This question comes from the first chapter of Robert Stoll's book, and it asks:
For each positive integer $n$, give an example of a set $A_n$ of $n$ elements such that for each pair of elements of $A_n$, one member is an element of the other.
I was able to solve for the case where, for a pair $a,b \in A$ we have $a \in b$ without necessarily having $b \in a$.
It is only when I tried to make both membership relations be true that I ran into some doubts. In particular, my solution was to define the $n$ elements in this way:
$$ S_0 = \{S_1, S_2, ..., S_n\} $$ $$ S_1 = \{S_0, S_2, ..., S_n\} $$ $$ S_2 = \{S_0, S_1, S_3, ..., S_n\} $$
and so on, and then define
$$ A_n = \{m_1, m_2, ..., m_n\}. $$
But I am uncertain on whether or not this is allowed, mostly because I am unfamiliar with rigorous, axiomatic set theories, but also because I know that, when we're dealing with infinities such as defined above, we often get counter intuitive results which are hard to see at first glance. In order to check, I tried to disprove this by deriving either a contradiction or an absurdity from this definition, but couldn't.
So my question is: is this allowed? If not, could you explain why not? Would this lead to either a contradiction or an absurdity?