$f: X \longrightarrow Y$ and $B \subseteq Y$ I am currentry trying to give an example where $f(f^{-1}(B)) \neq B$. I know, that for a function to have an inverse, it must be bijective. My thought process is, that the answer can only be equal to $B$ if the function $f$ is bijective, because only then can every element in $f^{-1}(B)$ be directly traced back to B. Here's where I struggle: How can $f(f^{-1}(B)) \neq B$ if the function needs to be bijective in the first place? Wouldn't this be a contradiction? Or is that exactly what I need to disprove here?
I haven't been able to do this exercise:     Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse   image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X \subseteq B$.   Since $X \subseteq B$, all the elements in $X$ belong to the codomain of $f$.  Since $f$ is surjective, it means that all elements in the codomain $B$ have some preimage in $A$. Given that $X \subseteq B$, all elements in $X$ must also have a preimage in $A$.  Have $\triangle = f^{-1}(X)$, $\triangle$ is now a set containing the preimages of the elements in $X$. Because of this, $\triangle \subseteq A$.  If we evaluate $f(\triangle)$, we...... nope, I don't know what I'm doing now.    What do you think?
Do I need to apply for a visa to enter South Korea as a tourist on an Australian passport? Do I need to apply for a visa to enter South Korea as a tourist for a few days when travelling on an Australian passport? Or will I be able to get one on arrival? Can't delete folder in Windows 7 Simplify result of $\int_0^{\infty} \frac{1}{1+x^n}dx$ It is quite easy to show that (by using residue theorem) $$\int_0^{\infty} \frac{1}{1+x^n}dx = \frac{2 \pi i^{1+2/n}}{n(e^{2 \pi i / n} - 1)} $$ for $$n \ge 2$$  Is there any possibility to simplify $$\frac{2 \pi i^{1+2/n}}{n(e^{2 \pi i / n} - 1)}$$ or it is best result?  Thanks in advance! How to merge vertices in Blender? I am attempting to create my first 3D character just for practice. Being one of my first 3d models I going through the learning curve needless to say. In doing so I find myself creating many extra vertices. Where we will say 10 vertices on a rounded shape are spaced out on the equator and on the other north pole end of the edges they are very close. The close end could all be one vertex. Is it possible to say merge all points to one point and delete the rest? Conditional clause (then) In this first instance the main clause only stands alone if you  omit (then).  But in the other example it clearly has a main clause. So would this be classed as an embedded main clause as it only works as a independent clause if it's adjusted and or if it's joined with a dependent clause.  If I have enough money, then I will buy a new car.  If it rains, the picnic will be cancelled.  The picnic will be cancelled if it rains. Invocable method Apex class is not visible in Process builder Is the Formula Logically Valid? I have a question for my exam and I find it hard to understand.  I have to prove that the following formula is logically valid:    The professor told me to "push" all the symbols inside the brackets, and use the deduction theorem.  But I don't know how to do it, because I can't find the identities to push the "exist" symbol inside the brackets.  Your help is appriciated, thank you.  Alan Markdown help page is broken (badly presented/formatted) on every site In the  I see:    Is that what the help is supposed to say and look like? Because it doesn't make sense to me.  As animuson points out in a comment, all of the highlighted example text is being shifted to the very left of each line. The entire page is broken. ArcGIS ModelBuilder iterator reports only last input shapefile? Using Arc 10.3.1, if I manually enter the names of multiple shapefiles (all from the same folder) into the Check Geometry tool, the (single) output table lists the errors that exist in all of the shapefiles.    That is, only one output table is created, even if multiple input shapefiles have errors.  For example, if two shapefiles have errors, both will be listed in the single output table. Here's a screen print of the input:    However, if I iterate those same shapefiles (using the Feature Classes option), the output table only lists the last shapefile that went through the iteration.  Here's a screenshot of the model:     I suspect that the iterator is running Check Geometry not once, but three times, with a single input shapefile going through each iteration, overwriting the previous output.    How do I get the iterator to list all of the shapefiles with errors, either in separate output tables or aggregated in a single output table? Wrong ordering in generated table in jpa Simulate INSERT key in Mac OS X Long story short, I am using Citrix to remote into my PC at work, and several applications require the use of the INSERT key, which, as everyone knows, Apple chose to eliminate from their keyboards. I've tried everything I can think of, and never been able to find an easy way to send this keycode via Citrix.  I'm sure there are other uses for this, and I've read quite a few people's questions on the subject, but no one seems to have offered a viable solution.  I have the Apple USB keyboard (full-size, with numeric keypad), and in the position where the INSERT key would normally exist, Apple placed the "fn" key. Now, I still need this key from time to time, but what I'd love to see is perhaps Control-fn or similar to send the keycode for INSERT.  If anyone knows how to achieve this, preferably without installing 3rd-party software (adware?), I'm all ears. Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis I'm taking a complex analysis course and doing some practice computing residues &amp; evaluating integrals. I pulled out an old book called     On page 196-197, there are some interesting integrals to evaluate. I'm at 5.4.3.10.: I was able to do question 1, but was stumped at how to even begin with question 2:     Evaluate the integral   $$\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx ,$$   where $n$ is an natural number.   The answer is simply $\frac{2 \pi}{n}$. Any hints? how can i forward requests to my own IP address to a local IP address in my network (possibly using named)? My setup  I have a linux server running in my basement. I currently have my router looking to that server for DNS lookup, so I can use named to spoof a bunch of domain names to point to my local server's local address, 192.168.0.111. This works great.  I also am running Apache and PHP on this server, and have forwarded port 80 on my router to this machine, so my public IP xx.xxx.x.xx is sent to the basement server.  This is a great development set up for me, because I can develop websites in my local network and pull them up on numerous devices with a *.mylocal.net address. I can also quickly share a website on my public IP with others that are outside of my network.  I have a static IP, so that makes this even easier.  The problem  Some applications need to have an absolute URL. My spoofed DNS doesn't work outside my network, and neither does my internal IP address.  I need to be able to access the sites at the same IP that they are accessed publicly, which is my public IP: xx.xxx.x.xx    For some reason, I can't access my own public IP address from within my own network.  My router's public IP is xx.xxx.x.xx, and when I hit xx.xxx.x.xx from outside my network I see exactly what I want -- my development website. But, when I hit xx.xxx.x.xx from inside my network, the connection times out.    I assume my ISP has a firewall that blocks requests from xx.xxx.x.xx to xx.xxx.x.xx where the origin and the destination are the same IP (or maybe this is impossible on any network?)  So how can I set up my network to forward the internal request to my public IP to the local IP of my basement server?   I would like to do this within the local DNS server I set up, or in my router, so the change can be applied to all devices in my network, and not just the basement server or the windows tablet/laptop, etc individually. How to choose the number of hidden layers and nodes in a feedforward neural network? Is there a standard and accepted method for selecting the number of layers, and the number of nodes in each layer, in a feed-forward neural network? I'm interested in automated ways of building neural networks. Units and Nilpotents If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit.   I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) = 1 = (u+a)x$. After tried several elements and manipulated $ua = au$, I still couldn't find any clue. Can anybody give me a hint? How to login to gamecenter at PewDiePie's Tuber Simulator? How do I login to the Gamecenter on PewDiePie's Tuber Simulator, if you have not done it when they asked? The separation between the two plates of a capacitor is increased I had the following question on a quiz recently:     A capacitor consisting of two parallel plates, separated by a distance $d$, is connected to a battery of EMF ε. What happens if the separation is doubled while the battery remains connected?    The correct answer was "The electric charge on the plates is halved." However, we were previously given a question in which a circuit consists of a capacitor of 6 μF and a resistor of 300,000 Ω connected to a battery, and the separation between the plates of the capacitor is quadrupled. However, in this circumstance, the charge on the capacitor immediately after the plates are separated remains the same, and the potential difference across the capacitor increases.   I am confused as to why these questions have different answers to them, despite having an extremely similar setup. My best guess is that the circuit in the second question also has a resistor, but I am not sure, and help would be greatly appreciated. Sufficient condition(s) for $f'$ to be Riemann integrable How do I install the latest  on Ubuntu? I've been looking around, and I can't find anything. Is there a Ubuntu package for node.js, or do I have to compile it myself? How to do string formatting with placeholders in Java (like in Python)?