Prove that $A-(B-C)=(A-B) \cup (A\cap C)$ I am trying to prove that $A-(B-C)=(A-B) \cup (A\cap C)$ For the first statement, I do: Let $x \in A-(B-C)$ means that $x \in A$ and $x \notin (B-C)$ $x \notin (B-C)$ means that $x \notin B$ or $x \in C$ $(*)$ Therefore $x \in A-(B-C)$ means that $x \in A$ AND ($x \notin B$ or $x \in C$) For the second statement, I do: Let $x \in (A-B)$ means that $x \in A$ and $x \notin B$ $x \in (A \cap C)$ means that $x \in A$ and $x \in C$ Therefore, $x \in (A-B) \cup (A\cap C)$ means that $x \in A$ AND ($x \notin B$ or $x \in C$) $(*)$ Those two are the same so these sets are equal. I'm not sure if I did the $(*)$ parts correctly. Can someone give me some guidance?
Show that $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$ Show that  $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$  This is an exercise I was trying to do ( not homework ) and I got stuck as follows:  Working from $A \setminus ( B \setminus C )$:  $x \in A \wedge x \notin ( B \setminus C )$ $x \in A \wedge ((x \in B \wedge x \in C) \vee (x \notin B \wedge x \in C) \vee (x \notin B \wedge x \notin C) )$  At this point I'm not sure how to proceed.  Working from the other side $ ( A \setminus B) \cup ( A \cap C ) $:  $ x \in ( A \setminus B) \cup ( A \cap C ) $  $ x \in ( A \setminus B) \vee x \in ( A \cap C ) $  $ ( x \in A \wedge x \notin B ) \vee ( x \in A \wedge x \notin C ) $  $ x \in A \wedge (x \notin B \vee x \notin C ) $  $ x \in A \wedge x \notin ( B \cap C ) $  $ A \setminus ( B \cap C ) $  I could also use some formating tips like how display logical not and how to get the lines to display properly without blank lines inbetween. :-)
Natural examples of deformation retracts that are not strong deformation retracts The question  asks for examples of deformation retracts which are not strong deformation retracts. A comment is given refering the asker to Exercise 0.6 in Hatcher. (The description of this space also appears in , which is about this very same exercise.)  This space, however, seems to be constructed specifically for this purpose.     Are there any naturally occurring (by which I mean not constructed specifically for this purpose, arising in some other way) spaces that are deformation retracts but not strong deformation retracts?   The main reason for asking is that it doesn't seem very likely to me that one would stumble upon such a space at random, so I don't know what to watch out for. One can never be too careful. How do I make ditto marks? How about a "Vote not to close" option to counter the "Vote to close"? My request is simple: Many times, there will be a question where some people feel like it should be closed. At the same time, I'll feel like it's a perfectly legit question, and it should therefore NOT be closed. Right now, I have to wait till it gets enough close votes (which it always does for some reason, as soon as 1 or 2 people vote to close, e/o else decides to vote as well) and then vote to re-open. How about, as soon as there's one vote to close, there should be an option to counter the close vote, something like "vote not to close" or whatever. This would then bring the "Vote to close" number down by one. Only when the vote to close outnumbers the vote not to close by 5, that's when a question is indeed closed. Any thoughts? If the imaginary part of an entire function is never zero, the function is constant Let $f : \mathbb{C} \to \{z\in\mathbb{C}:\Im(z)\neq0\} $ entire . Show that $f$ is constant.  I took $g(z)=\frac{1}{f(z)}$ and I think that g is bounded, therefore it is constant (due to Louville's theorem) but how do I prove it? Is $|g(z)|=\frac{1}{|f(z)|}=\frac{1}{u^2+v^2}\le M$ valid? How can I create an instance of the Java console inside of a GUI panel? I am using ubercart guest checkout module. I am able to submit  the order as guest etc works fine.   After Reviewing the order and submitting it throws me this error   Recoverable fatal error: Argument 2 passed to db_query() must be an array, string given, called in /var/www/html/dev.demo.net/sites/all/modules/ubercart_guest_checkout/uc_guest_checkout.module on line 263 and defined in db_query() (line 2342 of /var/www/html/dev.demo.net/includes/database/database.inc).   This is the query which is causing that issue    db_query('UPDATE {users} SET name = "%s", mail = "%s" WHERE uid = %d', $name, $mail, $account-&gt;uid);   I tried to change this query like this  db_update('users') -&gt;fields(array(     'name' =&gt; $name,     'mail' =&gt; $mail )) -&gt;condition('uid',$account-&gt;uid,'=')        -&gt;execute();   still no luck, it throws me same above specified error.    How can i resolve the query of above one in order to avoid the error. Any suggestions w.r.t query will be nice.Thanks in advance. Here's an example:  Double d = (1/3); System.out.println(d);   This returns 0, not 0.33333... as it should.  Does anyone know? I define $X_i$ as a random variable that is uniformly distributed between (0,1). What is the expected number of such variables I require to make the sum go just higher than 1.  Thanks How can I see the phone number dialed after I initiate a call? Horizontal Probability Tree with Level Labels Inductive proof of the closed formula for the Fibonacci sequence Why there is a delay between when a key is pressed and when it is accepted? Solder does not stick to tip of soldering iron? Basic question here, but it's driving me insane!  The solder does not seem to stick to the tip of my soldering iron. After the iron heats up, the tip appears black (sometimes on all sides and sometimes on one side), then either it doesn't melt the solder at all, or it I fiddle around with it for a while; it melts the solder, but the solder rolls into itself into a ball and doesn't stick to the tip.  I've seen this many times while soldering before and usually it's solved by cleaning the tip on a soldering sponge or cooling it down and scraping the tip off with a blade, but I have scraped this particular one many times and still it's the same problem: it heats up and gets black, then the solder doesn't stick to it.  It doesn't help that this soldering iron has a very fine tip and I have to use it to solder very tiny chips and wires under a microscope (I'm just starting to learn soldering under microscopes).  Any ideas on what I can do?    PS: Is scraping the tip a bad idea? I have been doing it with my other bigger soldering irons and it seemed to work (at least in most cases) but I am not sure whether it is a good solution or not. Related:   This is a subtly different question because I'm not asking whether there's a historical relationship between the two lineages.  I am asking this question because answers to these questions:      are making assumptions about the appearance of Padme while pregnant, the period of gestation, and appearance of newborn babies.  All of these presuppose that the "humans" of the Star Wars Galaxy are 100% biologically equivalent to us.   In fact, as far as I know, we don't actually know any of those facts from canon.   (Do we?)   All we know is that "human" children and adults look like us.  So, do we have any canon or out-of-universe commentary that tells us that Star Wars humans and Earth humans are biologically equivalent?  Another way to put this would be  If presented with a Star Wars human and a Milky Way human, would the best doctors or the best medical droids be able to distinguish them? Stop Edit/Rollback Wars: Limit Number Of Edits/Rollbacks Per User Per Post Per Day I came up with an idea for how to begin putting an end to the edit/rollback wars and then the ensuing comments battle...   What if you are only allowed to edit a post (not the author) once a day. What if you are also only allowed to rollback one edit to your post a day. There would not be a overall limit on the users ability to edit other posts other than the one edit per post per day. There would also not be a limit on the number of times a single post could be edited or rolledback (multiple users), but each edit/rollback would trigger the 24 hour wait... There is currently some threshold in place that allows a post to be edited after it is posted without calling it an edit.  This same threshold could apply to the edit in case a mistake is made (fat fingered, mis-tagged, etc.) which would allow the editor a window of time to correct a mistake before getting locked out.     If either the editor or rollerbacker feel that the current state of the post is not acceptable and they are out of mods for the day, then a moderator can get flagged through the current process. Adding objects algorithm in Objective-C Big tilde in math mode Filtering points in GeoServer layer I understand using phone while it is being charged by a normal charger is not advised. What about using phone while it is being charged by a portable charger (powerbank)?  I own a HTC one and i was told by a mobile technician HTC’s own portable charger only should be used for HTC one. How much truth is there to it as HTC powerbank is not available in my city and i am thinking of using Sony portable charger for charging my HTC One? I have video tutorials that I want to show in my presentation with beamer class. I don't want the video files get embedded to the PDF because they will make the PDF size very very huge.  If you know how the videos can be played on the HTML with a reference rather than embedding the video file to the HTML file, I want something like this to be applied to my PDF presentation. I just want to reference the video URL from my PDF presentation.   Shortly speaking, can we use media9 to show a video on a PDF but the video is not embedded to the PDF?  Uploading those videos to a web server might be difficult for me because internet connection is very slow and sometimes not available in the presentation room. :-)