Discrete-math 8

Dirichlet's function

\begin{matrix} D : R \to R, D (x) = {\begin{cases} 1 & x \in Q \\ 0 & otherwise \end{cases} \\ D [N] = {1} \\ D^{- 1} [N] = {1} \\ D^{- 1} [Z] = {0, 1} \\ (a, b) = {x \in R : a < x < b} \\ D^{- 1} [(0, 1)] = \emptyset \\ D [(0, 1)] = {0, 1} \end{matrix}

\begin{matrix} f^{- 1} [A ∖ B] = f^{- 1} [A] ∖ f^{- 1} [B] \\ X \subseteq f^{- 1} [f [X]], equality when f is injective \\ f [f^{- 1} [X]] \subseteq X, equality when f is surjective \end{matrix}