Proving f(f⁻¹(B)) = B for All B in Y

[The Captain]

Homework Statement

Prove that if f: $$X \rightarrow Y$$ is onto, then $$f(f^{-1}(B))=B$$ $$\forall B \in Y$$

Homework Equations

The Attempt at a Solution

 

[radou]

What does it mean for a function to be onto? What kind of inverse does f possesses iff it is onto?

 

[The Captain]

Onto means that for a function $$f:A \rightarrow B$$ if $$\forall b \in B$$ there is an $$a \in A: f(a)=b$$

The inverse means that if you take the $$f^{-1}(b)$$ that it should map back to a?

 

[radou]

Correct. But note that a right inverse exists if the function is onto. I.e., if g is a right inverse of f, then f(g(y)) = y, for every y in Y. What you need to prove is a direct consequence of this fact. (I used "g" rather than "f^-1" for the right inverse to avoid confusion leading to a conclusion that f^-1 is an inverse, i.e. both left and right).

 

[The Captain]

So I need to prove that if $$f(y)=Y$$ and $$f^{1}(Y)=y$$, that $$f(f^{1}(Y))=Y$$?