How to calculate a spectrum
Show that $R/I\otimes_R R/J\cong R/(I+J)$
Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable? Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)&gt;c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable?  There are a number of equivalent definitions for the measurability of a function and the most obvious one would be to show that $\{x\in E : f(x)&gt;c\}$ is measurable for all $c\in\mathbb{R}$.  Thus my strategy has been to consider an arbitrary irrational $y$ and use the density of the rationals in the reals to show that there exists some open set $O$ such that $m^*(O-\{x\in E : f(x)&gt;y\})&lt; \varepsilon$.  I would do this by choosing some rational $q&lt;y$, close enough to $y$, such that the set of values in $E$ which get sent under $f$ to values strictly greater than $q$ and less than or equal to $y$ is small enough.  But that doesn't in general seem possible since I could keep sending intervals of some fixed positive length to smaller and smaller regions just beneath $y$.  However conversely, such a function would not be enough for a counter example, since finding such a function only rules out this strategy, but doesn't necessarily entail non-measurability.  Is there a better definition of the measurability of a set I should use? or does the statement actually not imply measurability?  Thanks. Secretary problem - why is the optimal solution optimal? I have read about this problem:    But I want to see how it is proven that the "optimal" solution is indeed optimal. I understand how to prove that if the optimal solution is of the form "wait for $t$ candidates and then choose the next best one" then $t=n/e$ is optimal; but why is the best strategy of that form in the first place?  A complete proof is not required - a reference to a good text discussing this is good as well. I am trying to create the following trigger:      CREATE TRIGGER update_customers AFTER INSERT ON wp_usermeta     FOR EACH ROW      BEGIN             if new.wp_metakey = first_name then insert into test_customer(first_name) values (new.wp_meta_value);     END   I keep getting a 1064 error that says, "You have an error in your SQL syntax; check the manual that corresopons to your MySQL server version for the right syntaxt to use near " at line 5"  What am I doing wrong? A regular challenge that I face is balancing two (or more) logos. Whether they need to be equal or one needs to be given priority. Some logos however might be squares, others rectangles, circles, etc...   As a quick sample I snagged a nice high resolution photo from  and pretend its going to be an advertisement that needs to show some footwear brand logos on it. In this case I used Adidas and Saucony.  This is certainly not balanced even though they're height is the same:    Likewise this is not balanced even though they're width is the same:    So then the question becomes when are they in balance, is this balance:    Maybe a little more adjustment, is this balance:    And in these examples I'm only working with two Black Logos. Factor in color and it can get even more difficult to balance. Another issue being when there's tiny text as part of the logo.  So the question is, other than estimating / eye-balling it, what tips, methods or even tools are there to balance different sized logos? I've got a recipe that calls for some buttermilk, but none on-hand. Is there some way I could use some common ingredient(s) as a substitute? Which textbook is good for introductory group theory? Sorry for asking such a basic question - but is there an actual term for the numbers that appear before and after the decimal point?  Example:  25.18  I know the 1 is in the tenths position, the 8 is in the hundredths position. are there singular terms which apply to all of the numbers that appear both before and after the decimal point?  I'm building a billing system, storing dollars and cents as integers (in separate columns), but for other currencies "dollars" and "cents" are not the correct terms = ).  Thanks! Noticing that the "Human Verification" image did not show, I found that the src, , shows that sstatic.net is giving an SSL vertificate for *.stackexchange.com.. What is dependency injection? $\mathbb{Q}^{\times}$ is the group of rational number without $0$ under multiplication, and $\mathbb{Z}^{n}$ is the free abelian group of rank $n$. Show that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$. I tried to assume that there is an isomorphism and to get a contradiction.. didn't succeed How do you set, clear, and toggle a single bit? How do you set, clear, and toggle a bit? I have a Canon 550D with 70-200 f4 and 50mm prime lenses. Although the camera has built-in 9 focal points, I sometimes face a strange problem. Here's a sample photo:    In reference to the above pic, you can see that both of the faces are NOT in focus. I used 70-200 f4 for this shot at max aperture. Still, I couldn't get both of them in focus. Why? I also try decreasing the aperture to attain a greater DoF but still, I couldn't get both of them into focus.  Another thing: Although the cam has 9 focal points, why can't I use all of them to focus, say, 9 points in a portrait shot? In manual mode, it has the option to select one of the 9 focal points. And in auto mode, it sometimes select more than one focal points.  This is just an example image. I've faced this problem zillions of times. How to download Ubuntu over an unreliable connection How can I install a Nautilus script? For an architectural visualization, I would like to use MakeHuman models, but:   Only get them to render white with a black outline (on reflection surface too, if possible). Interact with light like normal human bodies in the scene (cast shadows, etc).   I can do the outline in freestyle, but any in-Cycles solution would be appreciated (OSL?) I know that there are questions asking like "intersection of a infinite collection of sets" and I can understand that the answer for that one is a null set, but I got a question here, in which all sets are finite and nonempty. Please take a look at the pic below.  It's a True/False question.  What I'm confused about is that the "infinity" sign in the interception. Since all sets are finite, does it mean that there are many sets in the chain are the same? (because the symbol used is for subsets not for proper subsets). Please give me some hints how I should think about this question.  Thank you Say person A and person B are having a discussion/argument.   Person A tells person B that his views are arrogant, or closed-minded, or some other derogatory adjective. But in fact it is person A's views that are arrogant, closed-minded, etc. Person B's views are usually the exact opposite of what person A is accusing him of.  This sort of situation happens a lot when discussing politics or religion, for example.  What do you call person A, for calling person B something that he himself is?  I guess, for the most part, person A doesn't actually realize that he is the one that the adjective applies to, and not person B. Maybe there is a different name if person A knows the adjective also applies to him. Can I preemptively pay my credit card bill before a charge is posted? How do I create a Java string from the contents of a file? How to integrate OpenCV Manager in Android App