Theorem

properties of preimages

Let $X: \Omega \to \Omega '$. Then the function $X^{-1}: \mathcal{P}(\Omega ') \to \mathcal{P}(\Omega)$ satisfies the following properties:

1. $X^{-1}(\emptyset) = \emptyset$, $X^{-1}(\Omega ') = \Omega$
2. complements: $X^{-1}(A'^{\complement}) = (X^{-1}(A'))^{\complement}$
3. unions: $X^{-1}(\cup A_{t}') = \cup X^{-1}(A_{t}')$
4. intersections: $X^{-1}(\cap A_{t}') = \cap X^{-1}(A_{t}')$