Equivalence relations and power sets.

Question

Let $\mathcal{A}$ be the class of all sets and define the relation $R$ on $\mathcal{A}$ as: $A\space R\space B$ iff there is a bijective function $f:A \to B$. Prove that $R$ is an equivalence relation on $\mathcal{A}$.

I know I need to prove bijection by proving injection and surjection, but I don't even know where to start. Any help would be greatly appreciated.

Answer 1

• There exists a bijection $A\to A\forall A\in\mathcal{A}$.

• If $A r B$, then $BrA$ since bijections have inverses.

• If $ArB,BrC$, then there exists a composition of bijections $A\to C$, which is in turn a bijection.

Now use the definition of equivalence relation.