Chapter 2.3, Problem 73E

Let S be a subset of a universal set U. The characteristic function $f_S$ of S is the function from U to the set $\left\{0,1\right\}$ such that $f_S\left(x\right)=1$ if x belongs to S and $f_S\left(x\right)=0$ if x does not belong to S. Let A and B be sets. Show that for $x\in U$ ,

   $f_{A\cup B}\left(x\right)=f_A\left(x\right)\cdot f_B\left(x\right)$

   $f_{A\cup B}\left(x\right)=f_A\left(x\right)+f_B\left(x\right)-f_A\left(x\right)\cdot f_B\left(x\right)$

   $f_A-\left(x\right)=1-f_A\left(x\right)$

   $f_{A\oplus m}\left(x\right)=f_A\left(x\right)+f_B\left(x\right)-2f_A\left(x\right)f_B\left(x\right)$