I am reading through Real Analysis by Fomin and Kolmogorov, and the book makes the statement that: "Isomorphism between partially ordered sets is an equivalence relation as defined in Sec. 1.4, being obviously reflexive, symmetric, and transitive".
So I have a couple questions regarding this:
- When equivalence relations are expounded upon earlier in the book, it is between members of a given set, but here we are speaking of a mapping between two distinct sets. I don't understand how to translate this into an equivalence relation when it is a mapping between two sets.
- How then does the isomorphism end up being symmetric? I understand the reflexivity and transitivity, as they are included in the definition of a set having a partial ordering, but how does the symmetry come into play?
Any help would be appreciated! Thanks!