A word for "lack of a word"?
What would you call a word that doesn't exist in or translate well into another language?
More homework help.  Given the function $f:A \to B$.  Let $C$ be a subset of $A$ and let $D$ be a subset of $B$.  Prove that:  $C$ is a subset of $f^{-1}[f(C)]$  So I have to show that every element of $C$ is in the set $f^{-1}[f(C)]$  I know that $f(C)$ is the image of $C$ in $B$ and that $f^{-1}[f(C)]$ is the pre-image of $f(C)$ into $A$.  Where I'm stuck is how to use all of this information to show/prove that $C$ is indeed a subset.    Do I start with an arbitrary element (hey, let's call it $x$) of $C$? and then show that $f^{-1}[f(x)]$ is $x$?  I could use a little direction here...  Thanks. Prove that $(\mathbb{Z}_n , +)$, the integers $\pmod{n}$ under addition, is a group. Prove that $(\mathbb{Z}_n , +)$, the integers  $\pmod{n}$ under addition, is a group.  To show that this is a group, I know I need to show three things (in our text, we do not need to show that addition is closed-- rather, we show these three items):  $(a)$ Associative Law  $(b)$ Existence of Identity  $(c)$ Existence of Inverse  This is what I have been working with so far:     $(a)$ Associative Law:   First, assume that $n$ is a composite number and that $a,b \in \mathbb{Z}$ s.t. $a=nx$ and $b=ny$ for some integers, $x$ and $y$. Then $a+nx+nx=n(x+y)$.     So, this is where I am stuck for the Associative Law. I'm not sure if I am headed down the correct path to show that $\mathbb{Z}_n$ is associative.      $(b)$ Existence of Identity:   Generally, when showing the existence of an identity we can use this information for addition :  $a + 0 = 0 + a =a$.    I am unsure how to insert the modular portion of the definition into the existence of this identity. I realize that for addition, we generally look at $0$ (rather than $1$ for multiplication).     $(c)$ Existence of Inverse: I need to show that for each element $a \in \mathbb{Z}_n$ that $a^{-1}=-a$. Again, I do not know how to insert the modular portion of definition into this aspect of the proof.   My question is how to incorporate this modular information into the proof that this is a group. We have not covered the integers modulo $n$, binary operations, nor groups in my intro to proofs course. I am trying to read my text to figure out how to prove this, but have a difficult time reading and interpreting math speak. Using restroom on airplane while fasten seatbelt light is on? This question about  over on  made me wonder what a passenger is supposed to do when they urgently have to use the restroom while the fasten seat belt light is on. FAA regulations require passengers to obey the seat-belt sign, even if flight attendants do not.  I saw this happen on a flight where an older gentleman asked the flight attendant if he could use the restroom, and she said "not until the light is off". The flight had been free of turbulence for 15 minutes or so by this time, and the flight attendants were walking freely throughout the cabin, collecting post-meal trash, etc. The gentleman who asked to use the restroom appeared to be in quite some discomfort, and practically ran to the restroom when the light went off 10 minutes later.  So what is a passenger supposed to do when he isn't permitted to use the lavatory, when he has to go and can't wait? Just go in their seat? Why do objects appear smaller when farther away? Simulate INSERT key in Mac OS X Long story short, I am using Citrix to remote into my PC at work, and several applications require the use of the INSERT key, which, as everyone knows, Apple chose to eliminate from their keyboards. I've tried everything I can think of, and never been able to find an easy way to send this keycode via Citrix.  I'm sure there are other uses for this, and I've read quite a few people's questions on the subject, but no one seems to have offered a viable solution.  I have the Apple USB keyboard (full-size, with numeric keypad), and in the position where the INSERT key would normally exist, Apple placed the "fn" key. Now, I still need this key from time to time, but what I'd love to see is perhaps Control-fn or similar to send the keycode for INSERT.  If anyone knows how to achieve this, preferably without installing 3rd-party software (adware?), I'm all ears. Why do my shortcut icons get broken on Windows 7? I have a strange problem. My icons seem to disappear randomly on my Windows 7. They were there a while back, now they ain't.  It happens often and quite randomly. Anyone know a fix? Can bug reports and feature requests requiring SE Dev attention be reported on per-site metas, or do they need to all go here? Construction of $\mathbb{CP}^2$ from $S^2\times S^2$ Show that $\mathbb{CP}^2$ is homeomorphic to the orbit space of $(S^2\times S^2)/\mathbb{Z}_2$. The action is given by $(x,y) \mapsto (y,x)$. My attempt: I can see that $S^2\times S^2$ is a subset of $S^5$. I am unsuccessfully trying to restrict the action of $S^1$ on $S^5$ to $S^2\times S^2$ to get the required map. Removing elements with Array.map in JavaScript Loading a properties file from Java package In linear regression, when is it appropriate to use the log of an independent variable instead of the actual values? Batch export Photoshop layers to individual PNG files $(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse? Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&amp;$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&amp;(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ as addition and multiplication, respectively. Do these operations form a field?  I have shown that every axiom of the field is satisfied except for the "existence of multiplicative inverses" axiom. This is where I need help. We either need to show that this axiom fails and so we don't have a field or that the axiom $aa^{-1}=1$ is satisfied for every $a\neq 0 \in \mathbb{R}$  The multiplicative identity in this problem is $(1,0)$.  Any solutions or hints are greatly appreciated. Wizard sickness I remember in an interview, or during a Q&amp;A, Rowling stating that Wizards live longer than Muggles. This got me thinking about Wizard deaths in general, and then deaths being attributed to illness.  So, I ask, has there ever been a Wizard, that has died from something like cancer? Is there anything, even briefly mentioning a Wizard becoming ill?  And finally, are there any illnesses that are exclusive to Wizards? How to install Windows 8 without the setup DVD that came with my notebook? How do I (or can I) SELECT DISTINCT on multiple columns? I need to retrieve all rows from a table where 2 columns combined are all different. So I want all the sales that do not have any other sales that happened on the same day for the same price. The sales that are unique based on day and price will get updated to an active status.  So I'm thinking:  UPDATE sales SET status = 'ACTIVE' WHERE id IN (SELECT DISTINCT (saleprice, saledate), id, count(id)              FROM sales              HAVING count = 1)   But my brain hurts going any farther than that. How do I reinstall Unity? I tested out Unity 5, and of course, it broke things, so I tried downgrading by    sudo apt-get install ppa-purge sudo ppa-purge ppa:unity-team/staging   It removed Unity completely.    apt-get install unity   seemed to work, but the login screen now offers only Gnome.  How do I get back to Unity 4? How to change Function (Fn) Key behavior on wireless Logitech keyboard Aiming at specific body parts PC Blue Screens and then instantly reboots. How do I stop this or find the error later?