This was part of a three part question where I was supposed to prove two sets have equal cardinality by finding bijections. I've created a bijection $f: \Bbb Z \Rightarrow 2\Bbb Z$ by $f(x)=2x$. I've created a bijection $g: (0,1) \Rightarrow (4,50)$ by $g(x)=46x+4$. I think those are both correct. My last question is finding a a bijection between (0,1) and [0,1]. I've seen this question several times on this board, but I've yet to understand them and I can't really go back and ask any questions to the original posters. I know a bijective function exists between (0,1) and $\Bbb R$, but I don't think that helps me here.
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.
'Possible duplicate' comment not appearing for questions with deleted comments containing the link What is the best way to represent show and hide in icons? I have a web page that has a list of columns of information.  All columns have names and then some of the columns have items in them.   I want to have an icon to represent   Hide all empty columns Showing all columns   All of the icons i see for hide look like "delete" so i want to see if there are any representations of "hide" icons that don't look like delete.  And for "show" the only thing i see is an eye graphic which doesn't seem very intuitive to me.  Does anyone have any recommendations for good "hide" and "show" icons? Do not allow giving minus one for new user questions I always see Stack Overflow users giving -1 to new users' questions. This is not how you welcome a new user to the community. A new user does not know the rules yet.    I suggest that if one wants to give -1 to a new user, you prevent that and ask the downvoter to write a comment instead. new user = 2 first questions    I see the credit rating on my question is now at -9?  My question, is she not well formed or formatted? If you don't agree with the content of a question do you down-rate it? If an Ubuntu user asks a question that might not be reasonable, will you also down-rate it?  No. You post an answer that will present your thinking and correct him? I think that is part of the problem. When you first go to a community like Facebook, and you don't know the faces, you don't feel like any of those people are your relations. Newbies exist here as well, especially in a professional community.    First, welcome the new user in more appropriate way. And most importantly, force them to see/read how to post a good question. Empty intersection and empty union Is it possible to combine processing power of 2 computers? Can a polymorphed creature use class features that don't require speech or hand gestures? The Polymorph spell states (in part):     The target’s game statistics, including mental ability scores, are   replaced by the statistics of the chosen beast. It retains its   alignment and personality.      The creature is limited in actions it can perform by the nature of its   new form, and it can’t speak, cast spells, or take any other action   that requires hands, or speech.   So, can a polymorphed Paladin use their Divine Smite class feature?  It expends a spell slot, but it is not a spell, and as far as we can tell does not require hands or speech.  We know (based on the 10th level Transmutation Wizard feature) that a wizard is supposed to be able to polymorph herself, and we can assume that she is also supposed to be able to maintain concentration needed to stay in the new form (otherwise it would be a pretty useless ability).  This is (as far as I can tell) not something any of the beasts eligible as polymorph targets can normally do.  So it is clear that the polymorphed creature maintains some abilities from her old form.  The question is which ones.    The Druid's Wild Shape ability explicitly says you can use class features, while Polymorph does not, suggesting that you can't use class features while polymorphed.  However, Wild Shape also explicitly says that you can continue to concentrate on a spell (see below, emphasis mine), and Polymorph does not, yet we believe that a polymorphed creature can continue to concentrate on the polymorph spell.  So the fact that Wild Shape explicitly allows something and Polymorph is not explicit about it does not necessarily mean that the polymorphed creature cannot do those things.  From Wild Shape rules:     You can't cast spells, and your ability to speak or take any action   that requires hands is limited to the capabilities of your beast form.   Transforming doesn't break your concentration on a spell you've   already cast, however, or prevent you from taking actions that are   part of a spell, such as call lightning, that you've already cast.   So, since Polymorph is silent on the question of whether or not class abilities can be used, and states only that the polymorphed creature cannot perform actions that require hands or speech, how can we best determine whether (RAI) polymorphed creatures maintain any of their old abilities beyond spell concentration? How to find files that contain one criterion but exclude a different criterion Are normal subgroups transitive? I was wondering how to create a theorem environment similar to the following example:   Specifically, the environment should resemble the box that contains "Claim -- The function g is linear".  This comes from a , who has his style files uploaded  I have tried to mimic what I believed to be the relevant code (as there's a lot of stuff there that isn't related to this), to no avail. I have attached my attempt below, but it may be better to ignore it.  \usepackage{amsmath} \usepackage{amsthm} \usepackage{mdframed} \usepackage{thmtools} \usepackage{tikz} \usepackage{xcolor} \usepackage[framemethod=TikZ]{mdframed}     \mdfdefinestyle{mdgreenbox}{%         linewidth=0.5pt,         skipabove=12pt,         frametitleaboveskip=5pt,         frametitlebelowskip=0pt,         skipbelow=2pt,         frametitlefont=\bfseries,         innertopmargin=4pt,         innerbottommargin=8pt,         nobreak=true,         backgroundcolor=SpringGreen!15,         rightline=false,         leftline=false,         topline=false,         bottomline=false,         linecolor=green,     }     \declaretheoremstyle[         headfont=\bfseries\color{OliveGreen},         mdframed={style=mdgreenbox},         headpunct={\\[3pt]},         postheadspace={0pt},     ]{thmgreenbox} \declaretheorem[style=thmgreenbox]{thrm1}   This returns an error, one for xcolor not recognizing the colors, and the following.  l.17        \mdfdefinestyle                       {mdgreenbox}{% The package mdframed has already been loaded with options:   [] There has now been an attempt to load it with options   [framemethod=TikZ] Adding the global options:   ,framemethod=TikZ to your \documentclass declaration may fix this. Try typing  &lt;return&gt;  to proceed.  Package thmtools Info: Key `mdframed' (with value `style=mdgreenbox') (thmtools)             is not a known style key. (thmtools)             Will pass this to every \declaretheorem (thmtools)             that uses `style=thmgreenbox' on input line 39. Package thmtools Info: Automatically pulling in `thmdef-mdframed' on input line 40. (/usr/local/texlive/2017/texmf-dist/tex/latex/thmtools/thmdef-mdframed.sty Package: thmdef-mdframed 2014/04/21 v66 ) \c@thrm1=\count282 (/compile/output.aux) \openout1 = `output.aux'.   I am looking for a way to mimic the environment more than to fix my methods.  Thank you in advance! Reputation widget crash I have an issue with crashing reputation widget when it's added to the home screen. When I add it for the first time, it says "Please login before using this widget" (even though the app itself is logged in already). So I open the app, go back to the home screen, add the widget again and it crashes (and it's not added to the home screen). When I'll try to add it again, it says "Please login before using this widget" again. And then it just repeats - "Please login...", execute the app, crash, "Please login...", etc.  Restart of the device didn't make any change.  I have found .  My set up:  Stack Exchange app version: 1.0.89 Android version: 5.0.1 Device: Samsung Galaxy S4 Badge icons missing in Achievements of mobile view The badge icon followed by the site icon in the Achievements dialog is missing in the mobile view. But the same is available in the full site view.  Screenshots for reference:  Achievements - Full site view:        Achievements - Mobile view: Evaluating $\int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx$ Let $\alpha \in \mathbb R$ and let's call $I:=[0,1]$. Evaluate   $$  \int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx.  $$   Well, the case $n=1$ is easy and the integral equals $\frac{1}{\alpha+1}$, for every $\mathbb R \ni \alpha \ne - 1$.  I've done also the case $n=2$ and, if I'm not wrong, it's $\displaystyle \frac{2}{(\alpha+1)(\alpha+2)}$.  My big problem is that I cannot understand how to deal with the general case. Any ideas?   Thanks in advance. Clearly, if Wolverine bleeds on the ground, a new Wolverine doesn't reconstitute himself from the drop. Presumably, if somehow his arm could be severed, that wouldn't happen there, either. If his head were cut off...probably not either.  So it seems that there's a minimum amount of contiguous body mass Wolverine has to have from which he can heal.  Maybe limbs are the limit...are there examples of Wolverine regenerating a lost limb? If so, well, this question. If not, is there anything else in the canon to suggest an answer? Chat messages should use consistent icon in iOS app It'd be nice if the chat notifications in the feed used the default Stack Exchange icon like the ones in the inbox do*:     Since items in the feed are grouped so that multiple items from the same site , the site icon helps delimit grouped items a bit less subtlely than the grey divider lines, and chat replies' lack of one causes them to sort of blend in to the previous group.  *Ideally it'd use the icon of the site the room was associated with, but I don't think that information is readily available How does the color of a car influence its inner temperature change over time when parked outside in windless, hot and sunny regions?  I know what's the common idea about that: black cars are supposed to build up high temperatures faster because black objects absorb radiations instead of reflecting them off. Is it true? And then, what happens during the steady state? Some claim the black car will stay hotter, some others claim it will be the same whatever the color. Do all cars have the same temperature inside in the long run, or are some of them hotter?  I've Googled it but the best I could find was . It claims to be based upon ~20 articles. But there's not even a mention of absorption or emission spectrums. A priori, a black car could absorb all visible radiations and emit it as infrared while reflecting all of the Sun's infrared. And it's still only qualitative. I've found nothing on Physics stackexchange, the closest being . I watch a lot of old movies, and I've noticed that American actors of the 1930s and 1940s often spoke in a quasi-generic-posh-British accent.  Katherine Hepburn's accent would be the perfect example. It seems exaggerated, and I imagine it was not common off the stage and screen. What was that? Was it just an accent or was it a dialect? Did real people actually speak that way? Book series my dad told me about. It involves space travel, clones, and an indigenous people with kangaroo tails My dad read a book series that he recommended to me last year. Now that I am ready to start it, neither of us can recall the title. He said there were several in the series but he can't recall how many.   The book is about traveling to a new planet, presumably with the goal of colonizing it. The crew are clones who have no memory. I think the goal was for the older original versions of the clones to transfer their consciousness into the clone bodies to be able to live on the new planet but something goes wrong.   There is an indigenous species on the planet who are humanoid but have kangaroo like tails.   The clone crew has markings on them that they don't understand until late. Apparently they are markings to indicate status or role on the new world. The person who they voted as there leader, it turns out, is marked as a servant or slave class. I have a Google Apps for your domain account for my domain which we will call DomainA  I have since replaced that domain with DomainB which is set up in the Google Apps account as an alias.   Since I want to let DomainA expire at some point in the future, I would like to change my primary domain to DomainB in my Google Apps account with DomainA as an alias.   How can I do this? Mutual set of representatives for left and right cosets: what about infinite groups? Let $G$ be a group and $H$ a subgroup of $G$. If $G$ is finite, then according to Philip Hall's "marriage theorem" there is a left transversal $T$ of $H$ in $G$ (that is, $T$ contains precisely one element from each left coset of $H$) such that $T$ is also a right transversal of $H$.   Does this theorem generalize to infinite groups? The original theorem treats only the case where $|H| &lt; \infty$ and $[G:H] &lt; \infty$.  What can we say when $|H| = \infty$ and $[G:H] = \infty$? What about $|H| &lt; \infty$ and $[G:H] = \infty$? Also $|H| = \infty$ and $[G:H] &lt; \infty$? The more I read about this, the more confused I become. Illuminance, as I understand it,  measured in lux, is the perceived brightness of light on a surface. It's what you'd measure with an incident light sensor. Luminance, then, is how much reflected light you'd perceive looking at that surface. This is what the light meter in a camera measures. Am I correct so far?  In both cases, the word "perceived" is important, because the scales are weighted using a  to map wavelengths to . However, in the case of illuminance, there's only actual perception if your eye happens to be the surface in question.  I can basically grasp this, but then I come across charts saying things like "family living room: 50 lux". Wait, hold on! Does that really mean that the typical lights in a house are that bright, or is it just confused and wrong, or am I confused and wrong?  If you're not taking pictures of light sources directly, why would an incident light meter reading ever be useful in photography? Reflected light recorded on film or by a sensor is usually what makes a photograph. So, if I have an incident light meter, how does that reading relate meaningfully to my camera settings?  Since incident light meters are sold and used, this implies that there must be some useful conversion. But this is where my brain explodes. Google tells me that 1 lumen equals 1 candela, so therefore 1 lumen/m² (i.e. 1 lux) must equal 1 candela/m2 (i.e. 1 nit). But clearly something is missing from this. There's something called a "steradian". Cones are involved. I'd never heard of this before; how does it fit in? I can see how it might be determined when calculating the usefulness of LED home lighting, but for a photo, I'm at a loss.  Some part of my (exploded, from the last paragraph) brain is trying to relate this to the difference between flash metering with TTL and with an incident meter. But how can the incident meter work without knowing the reflectance properties of the objects in the scene? Is this what the "C" in the  is all about? Is it just an average-this-probably-will-work value, or is there more to it than that? And if it is just an average, what knowledge is necessary for compensating for off-averages scenes? (As with the , where the photographer simply judges if the scene should be rendered brighter or darker than the average given by the meter.)  So yeah, so much confusion. In short:   What's the difference? Can one meaningfully convert between the two? When and how are illuminance / incident light measurements useful for photography?     Update: I appreciate Stan's answer, which covers the third point of when and how pretty well. And I think I've basically got the first point figured out, as described above. But I'd appreciate some answers covering the issue of conversion as well, both in the mathematical abstract and as the practical for photography. And I wouldn't mind more why and how, either.