What is the difference between $\emptyset$ and $\{\emptyset\}$?   I am reading from some notes and author denotes the empty set by $\{\emptyset\}$. I am getting confused as definition says      In mathematics, and more specifically set theory, the empty set or   null set is the unique set having no elements   I think $\{\emptyset\}$ is wrong notation as that implies that it has one element. More specifically it should be the notation of power set of $\emptyset$
Is ∅ equivalent to {∅}? I think they are, but I am not sure? If anyone could clarify, that would be great. Thank you!
How can I use BibTeX to cite a web page? I'm looking to cite a number of web pages using bibtex and I was wondering if there was a specific template of the form @&lt;template name here&gt; for doing that. If you could use the following website as an example that would be great Add warning when changing display name Delay in approval leads to missing out on being awarded an "Explainer" badge I am not sure if this is a bug or a design error, but I came across this issue on Robotics. I assume that the same code is employed across the various SE sites, so I am posting my question here. I submitted an answer to  at 16:26.    I then suggested an edit to the question at 17:03. However, it was not until 20 hours later, at 13:03, that the edit was approved, so I appear to have missed out on the &quot;Explainer&quot; badge, as the edit should be done within 12 hours.    I don't know the reason for the delay, as I do not have a low approval rating:    I'm not bothered about missing out on the badge, nor am I criticising the reviewers - after all, we all do this in our free time. However, it does seem that the algorithm for the badge relies on the approval time, in lieu of the submit time, which does not seem to be correct, as the badge then becomes dependant upon the reviewer's availability/responsiveness, rather than upon the actions of the answerer/editor.  TL;DR Should the badge also check if the edit was caused by a suggested edit and, in those cases, look at the time it was suggested rather than the time it was eventually approved? I am trying to solve this exercise from Royden's 3rd edition.     The question is as follows: Let $f$ be a real-valued function defined for all real numbers. Show that the set of points at which $f$ is continuous is a $G_{\delta}$.    Let $$A_n = \{y : \text{there is a }~\delta_y \gt 0 : |f(s)-f(t)|\lt 1/n ~ \text{whenever}~ s,t \in (y-\delta, y+\delta)\}\;.$$     Then by the definition of open sets, $A_n$ is open.     To complete the proof, I need help in showing that $f$ is continuous at say $x$ if and only if $x\in \cap A_n$.     If $f$ is continuous at $x$, the there is a $\delta \gt 0$ such that $|f(x) - f(a)| \lt 1/n$ whenever, $x\in (a-\delta, a+\delta)$. so $x \in A_n$ son it must be in $\cap A_n$.  Thanks. What do they do with all the plates and cups in Star Trek? Whenever someone in Star Trek creates food or drink in the replicator it is generated in the bowl, plate, or cup. It seems like after a few months they would have piles of used dishes from replicator food while making more for every meal. What do they do with all the dishes? Why no tag Badges for Good questions Why not make  but for questions too?  I love that everything is well though behind SO but I can't see no reason as to why great questions can't be prized with "Badges by subject". Microsoft has steadily improved its operating cash flows in the past three years. Microsoft has steadily improved its operating cash flows in the last three years. Microsoft has steadily improved its operating cash flows for the past three years. Microsoft has steadily improved its operating cash flows for the last three years. Microsoft has steadily improved its operating cash flows for the past three years. (&quot;for&quot; omitted) Microsoft has steadily improved its operating cash flows for the last three years. (&quot;for&quot; omitted) Microsoft has steadily improved its operating cash flows during the past three years. Microsoft has steadily improved its operating cash flows during the last three years. Microsoft has steadily improved its operating cash flows over the past three years. Microsoft has steadily improved its operating cash flows over the last three years.  Which of them are idiomatic and preferred? Are they interchangeable? One part of the render layer glowing? High Voltage Powerlines - clarification of energy loss Write down the sum of sum of  sum of digits of $4444^{4444}$ Let $A = 4444^{4444}$;  Then sum of digits of $A = B$; Then sum of digits of $B = C$; Then sum of digits of $C = D$;  Find $D$.  What should be the approach here? I see pictures online of everyone having all these Pokémon to catch being everywhere around them, but it seems like I can only ever see one or two. I live in a rural area, but even walking miles through the farmland, it's rare to find any Pokémon.  Why is this? How can I find more Pokémon? Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group. What updates should be done to older monsters to bring them in line with MM3 era monsters? I know that to better balance paragon and epic level players against similar level monsters, WotC changed the math used for monsters for MM3, and I believe that they changed some of the types of powers they gave monsters.   If I want to update monsters from MM1 or MM2 to be in line with MM3 era monsters, what changes should I make? How to calculate $\int \frac{x^2 }{(x^2+1)^2} dx$? I'm trying  to calculate $\int \frac{x^2 }{(x^2+1)^2} dx$ by using the formula: $$ \int udv = uv -\int vdu $$  I supposed that $u=x$ s.t $du=dx$, and also that $dv=\frac{x}{(x^2+1)^2}dx$, but I couldn't calculate the last integral. what is the tick here?           the answer must be: $ \frac{1}{2}arctan\ x - \frac{x}{2(1+x^2)} $ + C Let: $P(x) = a_nx^n + a_{n-1}x^{n-1} + ..... + a_1x + a_0$ where $a_0a_n &lt; 0$ I have to prove that the Polynomial $P(x)$ has at least one positive root how can I prove it? Any ideas? Number of bitstrings with $000$ as substring I have $F_n$ number of bitstrings that have $000$, How would I prove that for $n \ge 4$ , $a_n = a_{n-1} +a_{n-2}+a_{n-3}+ 2^{n-3}$?  Now there are many  ways to go about this but if I choose starting a $n-3$ bit string and append it with $000$ how would I show that? Multiple simultaneous downloads using Wget? Minimum value of $\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2$ Given $a,b,c \in \mathbb{R^+}$ such that $a+b+c=12$   Find Minimum value of $$S=\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2$$  My Try: By Cauchy Schwarz Inequality we have  $$\left(a+\frac{1}{b}\right)+\left(b+\frac{1}{c}\right)+\left(c+\frac{1}{a}\right)\le \sqrt{3}\sqrt{S}$$  $\implies$  $$\sqrt{3S} \ge 12+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$  Now by $AM \ge HM $ inequality we have  $$\frac{a+b+c}{3} \ge \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$ $\implies$  $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{3}{4}$$  hence  $$\sqrt{3S} \ge 12+\frac{3}{4}=\frac{51}{4}$$  hence  $$3S \ge \frac{2601}{16}$$  so $$S \ge \frac{867}{16}$$  is this approach correct and any better approach please share. How to install Wi-Fi driver for Realtek RTL8821CE on Ubuntu 18.04? Representation of electromagnetic wave?