Proving an equivalence relation on $\mathcal{P}(\mathbb{N})$

Question

I am trying to prove an equivalence relation $\frown \subseteq \mathcal{P}(\mathbb{N}) \times \mathcal{P}(\mathbb{N})$ with $X \frown Y$ iff $|X|=|Y|$. Now, it seems quite trivial to prove reflexivity, symmetry and transitivity. Pretty much just stating that if one side is the equal, the other side is, too. However, I am thinking maybe for symmetry I would need to give a bijection to show that $X$ and $Y$ have the same cardinality. Is my approach more complicated than necessary?

Answer 1

Yes, you are. Any binary relation of the type $x\sim y$ if and only if $f(x)=f(y)$ is an equivalence relation. And symmetry follows from the fact that $a=b\implies b=a$.