I'm studying for a test, and I have this two questions that I do not manage to solve.
The first question will be: Given a language ($\ \mathbb N, *, +, 1 $) show that the group $\{ 3^n : n \in \mathbb N\}$ is defineable in $\mathbb N$.
The second question will be: Define a language as you want, and find a closed formula $\phi$ s.t every model that supplies the formula $\phi$ is finite and is of size $\ n*(n-1) : n \in \mathbb N$.
Now I've tried to maybe consider a language with a function f, I will define f as a bijection by writing a formula that indicates it is a bijection, and will create f as a function $ f: A \mathrm X A -> |M| $, where A is some subgroup of the model M, now if I create a bijection, this means $|A \mathrm XA| = |M|$ but this only means that |M| must be of size $n^2$. I'm having a difficulty understanding how can I create $n*(n-1)$.