I'm looking at the question below from a past paper:
What is an equivalence relation? Say that two sets $X$ and $Y$ are related via the relation $\rho$ if $X$ and $Y$ have the same cardinality. Prove that $\rho$ is an equivalence relation.
Now, I know an equivalence relation is a relation that is reflexive, symmetric and transitive, but I thought they could only be defined between the same set? For example a relation $R \subset N \times N$ between $N$ and $N$. In this case you can define properties like reflexivity, but between two different sets surely you can't have the reflexive property? I assume $\rho$ is meant to represent the bijection between the two sets, in which case transitivity would be impossible as each member can only relate to one other member of the opposite set?