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Can anyone please help me with this homework question on automata from Peter Linz? Use induction on $n$ to show that $|u^n| = n|u|$ for all strings $u$ and all $n$.

Prove that that $U(n)$, which is the set of all numbers relatively prime to $n$ that are greater than or equal to one or less than or equal to $n-1$ is an Abelian ...

Mar 27, 2019 · I know the proof using binomial expansion and then by monotone convergence theorem. But i want to collect some other proofs without using the binomial expansion. *if you …

Right! I like that: $ (n-1)^2=n^2-2n+1=n (n-2)+1 \equiv 1 (\bmod {n})$. I was skeptical of the line "However, we know (I forgot the theorem's name) that the number of elements of order 2 is …

Thanks for the link @muzzlator. I've just had a look at it and it's very interesting (and seems a lot simpler), however it uses methods a little different to those that I have been using for the …

I was playing with my calculator when I tried $1.5!$. It came out to be $1.32934038817$. Now my question is that isn't factorial for natural numbers only? Like $2!$ is $2\\times1$, but how do …

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Yes, it is more natural to use the determinant, see the duplicate. There the same doubt was mentioned, that the proof with centers is not so "insightful". On the other hand, the definition of …

@NeilsonsMilk, ah, it did not even occur to me that this involves a step. See, where I learned mathematics, it is not unusual to first define when a sequence converges to zero (and we have …