Question

Let f : A → B be a function, and let X ⊆ A and Y ⊆ B. What is the relationship between X and f^−1(f(X))?
What about Y and f(f^−1(Y ))? Justify your responses with proofs.

Answer 1

If f is invertible, isn't it simply \[\Large f^{-1}(f(x)) = x\] ? or is it going to be more complicated?

Answer 2

\[f(f^{-1}(Y)) \subseteq Y\]
\[f^{-1}(f(X)) \supseteq X\]

Answer 3

@jim_thompson5910 what do you mean?

Answer 4

Now I'm not sure anymore. I might be thinking of something else