prove that for any function f : A (arrow) B and set Y (is a subset of) B, if f is onto, then
f(f^-1(Y )) = Y
well there are three steps to this:
1. Prove that f(f^-1(Y )) is a subset of Y
2. Prove that Y is a subset f(f^-1(Y ))
3. Prove using 1 and 2 that f(f^-1(Y )) = Y
Any insights on this?? THANKS
Alfredo Kraus
Favorite Answer
(1) f-1 (Y) is by definition the set D = { x: f(x) ε Y} so for every x ε D we have f(x) ε Y and therefore f(D) С Y
(2) Consider any element y of Y. As f is onto, there exists x ε A such that f(x) = y. But then f(x) ε Y and by definition of D= f-1 (Y) , x ε D. So all points in Y are images of some point in D and therefore Y С f(D)
(3) As f(D) С Y and Y С f(D) it follows that Y = f(D) = f (f-1 (Y))