How can I prove that $$(A\times B)\cap (C\times D)=(A\cap C)\times (B\cap D)?$$
How to prove $( A \times B ) \cap ( C \times D ) = ( A \cap C ) \times ( B \cap D )$ ?
I thought of this question right after I posted a tweet about a service upgrading me to a free student account since I am in college. I said "That really helps a broke college student out." I actually paused for a second while I was writing that to decide if I should say     That really helps out a broke college student.   or     That really helps a broke college student out.   Are there any prescriptive rules about splitting phrasal verbs like this? I know this breaks the "don't put a preposition at the end of a sentence" rule, but that "rule" has an exception for phrasal verbs.  To clarify:   Is there any rule that says phrasal verbs can't be split, even if it is just an imposed, prescriptive rule? Does splitting a phrasal verb to put a preposition at the end of a sentence fall under the phrasal verb exception to the prepositions at the end of a sentence rule? Does the alternating sum of prime reciprocals converge? $$\sum_{n = 1}^\infty \frac{(-1)^n}{p_n}$$ where $p_n$ is the $n$th prime.  I have computed this to 10000 rather than infinity. My results suggest that convergence does happen but it's very slow. But I can't even be sure about the first few digits: $-0.26959$?  I have looked at the "questions that may already have your answer" and some of the "similar questions," but they involve somewhat different formulas. I just started playing Dark Souls 2 without playing the first one.  Since I don't see any sell option in any merchant: Can I sell my old-unused-equipment for souls? If I cannot sell it, does it have any value to be keep until later? I am having some issues when rendering an image, my rendering screen is turning into grey even when making a full render. What am I doing wrong? Groups with Same Number Of Elements and Subgroups How do I return the response from an asynchronous call? reference variable In some text it is given that we can't assign constant values to a reference variable. When I executed such a program I could do it. Is there any condition we can't assign a constant value to a reference variable? If light is an electric and (magnetic field), how can it be absorbed? I was wondering how light or any electromagnetic radiation can be "absorbed" if it consists of electric and magnetic fields. For example if there is a charge at point A, and the  light reaches point A, it will cause that charge to move depending on the value of the electric field. However, why wouldn't this electric field continue on and influence others particles everywhere else? For example, if you have a static charge creating an electric field, then if you put a charge at point A it will feel its effects and then if you bring a charge at another location B it will also feel its effects, namely it will not have been absorbed. Please shed some light on this. Thanks  edit: Does this mean the electron that absorbs this light moves in such a way that it creates an electric field of its own canceling the initial one? Or am I way off? Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$ Let $n$ be a positive natural number, $n\ge 2$. Then $\displaystyle\sum_{k=1}^n \frac{1}{k^2} \lt 2 - \frac{1}{n}.$   The basis step was easy but could someone give me a hint in the right direction as to how to do the induction step?  I tried this:  $\displaystyle\sum_{k=1}^k \frac{1}{k^2} + \frac{1}{(k + 1)^2} \lt 2 - \frac{1}{k + 1}$  But it's getting me nowhere or I am doing something wrong. I am no expert so a clear explanation would be appreciated. Thanks. Proof of an elementary property of Projection Operators I'm asked to show the following:  Let $X$ be a linear space, and let $P : X \rightarrow X$ be a projection operator. Restricted to the linear space $range(P)$, the projection $P$ is the identity operator, that is, $Px = x$ for all $x \in range(P)$.  If anyone could offer a hint for how to show this. I'm rusty with linear algebra so don't really know what I'm supposed to bring in outside of what's specified in the lemma/definition of projection. What is so special about the range of P that would make this so? How can I install Skype on 32-bit Ubuntu 14.04? Why directory with large amounts of entries does not shrink in size after entries are removed? Close vote notice: While the linked duplicate asks same question, the comments and the only answer do not provide an answer which draws from authoritative source. The , does in fact provide the answer I've been seeking. Thus the proposed duplicate isn't helpful, however Thomas's answer here is proper.  According to the  and personal account , directories that have large amount of entries get to a size over 4096 bytes (which can be seen with ls -l output), but once the entries are removed - the number never goes down.   The question is "why" ? Is it due to how ext4 filesystem configured to retain directory metadata ? Obviously removing the directory and recreating it isn't a solution, since it deletes original inode and creates a new one. What can be done to decrease the number manually ? Given: a = 1 b = 10 c = 100  How do I display a leading zero for all numbers with less than two digits? This is the output I'm expecting: 01 10 100 Are there surface-mount pin header footprints for KiCAD? I'm designing a small module that breaks out a DIP SPI EEPROM to a 2x5 pin header socket(more info here: ). I want to mount the socket on the end of the board, with the pins lying parallel to the board.  I think two 5x1 surface-mount header footprints, on both sides of the board right near the end would do the job. Trouble is, KiCAD doesn't come with such a footprint(if it does, I haven't found it yet). Where might I find one? If one doesn't exist, that's OK; I'll have a shot at making my own. Cutting strands of wire to fit plug I am a hvac tech, but my question is I have 2awg stranded and my plug is 6awg, will it matter much if I cut some strands to fit my plug? the plug is 60amp 250vac for a piece of kitchen equipment if the matters. Is using Gravatar a security risk? This has been . It is unfortunate the accepted answer is so far from reality.   There are some facts:    's MD5 hash is an unsalted MD5 hash of an email address. A 3rd party could harvest every MD5 hash for every user's email on Stack Overflow.  The 3rd party could store all the hashes in a lookup table Email addresses are usually of a very particular format The attacker could start with a brute force of @gmail.com followed by @yahoo.com, etc. The attacker has the advantage that he/she could attack a huge set of emails in one go. So instead of needing to brute force every email individually the attacker can brute force in bulk.   The algorithm is quite simple, calculate a "random" MD5, look it up in the lookup table.       Is Gravatar a big enough security risk to warrant caching images locally?     Examples:    Someone could harvest about 50k email addresses quite easily and shoot off a bogus promotion from Stack Overflow to all the users. A spammer/stalker/generic evil guy can tie you identity to all the rest of the sites that use Gravatar. I've implemented AStar in Java and it works ok for an area with obstacles where the chosen destination is reachable.  However, when the destination is unreachable, the calculated "path" is in no way to the closest location (to the unreachable location) but is instead some random path.  Is there a feasible way to tweak AStar into finding the path to the closest location to an unreachable destination? Beam size and material for 15 feet long swing I need to create a 15 feet span beam for kid swing (300 lbs load / 150 lbs and 2 swings assumption). One end will bolt to the tree (just for convenience).  What "treated lumber" lumber size and material I should use? Or is it better / cheaper to use metal pipe (from the plumbing department)?  I guess 4x8x16 or two (2) 2x8x16 would do it according to this website . But I do like a clean look of metal. My wife would concern about big piece of lumber running on top of the kid (that looks heavy visually). How to rotate the curve but not the axes? I've discovered a new pattern.  Is this pattern well known or what is the opinion about it?  Basically, I have a hard time scrubbing up and down source files to figure out what module imports are available and so forth, so now, instead of  import foo from bar.baz import quux  def myFunction():     foo.this.that(quux)   I move all my imports into the function where they're actually used., like this:  def myFunction():     import foo     from bar.baz import quux      foo.this.that(quux)   This does a few things.  First, I rarely accidentally pollute my modules with the contents of other modules.  I could set the __all__ variable for the module, but then I'd have to update it as the module evolves, and that doesn't help the namespace pollution for code that actually lives in the module.  Second, I rarely end up with a litany of imports at the top of my modules, half or more of which I no longer need because I've refactored it.  Finally, I find this pattern MUCH easier to read, since every referenced name is right there in the function body.