As the title suggests I'm trying to determine if the following example is true or false. AND if it is false, to provide a counter example.   This is the example:  (A x B) $\cup$ (C x D) = (A $\cup$ C) x (B $\cup$ D)  Now here is what I have attempted:  For the L.H.S. (A x B) $\cup$ (C x D)  (x,y) $\in$ (A x B) OR (x,y) $\in$ (C x D) $\Leftarrow\Rightarrow$ x $\in$ A OR y $\in$ B OR x $\in$ C OR y $\in$ D  For the R.H.S. (A $\cup$ C) x (B $\cup$ D)  x $\in$ (A $\cup$ C) OR y $\in$ (B $\cup$ D) $\Leftarrow\Rightarrow$ (x,y) $\in$(A  $\cup$ C) x (B $\cup$ D)   To be honest I'm not really sure if what I have done is even correct. Given my working I guess it would be a FALSE statement?  But a counter example? I just need the expertise on this forum to help me fill a few gaps.   Thanks so much!  EDIT: It is slightly different from the linked solution as they asked $(A \times B) \cup (C \times D) = (A \cup B) \times (C \cup D)$  And I asked  (A x B) $\cup$ (C x D) = (A $\cup$ C) x (B $\cup$ D)  I realise that's not much of a difference but just thought for peoples future reference.
I think I have managed to work this out, however there is no solution provided with the question so I thought it best to check.   My working:  Let $(x,y) \in (A \times B) \cup (C \times D)$  $\Rightarrow x \in (A$ or $C)$ and $y \in (B$ or $D)$  $\Rightarrow (x, y) \in (A \cup C) \times (B \cup D)$  $\Rightarrow (A \times B) \cup (C \times D) \subseteq (A \cup C) \times (B \cup D)$  Now let $(x,y) \in (A \cup C) \times (B \cup D)$  $\Rightarrow x \in (A$ or $C) $ and $y \in (B$ or $D)$  $\Rightarrow (x, y ) \in (A \times B) \cup (A \times D) \cup (C \times B) \cup (C \times D)$  $\Rightarrow (A \cup C) \times (B \cup D) \nsubseteq (A \times B) \cup (C \times D)$  $\Rightarrow (A \times B) \cup (C \times D) \not= (A \cup C) \times (B \cup D)$  Is this correct?  Thank you.
iOS 11-13 - unable to paste into comments I’m able to copy-paste from any application to any application. I’m able to copy from a comment or from a question or from an answer. I’m able to paste to a question or answer. However, if I attempt to paste into a comment, nothing happens. I am able to subsequently paste into another application, confirming that the text is actually on the clipboard.   However, copying SE links will paste them successfully, but all other text - including other links - still fails. This leads me to believe that it’s a SE issue and not an Apple one.   This has been an issue since 11.0 beta and persists through iOS 13 release. Why does the set of all singleton sets not exist? How to change SQL Server collation If i try to update node revision like this:  $node = node_revision_load(59); $node-&gt;set('publish_on', null); $node-&gt;save();   or simple save:  $node = node_revision_load(59); $node-&gt;save();   I have a PHP memory error, (i have 1gb allocated) I am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all $n\in\Bbb{N}$    To prove is that both sequences converge and that they have the same limit. I don't know how to show this. I have spent 2 hours on this, no sign of success How can I prevent iPhoto/Photos/Image Capture from launching everytime I plug in my iPad/iPhone? What is the black thing on the circuit What is the black thing on the below circuit?   Is it used to hide the IC (or other components) that is used on the circuit?   What material is it? Let $X,Y,Z$ Banach spaces and $A:X\rightarrow Y$ and $B:Y\rightarrow Z$ linear maps with $B$ bounded and injective and $BA$ bounded. Prove that $A$ is bounded as well.   If I knew that $B(Y)$ is closed I'd have a bounded linear map $B^{-1}:B(Y)\rightarrow Y$ by the bounded inverse theorem. Therefore $A=B^{-1}BA$ is bounded. How to prove the claim if $B(Y)$ is not closed. I have been facing hard time understanding meaning of "random sample" as well as "iid random variable". I tried to find out the meaning from several sources, but just got more and more confused. I am posting here what I tried and got to know:  Degroot's Probability &amp; Statistics says:     Random Samples / i.i.d. / Sample Size : Consider a given probability distribution on the real line that can be represented by either a p.f. or a p.d.f. $f$. It is said that $n$ random variables $X_1 , . . . , X_n$ form a random sample from this distribution if these random variables are independent and the marginal p.f. or p.d.f. of each of them is $f$. Such random variables are also said to be independent and identically distributed, abbreviated i.i.d. We refer to the number n of random variables as the sample size.   But one of the other statistics book I have says:       In a Random Sampling, we guarantee that every individual unit in the population gets an equal chance(probability) of being selected.   So, I have a feeling that i.i.d.s are elements that construct random sample, and the procedure to have random sample is random sampling. Am I right?   P.S.: I am very confused about this topic, so I will appreciate elaborate reply. Thanks. What is the difference between “say” and “tell”? I really get confused when to use say and when to use tell. Which is appropriate in the following:   What did he tell? / What did he say? What are you saying? / What are you telling? How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer? $\forall m,n\in\mathbb Z$ , $m\ge1$ and $n\ge1$ how to prove that  $$\frac{(mn)!}{m!(n!)^m}$$ is an integer?  Thanks in advance. Composition of two Riemann integrable functions 32-Bit UEFI Boot Support Differences in meaning when the verb tense changes (headlines) What is the difference between the examples below?  Generally, in a newspaper, news is based on past tense (things which have already happened).   Then why are headlines written in ways that says: He Kills (shouldn't it be: He Killed)?  For example:  Parliament passes bill vs. Parliament passed bill  Also  He kills his friend vs. He killed his friend  Please help. Programmatically obtain the phone number of the Android phone Activity Monitor in Mavericks "Show Memory Usage" dock icon gone? I just purchased a new MacBook Pro with Mavericks installed and noticed that Activity Monitor doesn't have the "Show Memory Usage" dock icon option anymore.  I really liked this option since I used it to prevent disk swapping by closing applications when I had too many open.  What gives? Detect if an outgoing call has been answered Cannot find Background Images feature in Blender 2.8 Appending to the end of a file with NSMutableString Drush from GitHub stopped working I have a  that has been working since it moved there.  I did a git pull this morning, and am now getting errors:     $ drush      Warning: require(/home/mpd/drush/vendor/autoload.php): failed to open   stream: No such file or directory in   /home/mpd/drush/includes/bootstrap.inc on line 1007      Fatal error: require(): Failed opening required   '/home/mpd/drush/vendor/autoload.php' (include_path='.:') in   /home/mpd/drush/includes/bootstrap.inc on line 1007   How can I fix this?