Terence Tao Analysis 1 Exercise 3.4.1

Question

Exercise 3.4.1. Let $f : X → Y$ be a bijective function, and let $f^{−1}: Y → X$
be its inverse. Let $V$ be any subset of $Y$ . Prove that the forward image of $V$
under $f^{−1}$ is the same set as the inverse image of $V$ under $f$; thus the fact that
both sets are denoted by $f^{−1}(V)$ will not lead to any inconsistency.

Do I get it right that I have to prove $f^{-1} \circ f (V) = f \circ f^{-1}(V) $?
If so, what does $f (V) $ is equal to? $f$ is a function from $X$ to $Y$, but $V$ is a subset of $Y$.
My other idea is that I have to prove $f^{-1} \circ f (S) = f \circ f^{-1}(V) $, but it is not true, because on the left side is $S$ and on the right is $V$. So could anyone please explain what I have to prove?

Answer 1

No. What you have to prove is that, for each subset $V$ of $Y$, the sets:

• $\{x\in X\mid f(x)\in V\}$ (the inverse image of $V$ under $f$);
• $\left\{f^{-1}(v)\,\middle|\,v\in V\right\}$ (the forward image of $V$ under $f^{-1}$)

are equal.