Pre-Image Preserves Complement
Pre-Image Preserves Complement
For \(f : X \to Y\) and \(B \subseteq Y\)
\[ f^{-1}(B^{c}) = f^{-1}(B)^{c}\]
Proof
\[\begin{align*}
&x \in f^{-1}(B^{c}) \\
\iff &f(x) \in B^{c} \\
\iff &f(x) \notin B \\
\iff &x \notin f^{-1}(B) \\
\iff &x \in f^{-1}(B)^{c}
\end{align*}\]