# Math Help - Proving an equation

1. ## Proving an equation

My textbook says this is obvious, but I don't see how it is

How is this true?

Thanks

2. Originally Posted by 6DOM

My textbook says this is obvious, but I don't see how it is

How is this true?

Thanks
Proof of $f^{-1}(A^c)\subseteq(f^{-1}(A))^c$:

$x\in f^{-1}(A^c)\implies f(x)\in A^c\implies f(x)\notin A\implies x\notin f^{-1}(A)\implies x\in(f^{-1}(A))^c$

For the other direction, just flip the implication arrows. (This reasoning works in both directions.)