Q2 (10 points)
Let $U$ be a universe (i.e. any set) $A \subseteq U$ any subset. The characteristic function of $A$ is the function $\chi_A: U \to \{0, 1\}$ defined by
$\chi_A(x) = \begin{cases} 1, & \text{if } x \in A\\ 0, & \text{if } x \notin A \end{cases}$
That is, the characteristic function of $A$ indicates with 1 or 0 whether or not each element $x \in U$ is in $A$.
(a) [3 points] Prove that $\chi_{\overline{A}} = 1 - \chi_A$, i.e. for all $x \in U$, $\chi_{\overline{A}}(x) = 1 - \chi_A(x)$.
(b) [3 points] Let $B \subseteq U$. Prove that $\chi_{A \cap B} = \chi_A \cdot \chi_B$, i.e. for all $x \in U$, $\chi_{A \cap B}(x) = \chi_A(x)\chi_B(x)$.
(c) [4 points] Find and prove a formula for $\chi_{A \cup B}$ in terms of $\chi_A$ and $\chi_B$.
Note: Two functions $f: X \to Y$ and $g: X \to Y$ are defined to be equal if for all $x \in X$, $f(x) = g(x)$, i.e. they agree on every
input.
