Pre-Image Preserves Set Difference

For \(f : X \to Y\) and \(B_1, B_2 \subseteq Y\)

\[ f^{-1}(B_{1} - B_{2}) = f^{-1}(B_{1}) - f^{-1}(B_{2})\]

Proof

This follows from the fact that the pre-image of a function preserves itersections (1) and preserves complements (2) as follows:

\[\begin{align*} f^{-1}(B_{1} - B_{2}) &= f^{-1}(B_{1} \cap B_{2}^{c}) \\ &= f^{-1}(B_{1}) \cap f^{-1}(B_{2}^{c}) \tag{1} \\ &= f^{-1}(B_{1}) \cap f^{-1}(B_{2})^{c} \tag{2} \\ &= f^{-1}(B_{1}) - f^{-1}(B_{2}) \\ \end{align*}\]