undefined reference to a user-defined function
What is an undefined reference/unresolved external symbol error and how do I fix it?
Any Set with Associativity, Left Identity, Left Inverse is a Group Related Link:    Reference: Fraleigh p. 49 Question 4.38 in A First Course in Abstract Algebra     I will present my proof (distinct from those in the link) for critique and then ask my question.  $G$ is a set and $ \times $ is an associative binary operation.  Suppose that there exists a $e \in G$ such that, for all $a \in G$, $ea = a$ and $a^{-1}a = e$ for some $a^{-1} \in G$.  Show that for the same $e$, $ae = a$ and $aa^{-1} = e$.  $a^{-1}(aa^{-1})a = (a^{-1}a)(a^{-1}a) = ee = e = a^{-1}a = a^{-1}(a^{-1}a)a$  Since $a^{-1} \in G$, it has a left inverse; apply it to both ends, and we have $(aa^{-1})a = (a^{-1}a)a$.  As a result, $ae$ = $a(a^{-1}a) = (aa^{-1})a = (a^{-1}a)a = ea$.  For the right inverse, start with $aa^{-1} = a(a^{-1}a)a^{-1} = (aa^{-1})(aa^{-1})$.  Since $\times$ is a binary operation, $aa^{-1} \in G$ and has a left inverse; apply it to both ends, and we have $e = aa^{-1}$.  In second comment following the question in the link, Mr. Derek Holt pointed out that the requester did not word his/her question correctly.  Specifically, the identity in the second axiom is not well-defined.     Let $(G, *)$ be a semi-group. Suppose    1. $ \exists e \in G$ such that $\forall a \in G,\  ae = a$;    2. $\forall a \in G, \exists a^{-1} \in G$ such that $aa^{-1} = e$.    How can we prove that $(G,*)$ is a group?    This formulation makes the same technical error as many textbooks. The $e$ in your second axiom is not well-defined. "But obviously it's intended to be the same $e$ as in the first axiom" you reply. But the first axiom does not necessarily specify a unique element $e$. So should we interpret the second axiom as meaning "for some $e$ as in 1" or "for all $e$ as in 1"? –  Derek Holt Sep 17 '11 at 15:31   Was he saying that if, in axiom 1, we have $ae_1 = a, ae_2 = a$, but $e_1 \neq e_2$,  when we get to axiom 2, do we have $aa^{-1} = e_1, aa^{-1} = e_2$, or two different inverses so that $aa_1^{-1} =  e_1, aa^{-1}_2 = e_2$?  I think my wording eliminated the ambiguity.  It does not imply that $e$ is unique, but if $e$ is a left identity and produces left inverses, then it is also a right identity and produces right inverses. I tried really hard on this one; please kindly point out my mistakes. proof that function is constant I'm annoyed by quite a simple problem in calculus (I apologize in advance if I'm not using adequate terms in English, I don't take the course in English nor am I a native speaker):  Let $f:\mathbb R\longrightarrow \mathbb R$ be a function which is continuous at $0$ and satisfies $f(x)=f(2x)\,\forall x\in\mathbb R$.  My question is how do I formally prove that for every $x&lt;0$ $f(x)=c_1$, and for every $x&gt;0$ $f(x)=c_2$, where $c_1,c_2\in \mathbb R$? After proving those two statements (or at least one of them WLOG) it would be quite easy to prove that $f(0)=c_1=c_2$ by the definition of continuity of $f(0)$ and thus $f(x)$ is constant. Those two statements are just so obvious to see that I can't think of any formal way of proving so without "cheating". Thanks in advance! Is stats maths or not?   Given that it's all numbers, mostly taught by maths departments and you get maths credits for it, I wonder whether people just mean it half-jokingly when they say it, like saying it's a minor part of maths, or just applied maths.   I wonder if something like statistics, where you can't build everything on basic axioms can be considered maths. For example, the $p$-value, which is a concept that arose to make sense of data, but it's not a logical consequence of more basic principles. How to get full query text from performance_schema.threads? According to : Access to the threads table does not require a mutex  and has minimal impact on server  performance. The other sources have negative performance consequences because they require a mutex.   So yes, I want to use threads table to examine running queries. The trouble is, the PROCESSLIST_INFO column does not contain the full query. It just contains a truncated version of the query. There is a flag viz performance_schema_max_sql_text_length which controls how many bytes are allocated for the SQL query. But it seems to affect the events_statements_history_long table. There is no change in the thread table output. There is also the events_statement_current table. But it is not clear if accessing that table affects performance or not. How to Preload Tabs/Set Multiple Websites as Default in Firefox, IE, etc Any website can set as the homepage of Firefox which loads spontaneously. Is this for multiple tabs? Then BBC Health, CNN, Yahoo Health, msn.com would all extemporarily materialise (in four separate tabs); I then wouldn't have to open each website manually. Let $a$, $b$, and $c$ be complex numbers with $a\neq0$. Show that the solutions of $az^2+bz+c=0$ are $z_1,z_2=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, just as they are in the case when $a$, $b$, and $c$ are real numbers. Show $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$. Let $X=(0,\infty)$. Define two metrics on $X$ by   $d_{1}(x,y)=|x-y|$ and   $d_{2}(x,y)=|x-y|+|\frac{1}{x}-\frac{1}{y}|$   for all $x, y \in{X}$.   Let $(x_{n})$ be a sequence in $X$ and $x\in{X}$.   Show  $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$.  Note: $d_{1}$ and $d_{2}$ are then said to be equivalent metrics on $X$.  Since $d_{1}$ and $d_{2}$ are equivalent metrics, will it be sufficient to show $c*d_{2}(x,y)\leq{d_{1}(x,y)}\leq{C*d_{2}(x,y)}$?  So far I have the following: $d_{1}(x,y)=|x-y|\leq{|x-y|+|\frac{1}{x}-\frac{1}{y}|}=d_{2}(x,y)$. Therefore, $d_{1}(x,y)\leq{C*d_{2}(x,y)}$ where $C=1$. Is there a $c$ that would satisfy the left-hand side of the inequality?  Any help would be appreciated. (this is not homework) What does "official Ubuntu flavour" mean? What kind of support do they receive from Canonical or/and Ubuntu community? Who decides a distro should become official flavour? What privileges do "official flavours" have over unofficial derivatives of Ubuntu like Linux Mint, Elementary OS, Deepin OS,etc.? Assigning row number for sorted table by column in QGIS Field calculator has a great function under &quot;Records&quot; called &quot;$rownum&quot; which returns (assigns) the number of the current row. Is there a way to make it work together with sorting? Let's say I want to sort table by column &quot;length&quot; and get the order (sequence) of lines by length from shortest to longest. I know I can use some office spreadsheet software easily for this to sort by column and populate new column with order (sequence). But I would prefer to do the whole job in QGIS. As workaround there is also  - using mmqgis plugin and function modify/sort and after that use &quot;$rownum&quot; on the new saved file. I am tired of always trying to guess, if I should escape special characters like '()[]{}|' etc. when using many implementations of regexps.  It is different with, for example, Python, sed, grep, awk, Perl, rename, Apache, find and so on. Is there any rule set which tells when I should, and when I should not, escape special characters? Does it depend on the regexp type, like PCRE, POSIX or extended regexps? Why is there such a big difference between "Size" and "Size on disk"? Mathematical symbol for "and" I have found some pretty complete lists (I think) of mathematical symbols  and , but I don't see a symbol for the word "and" on either list.  A person could easily just write the word "and" or use an ampersand, but I was wondering if there was an actual mathematical symbol for the word "and".  Also, if anyone knows any lists that are more complete than the ones I have linked to please provide a link. How to explicitly discard an out argument? How do I post code in chat? I would like to be able to post code in chat. I know that I can post inline snippets of code by using backticks, but I don't know how to post blocks of code, with the entire chat message formatted as code, and indentation respected. I am aware of . After reading the comments, I found out that I could format in "fixed width", but I can't see a "fixed width" button anywhere. What should I do? Why do extract recipes call for a late addition of the bulk of the extract? My recipe calls for a pound of DME at the beginning of the boil and 6 pounds at 15 minutes to go.  What's the point of that? Dim screen past the minimum on Ubuntu My laptop's screen is far too bright at night, even if I set it to minimum brightness. I know that it is capable of being dimmer, because Windows will dim it further if I leave it idle. However, I can't figure out how to dim it lower to minimum on Ubuntu. Is there a way?  I've looked at  question, but it's talking about Windows. We have $n$ charged and $n$ uncharged batteries and a radio which needs two charged batteries to work. Optimizing logical sector size for physical sector size 4096 HDD Let's assume we have a table where it has two columns, one an ID and one an hstore column, which stores key/value pairs.  You could query that column as such:  SELECT name, attributes-&gt;'device' as device FROM products WHERE attributes-&gt;'edition'= 'ebook'   Can you create a view for this table where the keys/values appear as columns in the view instead of all in the hstore column?  Example: I have an hstore key where it might have the key of amenity.  I want to create a view showing the the ID and the amenity values in a new view. I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded into the Euclidean space of the same dimension. So I wonder if all manifolds can be embedded into a same dimensional Euclidean space when we cut out a measure zero set? Can anyone prove it or disprove it by giving me some counterexamples?