Problem: $\sim$ is the mark for bijection between two set. Let $A$, $B$, $C$ be sets. Then$$A \sim A\\A\sim B \Rightarrow B\sim A\\(A\sim B \land B\sim C )\Rightarrow A\sim C$$ I know that is not a relation because it is define on set that contains all set which is not set. But my professor told that if I write like this it is OK.
Let $A,B,C\subset U$ where $U$ is some universal set. Then$$A \sim A\\A\sim B \Rightarrow B\sim A\\(A\sim B \land B\sim C )\Rightarrow A\sim C$$ Because I define relation $\sim$ on $P(U) × P(U)$ Is this correct?