Definition 2.2.1 (Preimage Function).label Let $X, Y$ be sets and $P: 2^{Y} \to 2^{X}$, then $P$ is a preimage function if

1. (PF1)

   $P(\emptyset) = \emptyset$.

2. (PF2)

   For each $\mathcal{S}\subset 2^{Y}$, $\bigcup_{S \in \mathcal{S}}P(S) = P\paren{\bigcup_{S \in \mathcal{S}}S}$.

3. (PF3)

   For each $\mathcal{S}\subset 2^{Y}$, $\bigcap_{S \in \mathcal{S}}P(S) = P\paren{\bigcap_{S \in \mathcal{S}}}$.

and $P$ is total if

1. (T)

   $P(Y) = X$.