Does this proof work to show that 𝑓 is a bijection if the composite 𝑓 ◦ 𝑓 is a bijection? We are given that 𝑓 is a function from A → A. Injective: Let x, y ∈ A. Assume 𝑓(x) = 𝑓(y). By the composite function being a bijection, if 𝑓(𝑓(x)) = 𝑓(𝑓(y)), x = y. Then each z = 𝑓(𝑓(x)) = 𝑓(𝑓(y)) ∈ A has a y = 𝑓(x) = 𝑓(y) ∈ A, which has x = y ∈ A. Thus, when 𝑓(x) = 𝑓(y), x = y. Surjective: Let x, y ∈ A. Since the composite function is a bijection, each z = 𝑓(𝑓(x)) ∈ A has a corresponding x ∈ A such that 𝑓(𝑓(x)) = z. Then each z = 𝑓(𝑓(x)) also has a y ∈ A = 𝑓(x) ∈ A for which y = 𝑓(x). Thus for the function 𝑓, each y ∈ A has an x such that y = 𝑓(x), and 𝑓 is a mapping onto A. Since 𝑓 is both injective and surjective, it is a bijection. Does this proof make sense?
I keep going around in circles with this problem:  Given the functions $f:X\to Y$ and $g:Y\to Z$, if $g\circ f:X\to Z$ is bijective, then prove $f$ must be one-to-one and $g$ must be onto. I was able to prove this without difficulty. However the second part of the problem has me stumped:  a) Find functions $f,g$ where $g\circ f$ is bijective but $f$ fails to be onto.  b) Find functions $f,g$ where $g\circ f$ is bijective but $g$ fails to be one-to-one.  My attempt for (a): Let $f:\mathbb{R}\to \mathbb{R}^+$ be defined by $f(x)=e^x$ and $g:\mathbb{R}^+\to \mathbb{R}$ be defined by $g(x)=\ln(x)$. Then $g\circ f(x)=g(f(x))=g(e^x)=\ln(e^x)=x$ which is bijective, $g$ is onto and $f$ fails to be onto... or so I thought. Upon closer inspection $f$ $\underline{is}$ onto for the given codomain $\mathbb{R}^+$.  And now I'm stuck in a rut. Anyone out there have any hints?
Good color palette, usable for the color blind. I'm designing a chart right now, that will slowly have more items added to it. I'd like to create an list/order of colors for the items, that will be most usable for the color blind to differentiate between each item. Any ideas how/where to start? I just installed a fresh Ubuntu 12.04 (Precise Pangolin).   I noticed that, there is no screensaver settings, which is fine. I like to control the period of time after which the blank screen is shown,which automatically locks the screen.  I do not want to enable screensaver. I only wish to be able to change the idle time period required to lock the screen automatically. It seems to me, it is by default around 10-15 minutes. Updating the SubscriberKey on the All Subscriber table through the API I'm trying to update an existing Subscriber in the All Subscriber Table through the API; i've set up the AppId in the App Center and have connected and created Subscribers, but i'm still having issues updating.  What call do I need to be making to do this? These are the ones i've been trying. So I have seen the similar question and answers on here for $x^4 +1$, but I am having trouble extending anything there to this polynomial... I understand it is fairly trivial with Galois theory, but my class has just barely covered Field Extensions, so suffice it to say we have no Galois theory to play with.  I managed to prove it for the primes such that $p \equiv 1, 7 \pmod 8$, by noting that $2$ is a square modulo those primes and thus $x^4 - 2x^2 +1 = (x^2 -1 + 2qx)(x^2 - 1 - 2qx)$ for those $\mathbb{Z}_p$... however, trying to get a similar result for $3 \pmod 8$ and $5 \pmod 8$ has been stumping me for a long time, I am having a hard time making $q^2 = -1$ and $q^2 = -2$ give me something factorable...  I guess the worst part of all of this is that I don't think this solution is even particularly enlightening, in terms of abstract algebra. It's really just some number theory trickery. I don't think my course has prepared me theoretically for this problem, does anyone have an elementary approach to it? java.lang.LinkageError: javax.servlet.jsp.JspApplicationContext.getExpressionFactory While using the Pixel Inspector, whenever I click "copy all values" it only copies values that you can see on the screen. Anyone know a solution? I need all the values to count the number of times 1, 2, and 0 appear on the graph. I was watching the  tutorial by Andrew Price and I was able to follow along in Cycles until I hit the part where he mixes the day and night materials together so that the lights show up on the "dark side" of the earth.  How can I recreate this effect in cycles? I thought I could use the ray information to tell me if there was incoming light but I've found no such node to give me this information. explicit formula for recurrence relation $a_{n+1}=2a_n+\frac{1}{a_n}$ Prove that the difference between two rational numbers is rational This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt:     Let $a$ and $b$ be rational numbers. Therefore, \begin{align}  a=\frac{\lambda}{\beta},\:b=\frac{\xi}{\zeta},\:\ni\:\lambda,\beta,\xi,\zeta\in\mathbb{Z},\tag{1} \end{align} which gives us \begin{align}  a-b=\frac{\lambda}{\beta}-\frac{\xi}{\zeta}=\frac{\lambda\zeta-\xi\beta}{\beta\xi}.\tag{2} \end{align}   So I have shown that $a$ and $b$ are rational numbers which, by definition, can be represented by the quotient of two integers. But now how do I tackle the problem of the difference? By definition the difference between two integers is an integer. Does that require that this difference is thus rational?  Thank you for your time, I'm playing Minecraft on Xbox 360 and I can't find the . I have tried every trick I can think of, including the xray glitch, and recreating the world in Creative mode to dig out the entire stronghold and the surrounding areas. I have found nothing and I am totally lost on where it could be.  The seed is 1833758535681455132 I swear I read an ingame hint that said Workshops would share my stuff between them, but that doesn't seem to be the case. I have a settlement at Sanctuary and Red Rocket Truck Stop, but I can't access my items I added in Sanctuary at RRTS. Am I mistaken in remembering that you could do this, or is there something I need to do to get that to work? Strategies to Defeat Memory Editors for Cheating - Desktop Games I wrote a bash script as below:  #!/bin/bash echo "Enter a file name" read filename import -window root $HOME/Desktop/$filename.png   I expected this script to take a screen shot and save the image with the given filename on my desktop.  Problem:  My Ubuntu terminal is giving error:     import: Command Not Found in a topological space only finite subsets are compact sets If in a topological space only finite subsets are compact sets, is it then the discrete topological space?   Thank you. Proposing Windows as its own Stack Exchange site? Is there a reason why we have a "Unix and Linux" Stack Exchange and an "Apple" Stack Exchange but no "Windows" Stack Exchange? (Hell even Ubuntu got its own)  I couldn't find it anywhere so I'm asking.  Since Windows has many users it would be a potent Stack Exchange. How to read a csv file and use the values as x and y points in blender? How exactly does one “control for other variables”? Here is the article that motivated this question:   I liked this article, and it nicely demonstrates the concept of “controlling for other variables” (IQ, career, income, age, etc) in order to best isolate the true relationship between just the 2 variables in question.    Can you explain to me how you actually control for variables on a typical data set?      E.g., if you have 2 people with the same impatience level and BMI, but different incomes, how do you treat these data?  Do you categorize them into different subgroups that do have similar income, patience, and BMI?  But, eventually there are dozens of variables to control for (IQ, career, income, age, etc)  How do you then aggregate these (potentially) 100’s of subgroups?  In fact, I have a feeling this approach is barking up the wrong tree, now that I’ve verbalized it.  Thanks for shedding any light on something I've meant to get to the bottom of for a few years now...! Can multiple render systems be used? Why use getters and setters/accessors? AMPscript Error with XML I have a Partner sending some data to a Triggered Email/Data Extension which includes a column for XML data.   The trigger is erroring and not allowing the data to get to the Data Extension.   Below is what they are sending over. Is the formatting of the XML incorrect?  XML data that I've used in the past in Data Extensions starts with....   &lt;?xml version="1.0" encoding="UTF-8" standalone="yes"?&gt;   This version of XML starts with... {"XML":  Could that be it?  {"To": {"SubscriberKey": "test@acme.com", "ContactAttributes": {"SubscriberAttributes": **{"XML":** "&lt;root&gt;&lt;ITEM&gt;&lt;ITEM_PRODUCT_NAME&gt;dark chocolate nuts &amp; sea salt&lt;/ITEM_PRODUCT_NAME&gt;&lt;ITEM_PRODUCT_SKU&gt;15127&lt;/ITEM_PRODUCT_SKU&gt;&lt;ITEM_PRODUCT_URL&gt;https://www.test.com/15127.html&lt;/ITEM_PRODUCT_URL&gt;&lt;ITEM_PRODUCT_IMG_URL&gt;https://www.test.com/images/nuts.png&lt;/ITEM_PRODUCT_IMG_URL&gt;&lt;ITEM_QTY&gt;1&lt;/ITEM_QTY&gt;&lt;ITEM_SUBSCRIPTION_PRICE&gt;12.59&lt;/ITEM_SUBSCRIPTION_PRICE&gt;&lt;ITEM_TOTAL&gt;12.59&lt;/ITEM_TOTAL&gt;&lt;/ITEM&gt;&lt;ITEM&gt;&lt;ITEM_PRODUCT_NAME&gt;crunchy peanut butter&lt;/ITEM_PRODUCT_NAME&gt;&lt;ITEM_PRODUCT_SKU&gt;41800&lt;/ITEM_PRODUCT_SKU&gt;&lt;ITEM_PRODUCT_URL&gt;https://www.test.com/images/butter.png&lt;/ITEM_PRODUCT_IMG_URL&gt;&lt;ITEM_QTY&gt;1&lt;/ITEM_QTY&gt;&lt;ITEM_SUBSCRIPTION_PRICE&gt;11.37&lt;/ITEM_SUBSCRIPTION_PRICE&gt;&lt;ITEM_TOTAL&gt;9.55&lt;/ITEM_TOTAL&gt;&lt;/ITEM&gt;&lt;ITEM&gt;&lt;ITEM_PRODUCT_NAME&gt;pear cinnamon &amp; pecan&lt;/ITEM_PRODUCT_NAME&gt;&lt;ITEM_PRODUCT_SKU&gt;17227&lt;/ITEM_PRODUCT_SKU&gt;&lt;ITEM_PRODUCT_URL&gt;https://www.test.com&lt;/ITEM_PRODUCT_URL&gt;&lt;ITEM_PRODUCT_IMG_URL&gt;https://www.test.com/images/apples.png&lt;/ITEM_PRODUCT_IMG_URL&gt;&lt;ITEM_QTY&gt;1&lt;/ITEM_QTY&gt;&lt;ITEM_SUBSCRIPTION_PRICE&gt;12.59&lt;/ITEM_SUBSCRIPTION_PRICE&gt;&lt;ITEM_TOTAL&gt;12.59&lt;/ITEM_TOTAL&gt;&lt;/ITEM&gt;&lt;/root&gt;", "SHIPPING_LNAME": "Tester", "FIRST_NAME": "Test", "LAST_NAME": "Tester", "ORDER_TOTAL_SAVINGS": -5.97, "MERCHANT_ORDER_ID": "1810002352", "SHIPPING_ADDRESS_STATE": "NY", "ORDER_PLACE_DATE_LONG": "Fri 11 January 2019", "SHIPPING_ADDRESS_ZIP": "10004", "ORDER_SHIPPING": 0.00, "CARD_TYPE": "Visa", "ORDER_SUBTOTAL": 47.32, "ORDER_PLACE_DATE_SHORT": "01/11/2019", "SHIPPING_FNAME": "Test", "SHIPPING_ADDRESS_TOWN": "new york", "ORDER_TOTAL": 47.32, "CARD_EXPIRE": "12/2027", "CARD_HOLDER": "Test Tester", "SHIPPING_ADDRESS_1": "75 broad st", "SHIPPING_ADDRESS_2": "fl 23"}}, "Address": "acme@test.com"}}   AMPSCRIPT IN EMAIL Calling the XML....  %%[   var @LOOPINGAPPENDIX  set @LOOPINGAPPENDIX = AttributeValue("LOOPINGAPPENDIX")  set @productRows = BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM", 0)  ]%%   %%[   for @j = 1 to RowCount(@productRows) do       set @productXML = Concat("", Field(Row(@productRows, @j), "XML"), "")      set @ITEMPRODUCTNAME = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_NAME", 0), 1), "Value"))      set @ITEMPRODUCTSKU = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_SKU", 0), 1), "Value"))      set @ITEMPRODUCTURL = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_URL", 0), 1), "Value"))      set @ITEMPRODUCTIMGURL = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_PRODUCT_IMG_URL", 0), 1), "Value"))      set @ITEMQTY = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_QTY", 0), 1), "Value"))      set @ITEMSUBSCRIPTIONPRICE = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_SUBSCRIPTION_PRICE", 0), 1), "Value"))      set @ITEMTOTAL = Trim(Field(Row(BuildRowSetFromXML(@LOOPINGAPPENDIX, "/root/ITEM/ITEM_TOTAL", 0), 1), "Value"))      set @rowColor = Iif(Mod(@j,2) == 0, "#efefef", "#ffffff")   ]%%   %%[endif]%%   %%[next @j]%%