Question about union of cartesian product of sets
How to show $A\times (B\cup C) = (A\times B)\cup (A\times C)$
Can I completely disable Cortana on Windows 10? Name of the book/series based on magicians who tap into computer grid? Cartesian product of two $\sigma$-algebras is generally not a $\sigma$-algebra? When reading the book Measures, Integrals and Martingales written by R.L. Schilling, I saw a statement as below: $$\mathcal{A} \times \mathcal{B}:= \{A \times B: A\in \mathcal{A}, B\in \mathcal{B}\}$$ where $\mathcal{A}$ and $\mathcal{B}$ are $\sigma$-algebras, is not $\sigma$-algebra in general.  However, I cannot construct a counterexample. Could anyone offer help here?  Kind regards Can any one help me with this?  Determine all integer triples $(x,y,z)$ such that $1 ≤ x ≤ y ≤ z$ and $x + y + z + xy + yz + xz = xyz − 1$.  I thought of Vieta's formula but don't let me lead you into a dead end. Nonzero nilpotent elements in Zn So I've been stuck on this proof for the past threes days now. I keep going back and forth and making no progress.     Let $n\in \mathbb{Z}_{\geq 2}$. Show that $\mathbb{Z}/n\mathbb{Z}$ has a nonzero nilpotent element iff there is a prime $p$ s.t. $p^2$ divides $n$.   $(\Rightarrow)$ I know $a$ can be factored into distinct primes, so $a=p_1^{e_1}\cdot ...\cdot p_m^{e_m}$ then $a^k=p_1^{e_1k}\cdot ...\cdot p_m^{e_mk}$.  Also since $n|a^k$ so $a^k=nd$ for some $d\in \mathbb{Z}$.  $a^k=p_1^{e_1k}\cdot ...\cdot p_m^{e_mk}=nd$. So $nd$ has that distinct prime factorization.  Then I get lost on what to do. Meaning of "stackexchange" What does stackexchange mean? I have checked out this  to read about the Stack Exchange Network. Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable. Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable.  Can anyone help how can I solve the above problem? An irreducible polynomial of degree coprime to the degree of an extension is irreducible over this extension I'm having a hard time showing this:     If $K$ is an extension of $\mathbb{Q}$ with degree $m$ and $f(x)$ an irreducible polynomial over the rationals with degree $n$, such that $\gcd(m, n)=1$, then $f(x)$ is irreducible over $K$.   I have tried it by writing $f(x)=a(x)b(x)$ and then looking at the coefficients of those polynomials (some of them must belong to $K-\mathbb{Q}$ which could possibly result in a contradiction) to no success. I have no idea where to use the hypothesis of m and n being relatively prime.  Any help would be appreciated. How to access "The grimm Troupe" and "Hidden dreams" content in Hollow Knight? In these two official announcements from Team Cherry, it's said:       Grimm’s quest, and all other new features can be accessed at any point   in your Hollow Knight journey.        Hollow Knight: Hidden Dreams content is accessible at any point in the   game, whether starting fresh or scouring for completion.   I still haven't find any way to start the quests from the menus or during beginning of the game. Some sources say the content can only be accessed late in the game from specific locations.  So, is Team Cherry simply lying in our faces? How can I SELECT rows with MAX(Column value), PARTITION by another column in MYSQL? My table is:     id home datetime player resource     1 10 04/03/2009 john 399   2 11 04/03/2009 juliet 244   5 12 04/03/2009 borat 555   3 10 03/03/2009 john 300   4 11 03/03/2009 juliet 200   6 12 03/03/2009 borat 500   7 13 24/12/2008 borat 600   8 13 01/01/2009 borat 700     I need to select each distinct home holding the maximum value of datetime. Result would be:     id home datetime player resource     1 10 04/03/2009 john 399   2 11 04/03/2009 juliet 244   5 12 04/03/2009 borat 555   8 13 01/01/2009 borat 700     I have tried: -- 1 ..by the MySQL manual:   SELECT DISTINCT   home,   id,   datetime AS dt,   player,   resource FROM topten t1 WHERE datetime = (SELECT   MAX(t2.datetime) FROM topten t2 GROUP BY home) GROUP BY datetime ORDER BY datetime DESC  Doesn't work. Result-set has 130 rows although database holds 187. Result includes some duplicates of home. -- 2 ..join  SELECT   s1.id,   s1.home,   s1.datetime,   s1.player,   s1.resource FROM topten s1 JOIN (SELECT   id,   MAX(datetime) AS dt FROM topten GROUP BY id) AS s2   ON s1.id = s2.id ORDER BY datetime   Nope. Gives all the records. -- 3 ..something exotic:   With various results. Proof of the equality $\sum\limits_{k=1}^{\infty} \frac{k^2}{2^k} = 6$ Show that for $k$ running over positive integers  $$ \sum_{k=1}^\infty \frac{k^2}{2^k}=6 .$$ We can use finite calculus. I just got a question in the review queue that, though poor, I thought was potentially salvageable via editing.  However when I clicked the Edit button I got the deleted post error.  Since the post was gone, I clicked I'm Done (what is there do do with a deleted post?) only to get the review audit failed message instead.  The current behavior is also problematic in that, now that I know about it, it's a risk-free way to check if a question is an audit and not an actual review. That allows me or anyone else who knows about it to detect audits without having to recognize that the question is bad. As the question states... In Windows 7 with a single screen, you can drag a window to the right and a window to the left to maximize them both at half the screen, letting you quickly and neatly see two windows simultaneously.  If I have 2 monitors, is there a way to do this with 4 windows, where each monitor gets 2 windows split across each screen? Create an entry in index without creating a Chapter or Section title There are 100 ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends. What is the expected number of loops at the end of the process? How do injuries work with Polyjuice Potion? We know that if you (using Polyjuice), take the form of someone who had been injured, you assume that injured form (e.g. Barty/Moody). If you are injured, presumably you would be uninjured while under Polyjuice. My question is, is this in real-time? That is, if you take a Polyjuice, and while you're still under its effects, the person being transformed into gets injured (or even dies), will that injury be transferred onto you? And conversely, if you get injured while in someone else's body, will that transfer over to your real body after the Polyjuice wears off, will it disappear, or will it go to the person you turned into? Recently I created a website with hundreds of thousands of unique pages. A week ago, Google had indexed about 250,000 pages. It has now decreased to 90k URLs (using a site:domain.com query).   What should I do to have Google index all the pages again? Why did all these pages get de-indexed from Google?. How can I type a Unicode character (for example, em-dash —?) The title pretty much says it all: how can I type Unicode characters - specifically the em-dash —? Use LaTeX to produce Epub I'm asking here a question which was asked on the italian TeX group forum.  Is there a way to produce Epub output automatically from a LaTeX source?  I should remark that Epub is just HTML+a subset of CSS+some XML for metadata, packed in a ZIP file. So one can settle with the existing tools for HTML conversion. Still something better could be done for Epub output.  In the first place, Epub accepts only a subset of CSS. Then it would be nice to produce the file with the metadata from the LaTeX source too. Finally, I'm not sure how it works, but Epub files allow for automatic hyphenation of words. By this I mean that hyphenation can be specified in the Epub file itself, so that a reader will not need to know hyphenation rules. Note that since Epub files are resizable, all possible points of hyphenation must be specified. Fastest way to try all passwords Suppose you have a computer with a password of length $k$ in an alphabet of $n$ letters.  You can write an arbitrarly long word and the computer will try all the subwords of $k$ consecutive letters. What is the smallest word that contains all combinations of $k$ letters as subword? (i.e. the fastest way to hack the computer :) )  The smallest word that contains $n^k$ subwords of size $k$ has length $k-1+n^k$ and based on some easy cases, we would like to prove that it is in fact possible to find a word of such length that contains all possible passwords. The problem can be translated into a problem in graph theory, by taking as vertices all words of length $k$.  We tried $k=2$, where you can prove the conjecture by induction. For $n=2$ and small $k$ it also works.