I have the following problem:  Given a non-empty set $U$ and the power set $P(U)$. Proof $(P(U);\triangle, \cap)$ is a commutative ring. Which subset is the neutral and which one is the identity?   I have an idea of how to check if it's a ring (by checking all properties one by one) but I don't know how to deduce the neutral set and the identity.
I'm studying for an abstract algebra exam and one of the review questions was this:     Let $X$ be a set, and $\mathcal P(X)$ be the power set of $X$. Consider the operations $\Delta$ = symmetric difference (a.k.a. "XOR"), and $\bigcap$ = intersection.   a) Does $\Delta$ and $\bigcap$ make $P(X)$ into a ring?   b) If so, is it a ring with unity?   c) Is the ring commutative?   d) Is it a field?   For parts a) and b), I think it does form a ring with unity, but I'm not quite sure how to get started on proving it.   For part c), it is a commutative ring since $\mathcal P(X)$ is closed under symmetric difference and intersection, right?  Not even sure how to get started on d).  I'd really like to understand this question fully, so any kind of input would be tremendously helpful. Thank you!
The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ Someone told me I need a ticket out of Thailand in order to enter.  I am traveling to Thailand, on March 18th. I have a Ticket from Bangkok to Sweden booked for June 4th.   I will be staying in Thailand for 2 weeks when I arrive there, then traveling around Cambodia, Laos and Vietnam until June 4th when I leave.   With that ticket out of Thailand, will I be able to get in and get a (30 day) Visa-On-Arrival (or not because the ticket out is not within the first 30 days). Or do I need to show proof that I am leaving within 2 weeks of being there at first? If you are falling, can you Misty Step to the ground to avoid damage? If I fall off a cliff - say, 1000 feet - can I wait (readied action?) until I am near enough to the ground (my understanding is you fall 500 ft/round) then  safely to the ground? Or, does the momentum from the fall carry over through the teleport and I still splatter across the rocks? Object emit light distance I created an object, and I wanted it to glow. I used to just place points of light around it, and parent them to it, but using the emit option in the materials seemed so much easier.  But when I gave it an a higher emit than 0, it went only a small distance before the light stopped. I turned it up to 200, and the object was white with light, but the light didn't go any farther. Oh and I changed the color, and nothing happened. What does functools.wraps do? In a comment on this , someone said that they weren't sure what functools.wraps was doing. So, I'm asking this question so that there will be a record of it on StackOverflow for future reference: what does functools.wraps do, exactly? Examples of infinite groups such that all their respective elements are of finite order. I am in need of examples of infinite groups such that all their respective elements are of finite order. Genetic linkage greater than 50 centimorgans Classically, the linkage between two loci can be measured in centimorgans (cM), which represents the percent chance that these two loci will recombine an odd number of times (generating a recombinant genotype).  Due to independent assortment, markers on different chromosomes are expected to recombine 50% of the time. As I understand, if one conducts a classic experiment and counts various progeny, then deduces the apparent linkage, values of 50 cM or more are interpreted as "impossible to determine whether on separate chromosomes or just recombines very often". See (), with grammar corrected:     The final point that we need to make regards the maximum distance that we can measure. Because of the way in which the calculations are performed, we can never have more (than) 50% recombinant gametes. Therefore the (maximum) distance that two genes can be apart and still measure that distance is just less (than) 50 cM. If two genes are greater than 50 cM apart, then we can not determine if they reside on the same chromosome or are on different chromosomes.   Wikipedia gives an  and also remarks that (d is physical distance):     The probability of recombination is approximately d/100 for small values of d and approaches 50% as d goes to infinity.   However, what happens if the linkage is greater than 50 cM? It seems like classical experiments aside, such a situation is possible in reality.  For instance, , and . Let's take two genes on Chr.IV:    is at about +631 kb.  is at about +1370 kb.   The physical distance between these is about 734 kb. According to the cM/kbp value, the centimorgan distance is $734\text{ kb} \cdot 0.31\text{ cM}/\text{kb} = 228 \text{ cM}$ . How to interpret this 228?  If I mate a haploid yeast which is DNF2 TOM1 with another which is dnf2 tom1 (lowercase indicating minor allele rather than deletion), then sporulate them, what is the chance of getting spores with DNF2 tom1 or dnf2 TOM1 genotypes?  I realize that the cM/kbp value is only a simplification and that in practice linkage is a more complex phenomenon. Nevertheless, it seems plausible that several crossover events can happen on Chr.IV since it is so big. This includes the possibility of 1, 3, 5 and more crossovers which would create a hybrid (assuming they all happen between these two loci), as well as the possibility of 2, 4, etc crossovers which would create a non-hybrid spore (at least as far as our chosen markers are concerned). I want to use the "BackSpace" button as a shortcut key on nautilus. I want to return to the previous folder whilst browsing the Home folder, just like windows 7.   Also I want the same thing while using Firefox, In a way that when I press the shortcut key I will return to the previous web page. How do I install the Intel Graphics driver in my system? Can someone help me out and explain or point me in the right direction on how to check video drivers and see if my video card running okay?  I had 10.04 installed on my Thinkpad r61 with Compiz Manager and life was great, until the machine took water damage. I bought an ASUS (X54H) since and am trying out 12.04, but the desktop just doesn't look right.  I always struggled with video driver installation. There are no proprietary drivers available in the hardware manager.  When I run lspci | grep VGA:  00:02.0 VGA compatible controller: Intel Corporation 2nd Generation Core Processor Family Integrated Graphics Controller (rev 09)   I will greatly appreciate your help. I want to use Linux more, but like I said video drivers appear to be my biggest concern. I have also tried 12.04 on my desktop PC, but again failed to configure video card, so switched back to Windows 7. Consider a database table holding names, with three rows:  Peter Paul Mary   Is there an easy way to turn this into a single string of Peter, Paul, Mary? I have a problem with the interpretation of a test result in which the p-value is 8.6e-28. How should it be interpreted? What is the e for?  Specifically, I used White's Test for the homoscedasticity assumption of a linear regression:  White's general test statistic :  150.0535  Chi-sq( 9)  P-value =  8.6e-28 Problem with IPv6 sudo apt-get update/upgrade I am using Ubuntu 16.04 Beta 2 at the moment and every time I try to update at home via sudo apt-get update or upgrade trough sudo apt-get upgrade I wont get any feedback. Even installing of anything will have the same result. It will just "hang up" on 0% while fetching data.  $ sudo apt-get update [sudo] password for niclas:  0% [Connecting to archive.ubuntu.com (2001:67c:1560:8001::11)] [Connecting to archive.canonical.com (2001:67c:1360:8c01::1b)]   I have no idea how to solve the problem. I already tried to ping6 archive.ubuntu.com with 100% packet loss while just ping wont have any loss.  I changed the DNS-Server (IPv4 &amp; IPv6) to Google's one no solution either. Partial vs. Full Volume Wort Boil The starting position for most new extract brewers is to use a partial boil set up.  Defined as using a pot smaller than the intended batch size, all the ingredients are boiled in a smaller volume.  Then the wort is diluted to the final desired batch size.  (Often the dilution step is used as part of a cooling step)  What are the Ups and Downs of both Partial vs. Full boils? How do I programmatically log in a user with a POST request? Do anyone have a working REST Login on Drupal 8? This is what I tried. POST /user/login HTTP/1.1 Host: 8.d8.local Content-Type: application/x-www-form-urlencoded Accept-Encoding: http://nikhilmohan.in Cache-Control: no-cache  name=test&amp;pass=password&amp;form_id=user_login_form  It returns me HTML instead of JSON. Breadcrumb hyperlink header What's the best way to take a segment out of a text file? What's a good way of extracting say, lines 20 -45 out of a huge text file. Non-interactively of course! As much as I like having the hardcover core rule-books in front of me, usually I read or reference most information for RPGs on my computer.  Unfortunately, my google-fu has failed me at finding places to buy a Player's Handbook PDF, or even whether said PDF exists at all.  My question is this: Is there any place I can go to legally buy a D&amp;D 5e core rulebook PDF?  I'm not looking for the starter set or basic rules, since those are incomplete (and free), I'm just looking for the core books in PDF format. Get selected value in dropdown list using JavaScript After beating the final boss, can I return to the original castle? Introduction to AdS/CFT