Maybe. Or at least insist that questions are not posted in image files, unless there are accompanying diagrams or something.
I don't really see a better way to describe the classes other than to reuse the language of the problem. Something like,
For $\displaystyle A \in \mathcal P (\mathbb N),~ [A] = \{ B \in \mathcal P (\mathbb N) ~:~ |A| = |B| \}$
A~B iff A and B have the same cardinal.
therefore, there are countable many equivalent classes:
1-class: the collection of all sets that contains only one element.
2-class: the collection of all sets that contains two elements.
...
n-class: the collection of all sets that contains n elements.
...
infinite-class:the collection of all sets that contains countable infinitely many elements.
A further question: What is P(N)/~? I leave it for you to solve it. Solve it, and you will understand the cardinality better.