Kenneth Rosen Edition 7 Exercise 2.3 Question 71 (Page No. 155)

Question

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 $f_{S}(x)=1$ if $x$ belongs to $S$ and $f_S(x)=0$ if $x$ does not belong to $S$. Let $A$ and $B$ be sets. Show that for all $x$ $\epsilon$ $U,$

1. $f_{A \cap B}(x) = f_{A}(x). f_{B}(x)$
2. $f_{A \cup B}(x) = f_{A}(x)+f_{B}(x) – f_{A}(x).f_{B}(x)$
3. $f_{\sim A}= 1-f_{A} (x)$
4. $f_{A \oplus B}(x) = f_{A}(x) + f_{B}(x)- 2 f_{A}(x) f_{B}(x) $