inverse image question, is it correct?

Let $\displaystyle X$ and $\displaystyle Y$ be sets , let $\displaystyle A \subseteq X$ and $\displaystyle B$ $\displaystyle \subseteq Y$, and let $\displaystyle f:X \times Y \longrightarrow X$ be a projection map.

Show that $\displaystyle f^{-1}(A) = A \times Y$.

I wonder if the question is correct. For example, let $\displaystyle X $= {1, 2} and $\displaystyle Y$ = {a,b}. Then, $\displaystyle X \times Y$ = {(1,a), (1,b), (2,a), (2,b)}.

Suppose $\displaystyle f(1,a)= f(1,b)=f(2,a)=2,$ and $\displaystyle f(2,b)=1$.

Suppose $\displaystyle A $= {1} $\displaystyle \subseteq$ {1,2} = $\displaystyle X$.

Then $\displaystyle f^{-1}(A) $= {(2,b)}$\displaystyle \neq A \times Y$ = {(1,a), (1,b)}.

Please check whether I am right that the proposition was wrong from the question.