# Is equipotent $\sim$ relation?

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?

• Yes, that's correct. – Floris Claassens Mar 19 at 11:38
• It is a relation on $\wp(U)$ or equivalently a subset of $\wp(U)\times\wp(U)$. – drhab Mar 19 at 11:41