Complement of a set and inverse image.

We need the

Definition of $f^{-1}(C)$:

$$f^{-1}(C)=\{x\in E|\ f(x)\in C\}.$$

Then we just compute both sides and compare.

On one hand, $$f^{-1}(A^c)=\{x\in E| f(x)\in A^c\}=\{x\in E| f(x)\in \mathbb{R}\setminus A\}$$

On the other hand, $$E\setminus f^{-1}(A)=E\setminus \{x\in E| f(x)\in A\}=\{x\in E| f(x)\in \mathbb{R}\setminus A\}$$