I was asked to prove if $X = Y$, then $f^{-1}(X) \neq f^{-1}(Y)$ . I was only able to prove this to be true for the inverse but the comment I got back said I was confusing inverse with pre-image, and that the equality doesn't hold if $f^{-1}$ denotes the pre-image rather than the inverse.
Now, I thought it would be the same because they are represented in the same notation, but I lost points on the actual question. Could someone explain the case for pre-image? What couterexample would work?
My Proof For The Inverse:
Let $x\in f^{-1}(X)$
Then $f(x)\in X$
Since $X = Y$, $f(x)\in Y$, so
$x$ $\in$ $f^{-1}(Y)$, so $f^{-1}(X)$ $\subset$ $f^{-1}(Y)$.
Similarly, go opposite way to prove $f^{-1}(Y)$ $\subset$ $f^{-1}(X)$ and conclude they are equal because both sides are subset of each other.