Let $k\in\mathbb{N}$ and let $f$ be a real $k$-periodic sequence, i.e., $f(n)=f(n+rk)$ for every natural numbers $r$ and $n$. Denote by $A$ the set o multiples of the number $k$. To describe this periodic property, what are the differences between the notations:
- $\forall n\in\mathbb{N}:\forall s\in A:f(n)=f(n+s)$;
- $\forall s\in A :f(s)=f(s+n)$ and $n\in\mathbb{N}$;
- $\forall s\in A :f(s)=f(s+n)$, $\forall n\in\mathbb{N}$;
- For all $n\in\mathbb{N}$ and all $s\in A$, $f(n)=f(n+s)$.
All four statements can be used to express the same idea? These are things that I use without attention, but I know that there might be logical issues there.
Can someone give me clarifications about it?
Note that to require $f(n)=f(n+s)$ for all $n\in\Bbb N$ and $s\in A$ is not the same as requiring $f(s)=f(n+s)$ for the same $n,s$.
Statements (1) and (4) are equivalent, statements (2) and (3) are equivalent, but (1) and (2) are not. Let $f(n)=(3+(-1)^n)/2$, which has period $2$. It clearly obeys (1). But according to statement (2), $f(2)=f(2+n)$ for all $n$, which is clearly not the case for $n=1$ since $f(2)=2$ but $f(3)=f(2+1)=1\ne f(2)$.
You probably meant to write $f(n)=f(n+s)$ in formulas (2) and (3), but the intricacies of your notation mixed you up.