Equivalence Relations and 1-1 Correspondences

Question

I have an engineering instructor who has claimed that "equivalence relations and one-to-one correspondences are pretty much the same thing".

However, I believe the answers to both of the following statements are decided false:
1. If $R_1$ is an equivalence relation on a (possibly infinite) set $A$, then $R_1$ is a one-to-one correspondence from $A$ to $A$.
2. If $R_2$ a one-to-one correspondence from $A$ to $A$, then $R_2$ is an equivalence relation on $A$.

Unfortunately I cannot provide more context to the instructor's statement; frankly, I have no idea why he even brought this up. But can his statement actually make sense, if it is put on a more precise footing?

Answer 1

Are you sure he didn't say equivalence relations and partitions are the same thing? (Wikipedia)

However, it is certainly false that equivalence relations are the same as bijections from a set to itself. If nothing else, consider that for example, there are 5 equivalence relations on the set $X=\{a,b,c\}$, and 6 bijections from $X$ to itself.

Answer 2

$A=\{a,b,c\}$. $R_1=\{(1,1),(2,2),(3,3), (1,2), (2,1)\}$ is an equivalence relation on $A$, but not a function.

$R_2=\{(1,2),(2,3), (3,1)\}$ is a one-one correspondence from $A$ to $A$ but not reflexive.