I cannot prove this rather simple theorem in set theory.      Let $A$ be a finite set. If $f: A \to A$ is injective, then $f$ is also surjective.    My Attempt. Suppose for contradiction that this isnt the case. So there exists a mapping $f: A \to A$ that is not surjective. Then the image of $A$ under $f$ is a proper subset of $A$, i.e., $\text{Im}(f) = C$, where $C \subsetneq A$. Define the mapping $g: C \to A$, where $c \mapsto f^{-1} (\{c\})$. By construction, $g$ is not onto, but since $f$ is injective, it is well-defined, since every $c \in C$ has a unique preimage in $A$. Hence, every $c$ is sent to exactly one $a \in A$. But, if that is the case, $|C| = |A|$, as a function is by definition defined on its entire domain. With this contradiction, we conclude that no such function $f$ can exist.
Let S be a finite set.Let F be a surjective function from S to S.  How do I prove that it is injective?
How to prove or disprove statements about sets Prove or disprove:  The set: $\{ \emptyset\}^{\Bbb N}$ has only one element.  The set $\emptyset^{\Bbb N}$ is empty.    I'm pretty sure both are true, for 1. all the natural numbers go to the empty set and that is the only element. Where as 2. all the natural numbers go to nothing so that's why it's empty. (Same as $ {\Bbb N}^{\emptyset} $ nothing goes to something is nothing (I think)). The problem is I have no idea how to show this so advice would be appreciated.  Edit: My defintion of a function is: $f:B\to A$ if  $(a_1,b),(a_2,b)\in f \ \Rightarrow \ a_1=a_2$ So for 2. I can simply say that: $(1,\emptyset),(2,\emptyset)\in f \Rightarrow a_1\neq a_2$  ? Excel function that evaluates a string as if it were a formula? Let $F$ be a field and let $a \neq 0$ be a zero of the polynomial $a_0 + a_1x + . . . +a_nx^n$ in $F[x]$.  I want to show that $\frac{1}{a}$ is a zero of the polynomial $a_n + a_{n-1}x + . . . + a_0x^n$  How can I do this? Showing a set is closed under inverses Let $G$ be a finite group, say a group with $n$ elements, and let $\emptyset \neq S \subseteq G$. Suppose $e \in S$, and that $S$ is closed with respect to multiplication. Prove that $S$ is a subgroup of $G$.    Let $G = \{a_1, \ldots, a_n\}.$ Let $a_i \in G.$ Then any number of $a_ia_1, a_ia_2, \ldots, a_ia_n \in S,$ since $S$ is closed under multiplication. One of those $a_ia_j$ is $e$ since $e \in S.$ Then either $a_i$ or $a_j$ is $a^{-1}.$ Would this argument work? Given I have Drupal 9 composer installation with composer-patches plugin and given a contrib module with a stable v8 release, but no v9 release (not even dev branch) and given that contrib module has a working v9 patch in the issue queue  is there any method to install that module + patch in composer? Even if I manually add both, the package and patch, to my composer.json, I still can't require or update this module with composer due conflicting versions. I really want to avoid duplicating /contrib code into my project's /custom codebase. My current workaround is:  forking that module to my own, private git repo applying patch there creating a new composer.json in my private git, and changing the package vendor to my custom_private_vendor adding my private git as VCS repo in the D9 project's composer.json and then composer require custom_private_vendor/contrib_module  This fulfills my goal of not duplicating the contrib module in my project's custom codebase, but every time I do this I feel the urge to wash my dirty hands. Is there something more elegant like composer require drupal/contrib_module --apply-patch-first or can I somehow target drupal.org's git with a specific patch included? Creating your own Tinyurl style uid XGBoost Loss function Approximation With Taylor Expansion Biblatex: Don't show n.d. when no year is given I am using biblatex/biber with several bibliographies for my book, and the misc-type, which is also a single bibliography (as a chapter), for legal documents. These documents often do not have a proper date or rather the date is part of the title. However, Latex gives me a "n.d." output, and I am wondering what I can do for it to not give a "n.d." when there is no year given?    To make clear, the reason for why I don't want to use the "year" field is that I deal with very different kinds of documents, which may either have no date or the date is in the title, because of which I prefer to simply "hard-copy" the whole thing in the "howpublished"-section and need to suppress the year, if none is provided.  MWE:  \usepackage[autocite=footnote, language=german, style=authoryear-ibid, sorting=nyt, backend=biber]{biblatex} \usepackage[bottom]{footmisc} \bibliography{lit.bib} \makeindex \begin{document}  Test\autocite[2]{draftnotice}.  \printbibliography[title={EU Legal Documents}, type=misc1]  \end{document}   My lit.bib is as follows:  @misc1{draftnotice,     Author = {{COMMUNICATION FROM THE COMMISSION}},     Howpublished = {Draft Commission Notice of 2014 on the notion of State aid pursuant to Article 107 (1) TFEU}}   Check out my MWE on overleaf Cancel/kill window.setTimeout() before it happens I have a few places where I use the line below to clear out a status, for example.  I have a few of these that hang out for 10 seconds or more and if the user gets clicking around the action can occur at incorrect time intervals.  window.setTimeout(function() { removeStatusIndicator(); }, statusTimeout);   Is it possible to cancel or kill this with some jQuery or JavaScript code, so I don't have this process hanging around? I want to show the IP address like below  lo : 127.0.0.1  eth0 : 192.168.5.123 eth1 : 192.172.0.212 wlan0 : 10.1.0.124   I'm able to print all IP address by ifconfig | awk '/inet addr/{print substr($2,6)}'. But it is only printing the IPs. Every system have their own interface names and addresses. So my script has to show their interfaces with respect to IP addresses. Implementing auto correct/complete on Stack Overflow Since I'm confused with these two phrases, I did a search in Google Books hoping to find some examples. But then I found :     Two private spaces or replicas cannot contain the same object with two distinct references. For example, during a disconnected phase, the pump can be present in two different private spaces and still be designated with the same name. During a meeting, two replicas use a same name to access the pump object(local or distant).   This made me more confused. I don't really understand why "the same name" was used in one sentence while "a same name" was used in another. Are there any differences between them? Extreme cursor delay with Acer Aspire E15 on Ubuntu 16.04 Occasionally my laptop will enter a state where the cursor starts lagging so bad that it's unusable.  Oddly during this state, everything other than the cursor is still completely responsive.  I.e. anything that only involves the keyboard (opening a terminal, typing, etc) is unaffected.  Once it enters this state, on successive reboots and shutdowns it re-enters this state within seconds.  However, if I shutdown and leave it alone for a few minutes, it does not immediately re-enter this state.  Based on search results for "acer aspire ubuntu freeze" I have tried setting the intel_idle_max_cstate field in grub to 0, 1, and 2, but this did not seem to help.  To be clear - I am not experiencing any crashes, the system is completely responsive except for the cursor is delayed to the point where it would take several seconds to go from one edge of the screen to the other. This is a question that keeps bothering me: much of mathematics was created by extending rules to new domains e.g. negative numbers were created by consistently extending arithmetic beyond positive integers. However, with number bases it seems impossible to do this. Number bases are integers, but could they be rational numbers? I've had previous discussions about this, see . One problem can be seen with this example: in base 2.5, 0.22 is greater than one. That's not what we want! What, if anything, can be done about this?  EDIT: To put it another way - if every positional number is less than any digit to its left, why doesn't this apply to fractional base positional numbers? How to count derangements? I was going through .Suppose that there are n persons who are numbered 1, 2, ..., n. Let there be n hats also numbered 1, 2, ..., n. We have to find the number of ways in which no one gets the hat having same number as his/her number. Let us assume that the first person takes hat i. There are n − 1 ways for the first person to make such a choice. There are now two possibilities, depending on whether or not person i takes hat 1 in return:  A) Person i does not take the hat 1. This case is equivalent to solving the problem with n − 1 persons and n − 1 hats: each of the remaining n − 1 people has precisely 1 forbidden choice from among the remaining n − 1 hats (i's forbidden choice is hat 1).  B) Person i takes the hat 1. Now the problem reduces to n − 2 persons and n − 2 hats.  From this, the following relation is derived: !n = (n - 1) (!(n-1) + !(n-2))  I have a hard time understanding point B and the equation derived. I was trying to understand with an example of 4 people named A, B, C and D and the corresponding hats A, B, C and D. Everyone of them are supposed to wear any hat except their own matching hat i.e. A can wear B, C or D.  If suppose A wears B hat so we will be left with only 3 people and 3 hats. So I can understand "(n-1)!(n-1)" part but what i didn't understand is part B(marked above). Can anyone help? Proof by induction that $ 169 \mid 3^{3n+3}-26n-27$ $ 169$ | $3^{3n+3}-26n-27$ ?  Fulfilled for $n=0$. Induction to $n+1$: An integer $x$ exists so that  $ 169x= 3^{3n+6}-26n-27-26$ $ 169x= 27*3^{3n+3}-26n-27-26$ $ 169x= 26*3^{3n+3}+3^{3n+3}-26n-27-26$  An integer $m$ exists so that   $ 169x= 26*3^{3n+3}+169m-26$ ($ 13x= 2*(3^{3n+3}-1)+13m$)  Now I'm stuck although it looks simple. Thanks for any input in advance. So, for fun, I am taking some of the data of calls from the call center I work at and trying to do some hypothesis testing on them, specifically the number of calls received in a week, and using a Poisson distribution to fit it. Due to the subject matter of my job, there are two types of weeks, let's call one of them on-weeks where I hypothesize there are more calls, and off-weeks where I hypothesize there are fewer.  I have a theory that the $\lambda$ from on-weeks (let's call it $\lambda_1$) is larger than that of the one from off-weeks (let's call it $\lambda_2$)  So the hypothesis I want to test is $H_0: \lambda_1 &gt; \lambda_2, H_1: \lambda_1 \leq \lambda_2 $  I know how to test for one parameter (say  $H_0: \lambda_1 &gt; 1, H_1: \lambda_1 \leq 1 $) but not so sure how to go about doing 2 given a data set. Let's say I take two week's worth of data from each one $X_1 = 2$ and $X_2 = 3$ for the off-week and $Y_1 = 2$ and $Y_2=6$ for the on-week. Can someone help walk me though this simpler version such that I can apply it to a larger data set? Any help is appreciated, thank you. Help with completing the proof: Any unbounded sequence is either infinitely large or has a convergent subsequential limit Given $\{x_n\}$ is an unbounded sequence, prove:   $$ \lim_{n\to\infty}x_n = \infty\ \  \text{or}\ \ \lim_{n_k \to \infty} = A $$   In other words any unbounded sequence either infinitely large or has a convergent subsequence.     Everything before udpate section is wrong.  Part 1. Proving $|x_n| &gt; M \implies \lim_{n\to\infty} x_n = \infty$  $\Box$ Start with the definition of an unbounded sequence: $$ \forall M &gt; 0 \in \Bbb R \ \exists N \in \Bbb N: \forall n &gt; N\implies |x_n| &gt; M $$  Suppose that: $$ \lim_{n\to\infty}x_n \ne \infty $$ That means: $$ \forall n\in\Bbb N\ \exists C &gt;0 \in\Bbb R : |x_n| &lt; C $$  Now taking $M^\prime = \max\{M, C\}$ we have that: $$ \begin{cases} \begin{align} \exists N \in \Bbb N : &amp;\forall n&gt;N \implies &amp;|x_n| &gt; M^\prime \\ &amp;\forall n \in \Bbb N: &amp;|x_n| &lt; M^\prime \end{align} \end{cases} $$  So we have arrived to a contradiction which means the assumption is wrong and: $$ \lim_{n\to\infty}x_n = \infty $$ $\Box$     Part 2. It's not clear to me how to prove that if unboundedness does not imply infinity limit then it must imply the existence of a convergent subsequence.   Two questions is my mind:   Is part 1 correct?  How do I proceed with the second part?     I'm using the following definition. $$ \lim_{n\to\infty}x_n = +\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &gt; \epsilon \\ \lim_{n\to\infty}x_n = -\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &lt; -\epsilon \\ $$ A sequence is considered infinitely large when: $$ \lim_{n\to\infty}x_n = \infty\ \ \text{when}\ \ \lim_{n\to\infty}x_n = + \infty \ \ \text{or} \ \ \lim_{n\to\infty}x_n = - \infty $$    Update.   Looks like i've messed up a lot of things. I will try once again. As shown in comments and answers the statement holds only in case: $$ \lim_{n\to\infty} |x_n| = \infty \iff \forall \epsilon &gt; 0 \exists N \in \Bbb N: \forall n &gt; N \implies |x_n| &gt; \epsilon $$  Consider the following: $$ \lnot P = \lnot\left(\lim_{n\to\infty}|x_n| = \infty \right) \iff \exists \epsilon &gt; 0\ \forall N_1 \in \Bbb N : \exists n &gt; N_1 \land |x_n| &lt; \epsilon $$  But on the other hand it is given that $x_n$ is unbounded: $$ Q = \forall M &gt; 0\ \exists N_2 \in \Bbb N : |x_{N_2}| &gt; M $$  Construct a negative expression for boundedness: $$ \lnot P = \exists M &gt; 0\ \forall N_2 \in \Bbb N : |x_{N_2}| &lt; M $$  If $S = \lnot P \implies \lnot Q$ then $S = P \lor \lnot Q$  Let $\epsilon = M$, choose $N = \max\{N_1, N_2\}$ then both statements are true and: $$ \exists \epsilon &gt; 0\ \forall N = \max\{N_1, N_2\}: \exists n &gt; N \land |x_n| &lt; \epsilon $$  Now either $P$ is true, which would mean $\lim |x_n| = \infty$, or $\lnot Q$ is true which would mean the sequence is bounded and hence contains a convergent subsequence. What book features people living in a tree who move up it in a bubble? Are human bodies programmed to die? Following from this question: :  Is there any evidence that human bodies have systemic self-destruction built into their developmental program? I'm not talking about the cell death response, which I know is an important part of growth, development and cancer prevention.  I've read some things about telomere-shortening but don't know if this is a cause or an effect.