I cannot seem to find a formal definition for the following.  What's a "collection" in the context of set theory?
In school, I have always seen sets. I was watching a video the other day about functors, and they started talking about a set being a collection, but not vice-versa. I also heard people talking about classes. What is their relation? Some background would be nice.   It has to do with something called Russell's paradox, but I don't know what that is.   I think that the difference between a family and a set is that the former is a function and the latter is a set. Is this right?
Getting different answers when integrating using different techniques Simple bijection $\mathbb N_0 \times \mathbb N_0 \to \mathbb N_0$ I'm looking for a 'simple' bijection $$ \pi \colon \mathbb N_0 \times \mathbb N_0 \to \mathbb N_0, $$  where 'simple' in this context means that it should be as easy as possible to define and it should be self-evident that said function is in fact a bijection. It's a quest for convenience - not minimal complexity in any rigorous sense.  Here are a couple of examples I came up with - none of which satisfy both requirements:   Let $\pi(m,n) = c(2^m3^{n+1})$, where $c \colon \{2^m3^{n+1} \mid m,n \in \mathbb N_0 \}\to \mathbb N_0$ is the order isomorphism (or in fact Mostowski collapse) of its domain under the natural ordering, Let $\pi(m,n) = \langle n,m \rangle$ - Gödel's pairing function, Let $\pi(m,n) = m \oplus n$ - the number resulting from '' $m,n$ (where we imagine both $m$ and $n$ as in infinite decks of cards whose $i$th element is labeled with its respective $i$th digit). ... Summation by parts (finite calculus) of $\sum\limits_{i=0}^{n}i\cdot a^i$ I'm studying the calculus of finite differences and have read about summation by parts:  $\sum f(x)\Delta g(x) = f(x)g(x) - \sum g(x+h)\Delta f(x)$.  The tutorials I'm using go a bit woolly at the point of introducing this technique: they use examples where they set $\Delta g(x)$ to $2^x$, and don't really show how to use it for anything a bit harder. I'm at a loss for basic examples.  I wondered if it is possible to determine  $S.n = \sum\limits_{i=0}^{n}i\cdot a^i$  using summation by parts?  If so, how is it done?  Thanks in advance. Prove that $\lim\limits_{n\to\infty}1 + \frac{1}{1!} + \frac {1}{2!} + \cdots + \frac{1}{n!}\ge\lim\limits_{n\to\infty}(1+\frac{1}{n})^n$ So My professor assigned this question and I am really stuck on part B of the question.  For $n \in \mathbb{N}$ let {$T_n$} = {$1 + \frac{1}{1!} + \frac {1}{2!} + \cdots + \frac{1}{n!}$}.  (a) Prove that {$T_n$} is non-decreasing.  (b) Use {$S_n$} = {$(1+\frac{1}{n})^n$} to prove that {$T_n$} is bounded above and prove that $lim_n(T_n)\ge lim_n (S_n)$   Like I mentioned I can prove part (a) simply by supposing that $T_{n+1} \lt T_n$ then after some algebra concluding the statement is false and than with one other example proving that its not strictly increasing either.  Now for part (b) I have not a slightest clue about how to solve, I assume I will have to use the expanded form of $S_n$ but thats the best I have. So please give me some hints or nudges in the right direction. how to get the last character of a string? How to renew an expired keypair with gpg What is the best way to renew a gpg key pair when it got expired and what is the reason for the method?  The key pair is already signed by many users and available on public servers.   Should the new key be a subkey of the expired private key?  Should it be signed by the old (I could try to edit the key and change the date of expiration to tomorrow)? Should the new key sign the old? Which method performs better: .Any() vs .Count() > 0? in the System.Linq namespace, we can now extend our  to have the  and  extension methods.  I was told recently that if i want to check that a collection contains 1 or more items inside it, I should use the .Any() extension method instead of the .Count() &gt; 0 extension method because the .Count() extension method has to iterate through all the items.  Secondly, some collections have a property (not an extension method) that is Count or Length. Would it be better to use those, instead of .Any() or .Count() ?  yea / nae ? I was wondering if having less than 5-10GB of free space on an SSD would affect performance. Is it specific to SSD or is it more of an operating system aspect?  For example, on an Intel X-25 G2 that has TRIM support (on Windows 7). How can you draw more detailed/smoother images in pygame? I believe it is what the Germans call "". English itself has no such equivalent word. (Although it has been adopted as a .)  Does an idiom exist that describes it? Bounded sequence which is not convergent, but differences of consecutive terms converge to zero I have a question that says "Show that there is a bounded sequence $x_n$ which is not convergent but has the property that $x_n - x_{n+1} \to 0$ as $n \to 0$.  What does this mean? Do I need to come up with an example or does the problem actually want me to prove such proposition?  By the way, I see that this sequence looks like Cauchy because of $x_n - x_{n+1} \to 0$ as $n \to 0$, but it is obviously not. My pixel perfect picture gets blurry when compiled in LaTeX Identify duplicate lines in a file without deleting them? I have my references as a text file with a long list of entries and each has two (or more) fields.   The first column is the reference's url; the second column is the title which may vary a bit depending on how the entry was made. Same for third field which may or may not be present.  I want to identify but not remove entries that have the first field (reference url) identical. I know about sort -k1,1 -u but that will automatically (non-interactively) remove all but the first hit. Is there a way to just let me know so I can choose which to retain?  In the extract below of three lines that have the same first field (http://unix.stackexchange.com/questions/49569/), I would like to keep line 2 because it has additional tags (sort, CLI) and delete lines #1 and #3:  http://unix.stackexchange.com/questions/49569/  unique-lines-based-on-the-first-field http://unix.stackexchange.com/questions/49569/  Unique lines based on the first field   sort, CLI http://unix.stackexchange.com/questions/49569/  Unique lines based on the first field   Is there a program to help identify such "duplicates"? Then, I can manually clean up by personally deleting lines #1 and #3? Mouse pointer and keyboard Freeze when pluggin in external hard drive so as I say in the title I have this very big, annoying problem. On my new PC (HP Omen 2020, i7-10750h, Nvidia RTX 2070, 16GB RAM) when I connect an external hard drive (precisely an WD My Passport Ultra, 3TB) I get occasionally mouse and keyboard freezes. Practically, the system becomes unusable. The mouse and keyboards works, they even show in device manager, but from time to time (too often) they &quot;freezes&quot;: mouse pointer freeze, and it won't respond to my input, and keyboard too, if I type something it will not respond. First I thought this was a behaviour due to bad drivers, or something related to my new PC, the fact that it has USB 3.1 ports. But then I've noticed that this fact happens only when I connect my WD Hard Disk (which is USB 3.0) to my pc. I've tried some stuff online, suggesting to update drivers, change some windows settings, but with no success. My guess is that the hard disk and the mouse-keyboard does interfere with each other in some ways. I guess that it's a software problem, at driver levels maybe, not a physical one. When the mouse doesn't respond, I've tried to use the touchpad on the notebook, and it does work. So it's not a windows freeze, it's just the device (mouse-keyboard-receiver) that doesn't respond. The mouse and keyboard are from Logitech, bot connected with the Unifiying receiver (a wireless receiver they provide for the mouse and the keyboard). Model: MX Anywhere 2S for the mouse, K360 for the keyboard. Please, help me solve this problem, monday I start taking my engineering class lessons from MS Teams, and I need my pc up and running and well responsive (I need to access data from the hard disk while I take classes). PS: I've seen that there are some other questions like this in the forum, but none of them seems to provide a solution to the problem. I would also change the external hard drive in the future, but I need to know that the problem with this one won't show with another hard drive, or other devices. I've already sent this very same post to Microsoft community, I really need help with this thing otherwise it will be a problem following my online classes. Thank you if you will try to help me EDIT: On the notebook in question I have 4 USB 3.1 Ports, with two ports on each side, and a single Type-C Thunderbolt port (I'm not counting that one cause both the hard disk and the logitech receiver are USB 3.0 type A connectors). I've already tried switching USB port, connecting the WD Hard Disk to another port, doing the same on the logitech receiver, and I've even tried putting them one on the left side and the other to the right side, to minimize &quot;interference&quot; as someone suggested. But nothing, I still got this problem, which seems driver-related. Hope this will give more info Given a sample of n units out of a population of N, population median can be estimated by the sample median.  How can we get the variance of this estimator? Multiple outputs for beamer handouts I'm currently generating some presentations with beamer for students of my course. Some like to print the material, so I created a handout. Some students prefer to have 2,3,4,6 etc. pages per sheet of A4 paper. How can I go about to automatize the following code, so that it outputs 4 different files in 1x2,1x3,2x2,2x3 format?  I'd like to avoid having to compile it again everytime for the different formats.  \documentclass[a4paper]{article} \usepackage{pdfpages}  \begin{document} \includepdf[pages=1-last,nup=1x2,landscape=false,frame=true,             noautoscale=false,scale=1,delta=0mm 5mm]{Pres7Chap13-Handout.pdf} \end{document} How to remove/hide the recycling bin and All site content from the quick launch or remove the quick launch altogether? Ho can I remove the recycling bin and the 'All Site Content' link from the quick launch or remove the quick launch altogether How do I recover lost/inaccessible data from my storage device? grep'ping files for multiple strings (not necessarily on the same line) Merging two pictures Is there a tool in Ubuntu to automatically merge two images having a common strip? I need to scan a large image, which will not fit at once in the scanner. I was thinking about scanning it in two parts and hand-editing the result in Gimp to join them. But probably an automatic tool can do the same more quickly and better.  EDIT I think I may have not made my point clear. My difficulty is not aligning two images. My problem is that one has to compensate for the rotation or distortion introduced by the manual process of putting the sheet into the scanner, and an automatic tool should obtain more precise results at analyzing common patterns in the two images to rotate and align them seamlessly.