Given $\displaystyle A \in \mathcal{P}(X) $ define the characteristic function $\displaystyle \chi_{A}: X \to \{0,1\} $ by $\displaystyle \chi_{A}(x) = \begin{cases} 0&\text{if}\ x \not \in A\\ 1&\text{if}\ x \in A \end{cases} $.

(i) Prove that the function $\displaystyle x \mapsto \chi_{A}(x) \chi_{B}(x) $ is the characteristic function of the intersection $\displaystyle A \cap B $.

So $\displaystyle x \in A \cap B \Rightarrow x \in A \ \text{and} \ x \in B $. By definition, the characteristic function always has values of $\displaystyle 0 $ or $\displaystyle 1 $. Therefore the characteristic function is $\displaystyle \chi_{A} \chi_{B} $.

(ii) Find the subset $\displaystyle C $ whose characteristic function is given by $\displaystyle \chi_{C}(x) = \chi_{A}(x) + \chi_{B}(x) - \chi_{A}(x) \chi_{B}(x) $. Isn't this just $\displaystyle A \Delta B $?

Thanks