Leftrightarrow x \in B \lor x \in C ) \Leftrightarrow (\forall x)(x \in A \Leftrightarrow x \in B) $  But I feel like going nowhere. I assume this has something to do with the transitive property, but I'm not sure how to apply it here.
Prove or give a counterexample:  $$(A ∪ B = A ∪ C) \land (A ∩ B = A ∩ C) \implies B = C.$$  I think this is true, but I am not sure how to show it. I don't know if there are any manipulations with unions and intersections that allows me to move things around. Thanks in advance for any hints.
Normal subgroups of $S_4$ Can anyone tell me how to find all normal subgroups of the symmetric group $S_4$?  In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups? When is "seems to be" used instead of "seems"? The baby seems happy/comfortable. The baby seems to be happy/comfortable.   Are these both correct? If both are correct, what are the differences between them? How to remove zsh alias? I have an alias which is predefined by an oh-my-zsh plugin.  -&gt; % alias gcm gcm='git checkout master'  I'd like to remove it, i.e. I'd like alias | pcregrep "\bgcm\b" return 1. I've tried with alias gcm='' but after that the alias is still existent. How to prevent search engines from indexing a single page of my website? If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality I am trying to prove: If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality.  Set definitions here are from baby Rudin.   Attempt: There are 4 possible cases:  1.A is countable and B is finite.  2.A is countable and B is countable.  3.A is uncountable and B is finite.  4.A is uncountable and B is countable.  Then for each case I need to show that $A \cup B \sim A$. I can prove 1-3, but I am unsure how to prove case 4. And in general, it seems that my proof strategy is inefficient and there exist much faster proof then proof by cases. Is there a word that describes a person who dies on his/her birthdate? How long can eggs be unrefrigerated before becoming unsafe to eat? When is $-3$ a quadratic residue mod $p$? Prove that if $n \in \mathbb{N}$, then $$\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$$ where $d(n)$ is the divisor function. I have know only information is $d(n)=\sum_{d|n}{1}$. I've been playing the game a while now and have ranked up considerably. I'm wondering if there is any way to change to a different Google account and still keep all of my Pokemon/items etc, as I stupidly used my university gmail account when I first downloaded the app (which will be automatically deleted in September).   Is there any way to transfer my trainer/Pokemon/items etc to a different Google account? I would like to sell my Macbook Pro, but I want to make sure that there is no way to retrieve the deleted data from my SSD. Is there a way to completely delete all files on a hard drive so that even a forensic expert cannot recover anything? Reading the target of a .lnk file in Python? I'm trying to read the target file/directory of a shortcut (.lnk) file from Python. Is there a headache-free way to do it? The  is way over my head. I don't mind using Windows-only APIs.  My ultimate goal is to find the "(My) Videos" folder on Windows XP and Vista. On XP, by default, it's at %HOMEPATH%\My Documents\My Videos, and on Vista it's %HOMEPATH%\Videos. However, the user can relocate this folder. In the case, the %HOMEPATH%\Videos folder ceases to exists and is replaced by %HOMEPATH%\Videos.lnk which points to the new "My Videos" folder. And I want its absolute location. NetGear A6100 AC600 wireless adapter not working in Ubuntu 20.04. Worked fine in 18.04. Not same problem as previously Firstly, I know very well there are answers to this already. But, in my case, it is installed and shows in the terminal as such Bus 001 Device 007: ID 0846:9052 NetGear, Inc. A6100 AC600 DB Wireless Adapter [Realtek RTL8811AU]  It says it is installed in  as well.  I have tried downloading the driver for this adapter from several guides online, linking to GitHub, but no luck there either...  Any help is greatly appreciated! Thanks in advance!  EDIT: The answer from sudo dkms status command reads:  rtl8812au, 4.3.8.12175.20140902+dfsg, 5.4.0-31-generic, x86_64: installed  The other command  modinfo 8812au | grep 9052   spits out the following:  alias:          usb:v0846p9052d*dc*dsc*dp*ic*isc*ip*in* I have Ubuntu 16.04 with GNOME shell. I set ctrl+shift for the "Modifiers-only switch to next source" option.   This shortcut conflicts with apps hot-keys. I can't use ctrl+shift+i to open devtools in Chrome, it just switches the layout. In VS code IDE if I press shortcuts with ctrl+shift (for example ctrl+shift+f), shortcuts work but always switch a keyboard layout. When I used Windows this problem didn't exist.  Can I set the ctrl+shift shortcut in such a way that it causes a switch action only after the keys are released and only if this 2 keys were pressed (without additional keys)? A search on google for "iphone developer" turns up loads of websites which have "iphone" in them, a trademarked name by Apple.  So I'm led to believe that having a domain such as iphonedeveloper.com is ok? Well, you're still using Apple's trademark, but it would be hard to brand yourself otherwise. You're an IPHONE DEVELOPER... right?  Well, what if I want to provide a website where users pay to get a list of the best offers from Ebay? I might have a domain like ebaydeals.com (I don't... i'm just speculating!).  Now I've heard that places like Ebay are really hot on the trigger and fire out emails to people who register domains like that straight away. But whats the difference?  In both cases I'm making money from the trademark, effectively, so is it just down to how lenient the company who owns the trademark is? Or are there rules? Is there a specific "line" you don't cross?  Thanks android expandablelistview how to set space between groups items How can I change Win 10 protected registry entries? While I was looking at the values of the zeta function for the first natural numbers, I noticed that the sum of the values minus $1$, converge to $1$. Better put: $$\sum_{n=2}^{\infty} \left(\zeta(n)-1\right) = 1 $$ Furthermore, if you use only the even numbers for the zeta function, the sum will converge to $\frac{3}{4}$, or $$\sum_{n=1}^{\infty} \left(\zeta(2n)-1\right) = \frac{3}{4}$$  Leaving $$\sum_{n=2}^{\infty} \left(\zeta(2n-1)-1 \right)= \frac{1}{4}$$  This is probably common knowledge among mathematicians, but I couldn't find much about it on the internet. Is there a proof of this or perhaps even a simple explanation why this is so? How to change default viewport shading in 2.8? Is it possible to change the default mesh color/shading in 2.8 to be more like 2.79? I'm finding it hard to see finer detail in 2.8. On the left is 2.8 and the right is 2.79 What does echelon mean? When you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon?