I am trying to solve exercise 2.1 in Aluffi's textbook &quot;Chapter 0&quot;:  How many different bijections are there between a set $S$ with $n$ elements and itself?  Here is my attempt. My &quot;non-rigorous&quot; answer is that I can make the function injective by choosing $n$ elements for where to make the first element of $S$, $n-1$ possibilities for the second, and so forth, giving $n!$ functions. Then this is a function injective function from $S$ to $S$ which is finite, so it is immediately surjective. (I don't think surjectivity just &quot;follows&quot; without invoking this kind of obvious, though it sounds obvious that it's surjective, though it's somewhat tough to prove.) Here is my attempt at making the argument rigorous, but I'd appreciate feedback on this above sketch as well.  Since there are $|n|$ choices for where to send each $s_i \in S$, there are exactly $n^n$ functions $f: S \to S$. If $f(s_1) = f(s_2)$ for $s_1 \neq s_2$, then $\mathrm{Im}(f) \not \subset S$, i.e., its cardinality is $&lt; n$, so it is not surjective; furthermore, it is not injective. There are $n!$ mappings $S \to S$ with no repeated inputs or repeated outputs, which are the only bijections $S \to S$.  How does this look? Are there ways to make it more rigorous?
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 can I reset my menus? One of the topics that seems to come up regularly on mailing lists and online discussions is the merits (or lack thereof) of doing a Computer Science Degree. An argument that seems to come up time and again for the negative party is that they have been coding for some number of years and they have never used recursion.  So the question is:   What is recursion? When would I use recursion? Why don't people use recursion? Excel: I need a formula that will count months between two dates. If it's 1-15 it should be a half month. Any ideas? So I am trying to find a formula for excel. Bascically I have two dates and I need to know the number of months between them. But to add complexity, I need a rule in there that if the earlier date is the 1st to 15th it will be a whole month but if it is 16th to end of the month it will be a half month.  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When I import my layers to Global Mapper, datum of vector is "MGI Austria"   I need this projection: When can I say that $\cup H_i$ is a group? Let $G$ be a group and $\{H_i\}_{i\in I}$ a family of subgroups. I would like to find a condition that will imply that $\cup_{i\in I} H_i$ is a subgroup. I know that it's not true in general, I need help to find this condition.  Any help is welcome.  Thanks a lot. Example of a function $f$ on $X$ to $\mathbb{R}$ which is not $X$-measurable but $|f|$, and $f^2$ are measurable. Question  The following is from Bartle:  Give an example of a function $f$ on $X$ to $\mathbb{R}$  which is not $X-$measurable but is such that $|f|,$ and $f^2$ are measurable.  Relavant Information  A family $\textbf{X}$ of subsets of $X$ is a $\sigma$ algebra in case:   $\phi, \mathbb{R} \in \textbf{X}$ $X \setminus A \in \textbf{X}$ if $A \in \textbf{X}$ If $(A_n) \in \textbf{X}$, then the union $\bigcup_{n=1}^{\infty} A_n \in \textbf{X}$  Any set in $\textbf{X}$ is called $X-$ measurable.   A function is $X-$ measurable if for every real $\alpha$, the set $\{x \in X : f(x)&gt;\alpha\}$ belongs to $\textbf{X}$  Attempt  Consider the following function: $$f=\begin{cases} 1 &amp;\mbox{ if} \ x \not\in\mathbb{Q} \\                   -1 &amp;\mbox{ if} \ x \in \mathbb{Q}  \end{cases}$$ defined from $\mathbb{R}$ to $\mathbb{R}$ with $\textbf{X}$ being the Borel algebra (the $\sigma$ algebra generated by all open intervals $(a,b)$) Then if $\alpha \in [-1,1)$, the set $\{x: f(x)&gt;\alpha\}$ consists of the irrational numbers. 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