How to show that $A+ (B\cap C)=(A + B)\cap C$ if  $M$ is an $R$-module and $A,B,C$ are submodules of $M$ and $B$ is a submodule of $C$?
How to prove that if A, B, C are submodules of M module then $$ A \cap (B+C) = (A\cap B)+C \iff C \subseteq A $$  Any help would be appreciated!
Sarrus Rule with LaTeX Is there a way to create a new command with 9 arguments ({a_{11}},{a_{12}},{a_{13}},{a_{21}},{a_{22}},{a_{23}},{a_{31}},{a_{32}},{a_{33}}) that has as output a matrix with the Sarrus Rule? For instance: to obtain     just use the command that we will call sarrus:  \sarrus{-k}{k-1}{1}{0}{k-1}{k}{2}{0}{1}.   If it is possible i want to use the command sarrus in math mode just like matrix for istance. Easiest way to import access db (.accdb) into SQL Server 2016? I have a .accfb file from an access 2016 database and am trying to import this database into a SQL Server 2016 developer edition instance.   While I found posts online discussing using import/export data tools in SQL Server and also migration assistant for access, neither have worked out for me.   Does anyone know how to easily move a database from access 2016 to sql server 2016? More specifically an "accdb" file type?  All suggestions are greatly appreciated! Does commutativity imply Associativity? Does commutativity imply associativity? I'm asking this because I was trying to think of structures that are commutative but non-associative but couldn't come up with any. Are there any such examples?  NOTE: I wasn't sure how to tag this so feel free to retag it. Optimal way to ignore duplicate inserts? Background  This problem relates to ignoring duplicate inserts using PostgreSQL 9.2 or greater. The reason I ask is because of this code:    -- Ignores duplicates.   INSERT INTO     db_table (tbl_column_1, tbl_column_2)   VALUES (     SELECT       unnseted_column,       param_association     FROM       unnest( param_array_ids ) AS unnested_column   );   The code is unencumbered by checks for existing values. (In this particular situation, the user does not care about errors from inserting duplicates -- the insertion should "just work".) Adding code in this situation to explicitly test for duplicates imparts complications.  Problem  In PostgreSQL, I have found a few ways to ignore duplicate inserts.  Ignore Duplicates #1  Create a transaction that catches unique constraint violations, taking no action:    BEGIN     INSERT INTO db_table (tbl_column) VALUES (v_tbl_column);   EXCEPTION WHEN unique_violation THEN     -- Ignore duplicate inserts.   END;   Ignore Duplicates #2  Create a rule to ignore duplicates on a given table:  CREATE OR REPLACE RULE db_table_ignore_duplicate_inserts AS     ON INSERT TO db_table    WHERE (EXISTS ( SELECT 1            FROM db_table           WHERE db_table.tbl_column = NEW.tbl_column)) DO INSTEAD NOTHING;   Questions  My questions are mostly academic:   What method is most efficient? What method is most maintainable, and why? What is the standard way to ignore insert duplication errors with PostgreSQL? Is there a technically more efficient way to ignore duplicate inserts; if so, what is it?   Thank you! Is it possible to tell when statistics were last accessed by a query using T-SQL? I'm trying to ascertain what statistics are involved in a query, I know I can check the visual query plan but I wondered whether there was a table (or tables) with this information. It is a common assumption (and, as far as historical stock market data goes, also proven) that the stock market is always growing if you enlarge the time window long enough . In other words: The stock market always goes up in the long term. Is, therefore, the conclusion valid that you can totally invest as much as you want at any point in time you want and you can never go wrong, if  You diversify strong enough (invest &quot;in the world&quot;) You do not need to urgently sell your assets at any point in time  If yes, all that strategies that rely also on diversified, long-term investments (Dollar-Cost-Averaging &amp; Rebalancing, .. what others do you know?) are just to optimize the yiels, but you'd make profit even without those strategies. Is that correct? Would this hypothetical rule change ever matter? One of  is     104.3f If a player would both win and lose the game simultaneously, he or she loses the game.   . However, it seems to me like it "ought to be" the case that if there are only 2 players and  [A loses] ​ [B loses] ​ [B wins]  would occur simultaneously, then B should win, since  win-and-lose-simultaneously should be better than just-lose.  For that reason, it seems like the     Proposed Alternative : ​ ​ ​ If, simultaneously, all players would lose the game and      some players would win the game, then the players who would not win the game      lose the game, and the game is a draw between the remaining players. ​ Otherwise,      if a player would both win and lose the game simultaneously, he or she loses the game.   would make more sense. ​ However, I can't think of any situations in which that would matter.  Are there any possible situations in which my proposed alternative  would yield a different result from the actual rule? I am lost in normalizing, could anyone guide me please.  I have a minimum and maximum values, say -23.89 and 7.54990767, respectively.  If I get a value of 5.6878 how can I scale this value on a scale of 0 to 1. $\mathbb Z_{p^n}$ is not the direct product of any family of its proper subgroups I'm trying to solve this question of Hungerford's Algebra book:     $S_3$ is not the direct product of any family of its proper subgroups.   The same is true of $\mathbb Z_{p^n}$.   The first claim is easy: we note that every subgroup of $S_3$ is of order $2$ or $3$ by Lagrange, since its subgroups are of prime order, they are cyclic, then abelian, but $S_3$ is not abelian, contradiction because $S_3$ is not abelian while the direct product of abelian groups has to be abelian.  My problem is with the second claim, I need help. Complex zeros of the polynomials $\sum_{k=0}^{n} z^k/k!$, inside balls Density function for RV The density function for a random variable X is given in terms of a constant $c$.  Find the value of $c$.  What is the corresponding distribution function?  Sketch both the density and the distribution functions. Finally, find the probabilities.  5.1 $f(x)=0$ for $x&lt;0$ and $f(x)=\frac{c}{(x+1)^4}$ for $0&gt;x$   $P(X&gt;4)$  $P(X&lt;2)$  $P(1\le\ X&lt;3)$   I get $c=3$ by taking the integral, and then I am a little confused as to why  $F(x)=1-\frac{1}{(x+1)^3}$  Is that because in solving for $c$, I ended up with $1=\frac{c}{3(x+1)^3}$, so substituting c in I get $1=\frac{1}{(x+1)^3}$?  And, then  $P(X&gt;4)=1-P(X\le\ 4)$ $=1-(1-{1}{(1+4)^3})$ $=1/125$  $P(X&lt;2)=1-{1}{(1+2)^3}$ $=26/27$  I guess I am a little confused about what $F(x)$ and $f(x)$ actually MEAN? And thus that leads to my confusion about what the differences between $&gt;$ and $&lt;$ and greater than or equal to are, etc etc in terms of the equations. The ones above I solved by comparing to the book but I still don't understand what they mean really.  So as a result, I don't really get how to solve $P(1\le\ X&lt;3)$.  5.2 $f(x)=ce^x$ for $x&lt;0$ and $f(x)=ce^{-x}$ for $0&gt;x$  5.3 $f(x)=0$ for $x&lt;-1$ or $x&gt;2$ and $f(x)=cx^2$ for $-1&lt;x&lt;2$ Why have view counters been modified? Which is correct: I'll be moving next month or I'll be shifting next month? For changing one's home from one place to another, I've heard people in western part of the world using the sentence:      I'll be moving next month.   In India, even in the English news channels, I've heard many people using the phrase:     I'll be shifting next month.   Which one is the correct usage? Or are both of them correct and the difference in usage is regional? As a  developer I'm used to running through constructors:    class Test {     public Test() {         DoSomething();     }      public Test(int count) : this() {         DoSomethingWithCount(count);     }      public Test(int count, string name) : this(count) {         DoSomethingWithName(name);     } }   Is there a way to do this in C++?  I tried calling the Class name and using the 'this' keyword, but both fail. Which of the following is a correct sentence:   Does anyone know where's Linda?  Does anyone know where Linda is?    Does anyone know why and how to use above usage if #2 is correct. Also please let me know this kind of specific structures. I have a string variable that represents the name of a custom class. Example:   string s = "Customer";   I will need to create an arraylist of customers. So, the syntax needed is:  List&lt;Customer&gt; cust = new ..    How do I convert the string s to be able to create this arraylist on runtime? Training involved to ride 175 miles (281 km) in a day I want to be able to ride 175 miles (281 km) in a day (to do the ). I used to be able to do about 60 miles (96.5 km), only once hitting 80 miles (128 km), but I'm out of shape now. Wondering how I can systematically train to get to 175 miles at an average of 15 mph (24 km/h) Are there guides out there to help with this?  What's some advice?  [update] The  will happen each year, so my intention isn't to ride this year, rather what do I need to do to ride it in some future year? How does stuff glow in the dark? Square roots -- positive and negative It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.  I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\sqrt{9}$, does one assume that to mean only the positive root? And when simply talking about the square root of a number in general, would one be referring to both roots or just the positive one, when neither is specified? DNS subdomain (child) NS records If we have example.com we would setup things something like this: $ORIGIN example.com. @  1D  IN  SOA ns1.example.com. hostmaster.example.com. (                   2002022401 ; serial                   3H ; refresh                   15 ; retry                   1w ; expire                   3h ; nxdomain ttl                  )           IN  NS     ns1.example.com.           IN  NS     ns2.example.com.  since in the parent zone .com. there is already NS record for the example.com. child zone and A glue record for the ns1.example.com. for what purpose do we state NS records again in the child zone? and also, do we need to state A record for the name servers in the child zone itself?