Prop: If two functions $f,g$ are bijective, their compositions are also bijective.  Pf: Let $f:A\rightarrow B$ and $g : B \rightarrow A$. By definition, an injective function is invertible.  $f(g(x)) = f(g(y))$. Since $g$ is a bijection, $f(f^{-1}(x)) = f(f^{-1}(y)) \implies x = y$. Then, $f(g(x)) = f(g(y))$ is a bijection.   Conversely, $g(f(x)) = g(f(y)) \implies g(g^{-1}(x))=g(g^{-1}(y)) \implies x =y$.  $g(f(x))=g(f(y))$ is also a bijection as required.
Let f : A → B and g : B → C be bijections. Prove that g◦f : A → C is a bijection  Can someone show me the steps I should take to solve this problem?
Totally disconnected space that is not $T_2$ How to install multiple versions of the same software? I manage phone systems for multiple clients. Each system uses the same administrator software, but it runs on different versions depending on what version of firmware is installed on the system controller.   The software is downloaded directly from the system controller so it's the right version. For example, if the controller runs on version 5.0.2, you have to run version 5.0.2 of the administrator software. You can't administrate a 5.0.2 controller with a later versions of administrator software.   Bottom line, you have to have the right version of software to log into the controller. The software is not executable on its own, you have to install it. So every time I want to log into a different controller, I have to reinstall the right software.   Any way to get around this?   I'm running Windows 7 Enterprise x86. How do I delete a Git branch locally and remotely? I have only seen infinite vector spaces and the one finite vector space i.e the trivial vector space $\{0\}$. Is there any other finite vector space? Product of square of distances from vertices of a polygon of radius a I want to find out the following product.  $\prod_0^{n-1}  (r^2 + a^2 -2ra\cos(2k\pi/n - \theta))$ I have been trying to use complex numbers but did not work out. The book I am reading the result is directly stated as $r^{2n} + a^{2n} -2r^na^n\cos(\theta)$. Please help. Solve a system of equations with Euclid's Algorithm Rotate (group) objects outwards on spherical particle system? Prime divisors of the integer $n^2+n-1$ (using the Legendre symbol) What made me have a question is the following problem.     The prime divisors $p\not=5$ of the integer $n^2+n-1$ are of the form   $10k+1$ or $10k+9$.   I thought $\left(\frac 5 p\right) = 1$, because $5 \equiv 1 \pmod 4$.  So $\left(\frac 5 p\right) = (-1)^{p-1}\left(\frac p 5\right)=1$ by qudratic reciprocity.  Then $p$ should be of the form $2k+1$  But I think there is something wrong, and I have lost.  Is there someone to help me? Start ubuntu without a desktop environment but start an X application I want to do a small computer just for internet browsing. I dont want to load a desktop environment to keep RAM, CPU and GPU ussage as low as possible. The computer is going to run on a ARM CPU. I dont care with Distro I use, I thought about good old plain Debian. How to install driver for TP-Link TL-WN722N(EU)V2 on Ubuntu 17.04 Kylin I can not install the TP-LINK WN722N V2, which includes a 8188 EUS chip. Then I tried to install the backports v4.4.2,but   do not want to recognize either.  The TP-Link driver installation errors thrown out:  man@kman-livve:~$ cd '/home/kman/Documents/Tplink 722 N V2/rtl8188EUS_linux_v4.3.0.8_13968.20150417' kman@kman-livve:~/Documents/Tplink ...$ sudo make  "******************************************" "NO SKRC,we will use default KSRC" "******************************************" make ARCH=x86_64 CROSS_COMPILE= -C /lib/modules/4.10.0-15-generic/build M=/home/kman/Documents/Tplink 722 N V2/rtl8188EUS_linux_v4.3.0.8_13968.20150417  modules make[1]: Entering directory '/usr/src/linux-headers-4.10.0-15-generic' arch/x86/Makefile:140: CONFIG_X86_X32 enabled but no binutils support make[1]: *** No rule to make target '722'.  Stop. make[1]: Leaving directory '/usr/src/linux-headers-4.10.0-15-generic' Makefile:1367: recipe for target 'modules' failed   A method to know him, I do not know about. Thanks  Model:TL-WN722N(EU)_V2_161112_Linux.zip Chipset:rtl8188EUS_linux_v4.3.0.8_13 Bootcamp: The startup disk does not have enough space to be partitioned When clicking continue on the first Boot Camp screen,  I am presented with a dialog that displays:     The startup disk does not have enough space to be partitioned.   You must have at least 39 GB of free space available.     This is despite "About this Mac" reporting 271GB of free space:    And "Disk Utility" reporting 268GB of free space:    Searching online I found  which suggested it was the result of Time Machine which i recently enabled and suggests to run sudo tmutil thinlocalsnapshots / 999999999999 which I did, it ran sucessfully, and I restarted, yet to no avail.  What is the solution to this instance of Apple's poor engineering?  Running macOS High Sierra 10.13.4 (17E202) Something has changed in server 16.04 LTS, I can't find tty1, but I need to make an auto-login on tty1 (in console mode), in 14.04 LTS I can make this work.  I can't find /etc/init/tty1.conf  How to, or a link? Two envelope problem revisited I was thinking of this problem.  I believe the solution and I think I understand it, but if I take the following approach I'm completely confused. Problem 1:  I will offer you the following game. You pay me \$10 and I will flip a fair coin. Heads I give you \$5 and Tails I give you \$20. The expectation is \$12.5 so you will always play the game.  Problem 2:  I will give you an envelope with \$10, the envelope is open and you can check. I then show you another envelope, closed this time and tell you: This envelope either has \$5 or $20 in it with equal probability. Do you want to swap? I feel this is exactly the same as problem 1, you forgo \$10 for a \$5 or a \$20, so again you will always switch.  Problem 3:  I do the same as above but close the envelopes. So you don't know there are $10 but some amount X. I tell you the other envelope has double or half. Now if you follow the same logic you want  to switch. This is the envelope paradox.  What changed when I closed the envelope?? EDIT: Some have argued that problem 3 is not the envelope problem and I'm going to try and provide below why I think it is by analysing how each views the game. Also, it gives a better set up for the game. Providing some clarification for Problem 3: From the perspective of the person organising the game:  I hold 2 envelopes. In one I put \$10 close it and give it to the player. I then tell him, I have one more envelope that has either double or half the amount of the envelope I just gave you. Do you want to switch? I then proceed to flip a fair coin and Heads I put \$5 in and Tails I put \$20. And hand him the envelope. I then ask him. The envelope you just gave me has twice or half the amount of the envelope you are holding. Do you want to switch?  From the perspective of the player:  I am given an envelope and told there is another envelope that has double or half the amount with equal probability. Do I want to switch. I think sure I have $X$, hence $\frac{1}{2}(\frac{1}{2}X + 2X) &gt; X$ so I want to switch. I get the envelope and all of a sudden I am facing the exact same situation. I want to switch again as the other envelope has either double or half the amount. What's the difference between !, %, and @ in replacement tokens? The following is the residuals vs predicted scatter plot for a regression model with two IVs.   Initially, I thought it was evidence of heteroskedasticity. But, I reasoned that although there is a visible pattern in the plot, the variance across different levels of predicted values is same.  To clear my doubts, I saved the standardized residual and predicted values and ran a bi-variate correlation test. The correlation is zero, as expected. I am, nevertheless, intrigued by this observation. Any idea why I might have obtained this pattern?  PS:- My dependent variable is a sum of two likert-type items (Response scale: 1-5). So, it's theoretical range is 2-10 and it has no absolute zero value. Traveling without paying lodging fees in America I want to travel around the East Coast without paying for lodging.  I'd be willing to sleep in my car or even pitch a tent somewhere. Is this a problem?  Legal?  I really want to get to the beach this season but I’m afraid that lodging will be out of my price range.  Specifically Maryland, Delaware, maybe New Jersey or Virginia. Split a string by spaces -- preserving quoted substrings -- in Python A union of two subspaces not equal to the vector space. I need to calculate the SQRT of  $x$ to $y$ decimal places. I'm dealing with $128$-bit precision, giving a limit of $28$ decimal places. Obviously, if $\,y &gt; 28$, the , which I'm currently using, becomes futile, as it simply doesn't offer $y$ decimal places.  My question to you is, can I calculate the Square Root digit-by-digit? For example, calculating the $n$-th digit, assuming I already have digit $\,n - 1$. I'm well aware this may be impossible, as I've not found any such method through Google yet, just checking here before I give up in search of arbitrary precision methods. Would depositing large cash amounts cause problems? I saw  recently where a man withdrew $600,000 USD in cash from his bank.  I'm sure many folks here would advise against this move for obvious (and perhaps not-so-obvious) reasons.   My question: suppose you did this and then, some months later, changed your mind and wanted to re-deposit with a bank.  I assume the gent cannot simply walk into a bank with $600,000 USD and deposit it, right?  Wouldn't money-laundering alarms be raised?  Wouldn't the IRS like to know where the money came from?