EDB — 091

Exercises

1. [091] Let \(D,C\) be non-empty sets and \(f:D→ C\) a function. Let \(I\) a non-empty family of indexes, \(B_ i⊆ C\) for \(i∈ I\). Given \(B⊆ C\) remember that the counterimage of \(B\) is

   \[ f^{-1}(B){\stackrel{.}{=}}\{ x∈ D, f(x) ∈ B\} ~ ~ , \]

   Given \(B⊆ C\) we write \(B^ c=\{ x∈ C,x∉ B\} \) to denote the complement. Show these counterimage properties.

   \begin{align} f^{-1}\bigl(⋃_{i∈ I} B_ i \bigr) & = ⋃_{i∈ I} f^{-1}\bigl(B_ i \bigr)\\ f^{-1}\bigl(⋂_{i∈ I} B_ i \bigr) & = ⋂_{i∈ I} f^{-1}\bigl(B_ i \bigr)\\ f^{-1}\bigl( B^ c \bigr) & = f^{-1}\bigl(B \bigr)^ c~ ~ . \end{align}