What is the definition of function application in material set theories like ZF?  I know how functions are represented using functional relations (optionally paired with the codomain).  I'm aware of the definition of function application in other formal theories such as Martin-löf theory, but there it is a a primitive notion, defined as one of the natural deduction rules. In ZF-like set theories the sets representing functions are a derived notion.  Say I had an arbitrary functional relation $f$ representing some function and a a value $x$ in the domain of $f$. How is the operation that takes $f$ and $x$ and gives me the set $f(x)$ defined using the axioms and rules of ZF+FOL?
I am familiar with the set-theoretical definition of functions.   A function $f : A \rightarrow B$ from a set $A$ to a set $B$ is a subset $f \subseteq A \times B$ of the Cartesian product $A \times B$ such that the following conditions hold:   $\forall a \in A : \exists b \in B : (a,b) \in f$ $\forall a \in A : \forall b, b' \in B : (a,b), (a,b') \in f \rightarrow b=b'$   Is there a purely set-theoretical definition of the function evaluation $f(a)$ for every $a \in A$? I get as far as defining  $f(a) = \{ b \in B : (a,b) \in f \}$ but that defines rather the set of values of a multi-valued function.
What does the '!!!' syntax mean in javascript? Until recently  had one single reputation point, you can see it in this old screenshot:    Now it has over 2K reputation all of a sudden.. what has caused this? Has it affected "real" members as well? What should be done with questions that have been resolved as a comment instead of an answer? I'm playing on a Survival world with cheats on, "easy" difficulty, and at night in-game. I didn't have a sword or anything to make a sword with, and I was starving, so I switched to Creative to give myself a stack of food and a bed to set my spawn point. When I switched back to Survival, I couldn't destroy or place blocks. I'm on single player, and my Internet is fine, and I can destroy blocks still if I switch to creative. I've tried resetting the button one, and reloading the map, and it still doesn't work. I've even tried deleting everything I gave myself in Creative. This happened before on a different world and I ended up deleting it.  This only happens on this world. How do I fix it?  If I know I'm in the correct game mode, what else can cause this?  Are there different causes in multiplayer? Number theorems with chapter number I use declaretheoremstyle to define a theorem environment, and print a list of theorems in the end of my document:  \documentclass[12pt]{book} \usepackage[english]{babel} \usepackage[margin=2cm]{geometry}% just for the example \usepackage{xcolor} \usepackage{amsthm} \usepackage{thmtools} \usepackage{mdframed} \usepackage{etoolbox}  \declaretheoremstyle[   spaceabove=6pt, spacebelow=6pt,   headfont=\normalfont\bfseries,   postheadspace=1em,   notefont=\bfseries,   notebraces={(}{)},   bodyfont=\itshape,   shaded={bgcolor=yellow!20} ]{thmstyle}  \declaretheorem[style=thmstyle,name=Theorem]{theorem}  \begin{document}  \begin{theorem}[Somebody] ... \end{theorem}  \begin{theorem} ... \end{theorem}  \renewcommand{\listtheoremname}{List of Theorems} \addcontentsline{toc}{section}{\listtheoremname} \listoftheorems[ignoreall,show=theorem]  \end{document}   The theorems are numbered from the beginning to the end of the book, and from 1:    However, I would like to number the theorems with chapter number (like Figures or Tables), for instance, 1.1 (the first theorem in Chapter 1), 1.2 (the first theorem in Chapter 2)...  Does anyone know how to do it? Several badge descriptions contain raw HTML I was reading through some badge descriptions and noticed that Synonymizer's description needs to be updated. As it stands, there is raw HTML (an href) in the description:    EDIT: after reading comments by @ColeJohnson, I have come to the understanding that this post comes off as whiny. I want to make it very clear that I wasn't ever trying to complain. My first reaction was to fix the perceived error myself. When I realized I was unable to do so, I figured the best course of action would be to find someone who would be able to make the necessary edits. Hence the post here. I have an equation contained inside \[...\], which automatically makes a \sum with sub- and superscripts turn big--so that the summation sign looks awkward inside parenthesis.  Any idea how to make the parenthesis completely enclose the whole summation?  \documentclass{article} \begin{document} We have: \[ \sum_{i=1}^n i = (\sum_{i=1}^{n-1} i) + n = \frac{(n-1)(n)}{2} + n = \frac{n(n+1)}{2} \] \end{document} Facebook: Ways to preserve image quality of uploaded images? Magic leaves "smudges" and a sorceress becomes a book I could use some help identifying a book I read some time in the last year or so. The book goes between two different stories:  1) A student of magic finds a book in a cave that turns out to be a person transformed into a book. The book talks to the student and they share thoughts. At some point, the student runs away, chase ensues, and at the very end of the book, he manages to travel to another plane.  2) A girl in a religious society where magic is forbidden. She is the daughter of painters. She discovers she can see magic. Falls in love with some random painter, risks performing magic to not be pregnant, does it again so she can, and is arrested and sent to some prison thing.  The common element between the stories is that magic use leaves behind a smudge of some kind that can be visible.  Any assistance is appreciated. I have a one-dimensional array of strings in JavaScript that I'd like to turn into a comma-separated list. Is there a simple way in garden-variety JavaScript (or jQuery) to turn that into a comma-separated list? (I know how to iterate through the array and build the string myself by concatenation if that's the only way.) Why do all SSH-RSA Keys begin with "AAAAB3NzaC1yc"? How to integrate $\int x^2 \sin^2(x)dx$ I don't know how to integrate $\int x^2\sin^2(x)\,\mathrm dx$. I know that I should do it by parts, where I have  $$ u=x^2\quad  v'=\sin^2x \\  u'=2x \quad v={-\sin x\cos x+x\over 2}$$  and now I have  $$ \int x^2\sin^2(x)\,\mathrm dx = {-\sin x\cos x+x\over 2} x^2 - \int 2x{-\sin x\cos x+x\over 2}\,\mathrm dx\\ ={-\sin x\cos x+x\over 2} x^2 - \int x(-\sin x\cos x+x)\,\mathrm dx\\  $$  so I have to calculate $$ \int x(-\sin x\cos x+x)\,\mathrm dx=-\int x\sin x\cos x\,\mathrm dx+\int x^2\,\mathrm dx$$  I know that $\int x^2\,\mathrm dx = {1 \over 3}x^3+C$ but I don't know what should I do with $$\int x\sin x\cos x \,\mathrm dx$$ Should i use parts again or what? Please help. Only squares instead of letters and numbers are displayed in my calculator and other apps in Ubuntu 18.04 Does one work in or on an aeroplane? In an exam paper, there was a picture of an air stewardess in the aeroplane serving passengers. One of my pupils wrote the following:      The air stewardess works on an aeroplane.    Shouldn't it be the air stewardess works in an aeroplane? When searching for "Skype" I always see these 2 entries.  There is this duplicate blank Skype icon. Both of them seem to run Skype in the same way.    But I've checked both /usr/share/applications, and ~/.local/share/applications, and can't find the extra icon so I can delete it.    Also as described , sometimes similar ghost icons are present even after removing associated applications. Created user can access all databases in PostgreSQL without any grants Verbatim Environment in Figure Caption Teenagers find an ancient lizard creature in stasis and accompany him on adventures to an undersea city When I was a kid, I read a book (likely from the '60s) about a pair of kids who find an ancient lizard/fish man inside a cave. The cave has been recently exposed by an earthquake and the creature is inside a super technologically advanced stasis chamber. I think this creature's name is something like "Sslaa". He takes the kids on a quest to find his lost underwater city.  They end up finding the city, which is still populated by creatures like himself and protected from the ocean above by an immense diamond dome. Eventually, Sslaa has to drive an advanced mining vehicle deep into the crust of the earth, presumably to prevent his city's destruction by an impeding magma flow or other seismic event.  For extra obscurity bonus points, I could swear I managed to find this book on Google several years ago, and was shocked to learn that it had originally been written in German, and then translated to English.  Does anyone out there recognize this one? How to show $ \sin x \geq \frac{2x}{\pi}, x \in [0, \frac{\pi}{2}]$? I have tried the following: $$ f(x) = \sin x-\frac{2x}{\pi} \\ f'(x)= \cos x-\frac{2}{\pi} \\ f''(x) = -\sin x \leq 0 $$ But this doesn't seem to be heading in the right direction as it would appear $f'(x)$ is decreasing and has a zero within the domain, so I'm not sure how this might be used to prove the inequality. Is there any utility which will read a photo's date and time from its EXIF data and embed it into the corner of the photo?