EDB — 091

Exercises

1. [091] Let $D,C$ be non-empty sets and $f:D\to C$ a function. Let $I$ a non-empty family of indexes, $B_i\subseteq C$ for $i\in I$. Given $B\subseteq C$ remember that the counterimage of $B$ is

   $$f^{-1}(B)\stackrel{.}{=}\{x\in D,f(x)\in B\},$$

   Given $B\subseteq C$ we write $B^c=\{x\in C,x\notin B\}$ to denote the complement. Show these counterimage properties.

   $$\begin{array}{}\text{(1)}& f^{-1}({\bigcup }_{i\in I}{B}_i)& ={\bigcup }_{i\in I}{f}^{-1}(B_i)\text{(2)}& f^{-1}({\bigcap }_{i\in I}{B}_i)& ={\bigcap }_{i\in I}{f}^{-1}(B_i)\text{(3)}& f^{-1}(B^c)& =f^{-1}(B{)}^c.\end{array}$$