$A$ and $B$ are two non-empty subsets of the real set. Given that  $$(x \in A \text{ and } y \in B) \implies x\leq y$$  Show that $\sup A$ and $\inf B$ exists and that $\sup A\leq\inf B$
If $A, B$ are two non empty sets of real numbers and for every $a$ from $A$ and $b$ from $B$     $a &lt; b$,   how can I prove that      $\sup A \leq \inf B$?
How to prove that $P \rightarrow Q$ is equivalent with $\neg P \lor Q $? In my book about Logic, which is called 'Language, Proof and Logic', by the way, there is explained that the conditional $ P \rightarrow Q $ is equivalent with $\neg P \lor Q$.   that gives as answer: 'just compare the truth tables and you can see that they are equivalent'. However, I would like to know how to prove this using formal propositional logic. This should be possible, right? What does the decay constant mean? In my curriculum, the  is "the probability of decay per unit time"  To me, this seems non-sensical, as the decay constant can be greater than one, which would imply that a particle has a probability of decaying in a time span that is greater than 1.  Can someone explain this? Web colors: how to compensate for differences in monitors What do the different colors mean in ls? What do the different colours in Ubuntu's ls command mean? For example, when I type the ls command in one of my folders, I get one of the files in light green, the other (which is a folder) in blue with green highlighting.   What do those colours mean, and there is any manual about all the colours? What is the ancestral form of language used in the fable that David was studying during the flight? David studies linguistics during the flight to LV-223. We get to listen and read a fable. Is it written in a real language? What is the name of the language? Can the fable be translated?  Approximate quotation:     hjewis jasma hwaelna nahast akwunsez dad'kta, tam ghermha vagam ugenthe Inno Setup: How to manipulate progress bar on Run section? Over time, I've noticed an increasing practice among moderators of the smaller sites (i.e. not in the Trilogy) to move extended comment chains to chat, just because they're too long. Is this a good idea? In the past, I've taken the position on many occasions that I believe that comments serve two purposes only: clarification, and asking for clarification.  I take this position because as a mod I want the freedom to purge a comment trail that is clearly running off the rails without having to argue with anyone about politeness definitions. But that's not the same thing as two pedantic C++ language lawyers debating the finer points of &quot;undefined behavior.&quot;  I give wide latitude to such discussions, as long as the participants don't start throwing spears, because I believe they are instructive. The SE software even has accommodations for this.  Long comment chains are automatically collapsed, so as not to disrupt the Q&amp;A format of the post.  And moving a comment chain to chat is not even a move, really; to do an actual move, you have to perform the conversion, and then delete all of the copied comments from the post. Here is a recent example:  and it's accompanying . What I find amusing is that the OP's detailed POV about the term, where he floats a theory about why everyone thinks that &quot;bugged&quot; is not a thing (that conversation took place in the comments, and is now safely sequestered in the chat room) gets to stay prominently featured in his question, at the very top of the post. Now you could argue that the chat room is a better environment to have these discussions anyway, and you'd be right.  I could argue that if we're going to do that, then why not just automatically convert long discussions to chat rooms? The problem is not so much where the comments reside.  I don't understand why mods are taking it on themselves to fix something that isn't hurting the site (at least in my opinion).  In addition, every time I see this happen, the participants in the conversation immediately lose interest in it.  The only comment that was posted in that room after all the comments were moved is most likely this one:  It wouldn't surprise me if the folks using &quot;bugged&quot; to refer to a certain software quality are the same folks who refer to their programs as &quot;codes&quot; and who believe textspeak is the pinnacle of modern English.  Which nobody will ever see. :) Some mods are really aggressive about this.  I discussed this with a mod on Physics, and he told me that they routinely move discussions when they think they might run off the rails. When & how did Yoda come to know about the Rule of Two? I know it is so weird, but last night I set an image on my desktop background and delete the image, then I right-click on the desktop and press the desktop properties to change the style of background (center, stretch and etc) then the desktop properties stopped something like hanging on Windows.   Then a black window alerted something and I press yes without reading it. after that with right-click on the desktop I just see this:    Even some system applications don't work now. for example, I cannot take a screenshot.  Also, my pcmanfm-qt is dead completely, with the command line or clicking on the icon nothing happens.  I don't know what happens. and I have no idea how to fix it. I've got a small piece of code that is parsing an index value to determine a cell input into Excel.  It's got me thinking...  What's the difference between   xlsSheet.Write("C" + rowIndex.ToString(), null, title);   and  xlsSheet.Write(string.Format("C{0}", rowIndex), null, title);   Is one "better" than the other? And why? Can a parallel computer simulate a quantum computer? Is BQP inside NP? I know that if you create a custom object, with an Auto-number Name field with a format, e.g. MYC-{0000} you will be able to create records from MYC-0000 to MYC-9999.  The moment you create 10,000 records Salesforce happily extends the range and you can then create MYC-10000 to MYC-99999.  My question is, what is the limit to this - for obvious reasons its a difficult one to test. I know instances that have exceeded 999,999 e.g. MYC-1000000 but am interested to know how far this extends to.  EDIT: When the field has a data type of Text the maximum length is 80. Is it safe to assume that the same is true for Autonumber (though that would be a very big number)? If I click to open achievements from a question when I click the X to close the achievements view I end up on the answers tab of the question, I assume because the X is over the tab and it's tapping through.    It appears if I hit the X it's ok, but if I tap anywhere else in the top row (where the X is) then I can switch to the question or answer tab. Assuming I have an ArrayList  ArrayList&lt;MyClass&gt; myList;   And I want to call toArray, is there a performance reason to use  MyClass[] arr = myList.toArray(new MyClass[myList.size()]);   over  MyClass[] arr = myList.toArray(new MyClass[0]);   ?  I prefer the second style, since it's less verbose, and I assumed that the compiler will make sure the empty array doesn't really get created, but I've been wondering if that's true.  Of course, in 99% of the cases it doesn't make a difference one way or the other, but I'd like to keep a consistent style between my normal code and my optimized inner loops... Query a region within a larger region with GeoDjango REST API I have a working Django REST API which can query all the counties in a country. //My model class kenya_counties(models.Model):   objectid = models.BigIntegerField()   id_field = models.BigIntegerField()   county_nam = models.CharField(max_length=80)   const_code = models.BigIntegerField()   constituen = models.CharField(max_length=80)   county_cod = models.BigIntegerField()   shape_leng = models.FloatField()   shape_area = models.FloatField()   geom = models.MultiPolygonField(srid=4326)    def __str__(self):       return self.county_nam  //My Serializer class CountiesSerializer(serializers.ModelSerializer):   class Meta:      model = kenya_counties      fields = ('objectid', 'id_field', 'county_nam', 'const_code', 'constituen', 'county_nam', 'county_cod', 'shape_leng', 'shape_area')  //My view @api_view(['GET', 'POST']) def counties(request):    if request.method == 'GET':        data = kenya_counties.objects.all()         serializer = CountiesSerializer(data, context={'request': request}, many=True)         return Response(serializer.data)     elif request.method == 'POST':        serializer = CountiesSerializer(data=request.data)        if serializer.is_valid():            serializer.save()            return Response(status=status.HTTP_201_CREATED)                 return Response(serializer.errors, status=status.HTTP_400_BAD_REQUEST)  //My url     re_path(r'^api/gis/counties/$', views.counties),  I get . How can I query just one county and display it on a react-leaflet map as shown below? &lt;MapContainer center={position} zoom={13} scrollWheelZoom={false}&gt;     &lt;TileLayer       attribution='&amp;copy; &lt;a href=&quot;http://osm.org/copyright&quot;&gt;OpenStreetMap&lt;/a&gt; contributors'       url=&quot;https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png&quot;     /&gt; &lt;/MapContainer&gt; What is an undefined reference/unresolved external symbol error and how do I fix it? This  shows the dual cone of $K = \{(x,t) \mid \| \boldsymbol{x} \|_1 \le t \}$ is $K^{*} = \{(x,t) \mid  \| \boldsymbol{x} \|_{\infty} \le t\}$. Is it right? How is it proved? How can I switch between starting a coroutine vs running the routine all at once? I'm generating a level, and I have an optional delay to show the steps level generation one by one.  If my generationDelay boolean is true, I want to use StartCoroutine to spread the generation out over time. If it's false, I want to call the same method but without StartCoroutine.  The problem is that the method is type IEnumerator.  I can make two methods one type of IEnumerator and one not. But I wonder if there is a way to use one method.  In the first script top i did:  public bool generationDelay = false;   Then in a method:  private void BeginGame()     {         mazeInstance = Instantiate(mazePrefab) as Maze;          if (generationDelay == true)         {             StartCoroutine(mazeInstance.Generate(generationDelay));         }         else         {             mazeInstance.Generate(generationDelay);         }     }   Then the other script where the method Generate is: In the top:  public float generationStepDelay = 0.01f;   Then the method Generate:  public IEnumerator Generate (bool generationDelay) {          if (generationDelay == true)             WaitForSeconds delay = new WaitForSeconds(generationStepDelay);         cells = new MazeCell[size.x, size.z];         List&lt;MazeCell&gt; activeCells = new List&lt;MazeCell&gt;();         DoFirstGenerationStep(activeCells);         while (activeCells.Count &gt; 0) {             yield return delay;             DoNextGenerationStep(activeCells);         }     }   I'm getting an error on the line:   WaitForSeconds delay = new WaitForSeconds(generationStepDelay);      "Embedded statement cannot be a declaration or labeled statement"   And an error on the line:  yield return delay;      "The name 'delay' does not exist in the current context"   The reason I want to use a bool variable is that when i'm using StartCoroutine it's kind of slow, even if I change the value of generationStepDelay to 0 it's kind of slow. And if I'm not using StartCoroutine and not using the Generate method as IEnumerator it will work fast.  So i wonder what should I do and how? How do I install VMware player on a Mac os X? I went to their site, and downloaded a .bundle which automagically turned into a .txt file after downloading.   Long story short, I can't run the txt file because I guess Mountain Lion Mac os X isn't a strict Linux box and is missing some core components to do that type of installation.  Does anyone know how to install VMware? Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$ I'm learning calculus, specifically derivatives and applications of MVT, and need help with the following exercice:      Show that, for all $n &gt; 1$ $$\frac{1}{n + 1} &lt; \log(1 + \frac1n) &lt; \frac1n.$$    I tried to follow the below steps in order to prove the RHS inequality:   Proving that $f &lt; g$ on $I$ from $a$ to $b$:  Step $1$. Prove that $f' &lt; g'$ on $\operatorname{Int}(I)$.  Step $2$. Show that $f(a) \leq g(a)$ or that $f(a^+) \leq g(a^+)$  Following the above steps, let $f(x) = \log(1 + \frac1x)$ and $g(x) = \frac1x$, for all $x &gt; 1$. One has  $$f'(x) =  -\frac{1}{x^2 + x} \quad \text{and} \quad g'(x) = -\frac{1}{x^2}.$$  We note that, for every $x &gt; 1$, $f'(x) &gt; g'(x)$. Moreover, $f(1^+) = \log(2) &lt; 1 = g(1^+).$   My problem is that I got the wrong inequality sign in Step $1$.    Looking at the solution in my textbook, the author suggests using the MVT but I don't know how to apply it in this case.