Inverse image of $Y$ and cardinality.

Question

I need to solve the following exercise.

Let $X$ be a non empty set and $P_2$ the set whose elements are all the subsets of $X$ that have $1$ or $2$ elements.

We have $f : X^2 \rightarrow P_2$, $(x, y) \mapsto \{x, y\}$

Let $Y\in P_2$. Determine $f^{-1}(\left\{{Y}\right\})$ and find its cardinality depending on the cardinality of $Y$.

The only things that I've managed to figure out is that $f$ is surjective and that $\text{card}(Y) = 1$ or $\text{card}(Y) = 2$.

Answer 1

If $ card(Y)=1 $ then it means $Y$ has only one element $x$. Thus $f^{-1}(\left\{{Y}\right\})=\left\{{(x,x)}\right\} $.

If $ card(Y)=2 $ then $ Y=\left\{{x,y}\right\} $ for some $x,y \in$ X. Thus $f^{-1}(\left\{{Y}\right\})=\left\{{{(x,y)},{(y,x)}}\right\} $