In this week's discrete mathematics' class, we received the following 2 exercise subsections:   prove that the sets: $[a,b],(a,b)$ have similar cardinalities. prove that the sets: $(a,b),(0,1)$ have similar cardinalities. if we know that $A=\{x\in \mathbb N,2|x\},B=\{x\in \mathbb N, 3|x\},$ prove that $C=A\cup B$ has the same cardinality.   I'm lost, especially with the 2 first questions. In the third one, I have succeded in proving that $A$ and $B$ have the same cardinality, but can't seem to find a bijective function from $A$ to $C$.
How to define a bijection between $(0,1)$ and $(0,1]$?   Or any other open and closed intervals?   If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it): I make a function $f : (-1, 2)\rightarrow (-5, 4)$ of the form $f(x)=mx+b$ by \begin{align*} -5 = f(-1) &amp;= m(-1)+b \\ 4 = f(2) &amp;= m(2) + b \end{align*} Solving for $m$ and $b$ I find $m=3\text{ and }b=-2$ so then $f(x)=3x-2.$  Then I show that $f$ is a bijection by showing that it is injective and surjective.
Allow moderators to edit/delete comments directly from the per-user comment moderation page A while back, a new dashboard was added to the user moderation tools to .  It's great for scouring a user's comment history to find certain comments, but I can't do anything from this dashboard except filter by flagged1, deleted, or active comments, and visit the posts to which the comments are attached where I can then edit or delete them.  Can we have the ability to edit and delete comments from this dashboard just as we're able to from the flag queue? It gets particularly tedious to have to open 100 tabs just to delete 100 spam comments from the posts they're attached to (and no, I'm not making that number up2...).    1 .  2 Alright, fine, I made that up; the actual number of spam comments I had to delete was 111. What equipment is necessary/preferred for real estate photography? What equipment is necessary for real estate photography, and what is preferred?  I'm starting into real estate photography, and I'm trying to figure out what I need and what I will want to plan on adding to my kit as I go.  I'm thinking especially along the lines of lighting equipment and lenses (focal lengths).  For example, is an ultra-wide actually necessary? Or is a general walk around sufficient. I have a bunch of external and internal HDDs that I use on a Linux system. I only have Linux systems, so using a Linux file-system would only make sense, right? However I'm currently using NTFS everywhere, because it gives me the most usable space out of HDDs.  I would like to switch to Linux file-systems now though, mostly because of permissions and compability (e.g. I can't get my LUKS encrypted NTFS partition to resize under Linux, keeps telling me to chkdsk under Windows).  However when I formatted those HDDs I tried out a bunch of different filesystems and every Linux filesystem, even ext2 which as far as I know has no journaling, used a lot of space for itself. I don't recall exact values, but it was over 100GB that NTFS got me more on a 2TB HDD, which is a lot.  So my question is: Is there a way to make ext-filesystems use less space for themselves? Or is there another filesystem (I've tried ext2, ext3, ext4, NTFS and vfat - None of them came even close to the usable space NTFS offered me) with perfect Linux support and great usable space?  I'd love to hear about how and why filesystems (especially ext2 which has no journaling) use that much more space than NTFS and I don't know where else to ask. I'd mostly prefer a way to use ext4 without journaling and anything else that uses up this much space, if that's possible. Failed a Reopen Question audit — sure looks like "Not a real question" to me I just failed , which was shown as being closed as "Not a real question".  Seeing as the vote count was positive, I figured it was an audit, but I selected "Leave Closed" anyways because to me     It's difficult to tell what is being asked here.  This question is ambiguous, vague...   There's no code examples, and the phrasing is clunky at best.  This is my second failed audit in two days for difficult questions (first failed audit is , also now closed), so if I get temporarily suspended, it won't be the end of the world (I'll think of it as a forced vacation).    That said, is there a way to dispute these audits?  Or is posting the audit here sufficient? Suppose we have a ball floating in liquid A. Then we pour a liquid B on top of the ball (Assuming that the liquid is poured such that the ball doesn't oscillate) . The ball rises up. Why?  According to me, the liquid on top (liquid B) will exert a force downward on the ball and it should go lower. The particles in the liquid B will exert an normal force which is acting downwards on the ball. How can adding liquid B make it rise up, that is, provide an upward force?  Taking a specific case, density of B > density of ball > density of A   what would happen? Ubuntu with spyware? Richard Stallman states that Ubuntu "has installed surveillance code". See:   Is this true?   I agree with him that "Any excuse Canonical offers is inadequate....". A matrix and its transpose have the same set of eigenvalues/other version: $A$ and $A^T$ have the same spectrum Can someone please elucidate the difference between "many" and "many a"? In what context of usage should we add an extra "a" beside the word "many"?  For example:         Many times, I had seen that . . . .   Many a times, I had seen that . . . .      I do not know if the example is accurate enough to support my question, but I would like to know the difference between how they are used. How to partition a list in a specific way How can the pixel values of a raster image be extracted under a polygon fishnet? When does Bleeding Attack effect apply? I've designed for a campaign a duo of rogues. My character is as , and his follower is a Knife Mistress. Couple of question about them. Our party is about to ambush several . My Sniper will hide near their camp, and shoot the weakest one when fight begins. His follower will then rush the troll and slice him in several pieces. My Sniper has the Bleeding attack feat:  Bleeding Attack (Ex) Benefit: A rogue with this ability can cause living opponents to  by hitting them with a sneak attack. This attack causes the target to take 1 additional point of damage each round for each die of the rogue's sneak attack (e.g., 4d6 equals 4 points of bleed). Bleeding creatures take that amount of damage every round at the start of each of their turns. The bleeding can be stopped by a DC 15 Heal check or the application of any effect that heals hit point damage.  Emphasis mine. Thus, a normal enemy would bleed every turn, until he get cleaned of the bleeding effect. However, troll have a special ability, which is  5 (acid/fire)  Regeneration (Ex) A creature with this ability is difficult to kill. Creatures with regeneration heal damage at a fixed rate, as with , but they cannot die as long as their regeneration is still functioning (although creatures with regeneration still fall unconscious when their hit points are below 0). Certain attack forms, typically fire and acid, cause a creature’s regeneration to stop functioning on the round following the attack. During this round, the creature does not heal any damage and can die normally. The creature's descriptive text describes the types of damage that cause the regeneration to cease functioning.  So, the troll will automatically get rid of the bleeding effect during his first turn. My question is: Does the bleeding effect apply before the regeneration, or get neglected, because the troll will heal during his turn? Am I right to assume that, Bleeding attack effect getting triggered at the beginning of the troll turn, it will deal him the damages before the wounds heals? I just read a description of CVE-2020-1971, which indicates that a patch was released today for openSSL, the patched version is 1.1.1i -- my version of Ubuntu 20.04.1 LTS is running openSSL version 1.1.1f.  From last March. Software Updater says that my system is up to date.  I am hoping that this openSSL update is released by Canonical by tomorrow... can anyone confirm they are working on this? Pivot in sql server Tikzpictures side by side I would like to use the sentence     He was 'over-eager' to implement his plan, leading to failure.   I am looking for a single word which expresses over-eagerness in a negative sense.  TIA How do I size a UITextView to its content? Autokey (autokey-qt) freezes and won't insert text Export iMessages in human readable form for archival I have a long (several months' worth) iMessage conversation with a contact.  He lost these messages when he re-installed his OS, but I still have the archive.  There is some important information there that we'd like to keep.  Is there a simple way to export this conversation for archival in a human readable form, and also keep any images?  I can select parts of the conversation, copy it to the clipboard and paste it into a rich text document, but this discards all images.  I'm looking for a way that keeps the images.  Note: I use iMessage on OS X only, not iOS.  (Ideally I'd also like to export all non-image file attachments, but I'm not expecting that this will be possible.) RMSE vs Standard deviation in population ref to subsection number only is there a way to reference the subsection part only? Example:  \documentclass{article} \begin{document}  \section{This is my section} For further information see subsection \ref{subsecthree} within this section. \subsection{One} \subsection{Two} \subsection{Three}\label{subsecthree}  \end{document}   Usually gives (without the stars):  1 This is my section For further information see subsection **1.3** within this section.   But I want:  1 This is my section For further information see subsection **3** within this section.   Is that possible?  [Edit] To make it more clear: I am looking for a command that works like \ref, but prints (the arabic number of) the value of the subsection counter at the position of the \label.  ======================= == Solution in LaTeX ==  There is another solution below. This is a different solution I came up with  (I cannot post an answer to my own question, so I put it here):  \documentclass{article}  \newcommand{\labelsubseccounter}[1]{     \renewcommand\thesubsection{\arabic{subsection}}     \addtocounter{subsection}{-1}     \refstepcounter{subsection}     \label{#1}     \renewcommand\thesubsection{\thesection.\arabic{subsection}} }  \begin{document}  \section{This is my section} For further information see subsection \ref{subsecthree} within this section,  which is called \ref{onepointthree}, not \ref{onepointfour}. \subsection{One} \subsection{Two} \subsection{Three}\label{onepointthree}\labelsubseccounter{subsecthree} \subsection{Four}\label{onepointfour}  \end{document}   (I would like to insert the compiled code, but don't know how, sorry)