We want to construct a nonempty and proper subset $S$ of the set of non-negative integers. This set must have the following properties. For any $m$ and any $n$, if $m \in S$ and $n \in S$ then $m + n \in S$ and if $m \in S$ and $m + n \in S$ then $n \in S$. For each statement below, state if it is true or false. (i) 0 must be in $S$. (ii) 1 cannot be in $S$. (iii) There are only finitely many ways to construct such a subset $S$. (iv) There is such a subset $S$ that contains both $2015^{2016}$ and $2016^{2015}$.
We want to construct a nonempty and proper subset $S$ of the set of non-negative integers. This set must have the following properties. For any $m$ and any $n$,
if $m \in S$ and $n \in S$ then $m + n \in S$ \quad and \quad if $m \in S$ and $m + n \in S$ then $n \in S$.
For each statement below, state if it is true or false.
(i) 0 must be in $S$.
(ii) 1 cannot be in $S$.
(iii) There are only finitely many ways to construct such a subset $S$.
(iv) There is such a subset $S$ that contains both $2015^{2016}$ and $2016^{2015}$.