I've encountered the following exercise that's giving me a lot of trouble:
Let $X$ be a set. For any subset $A\subseteq X$, we can define a function $C_{A}:X \to \mathbb{Z}$ by: $C_{A}(x)=\begin{cases} 1 \text{ if } x\in A\\ 0 \text{ if } x\notin A \end{cases}$
Find formulas for $C_{A \cup B}$, $C_{A \cap B}$, and $C_{\overline{A}}$ in terms of $C_{A}$ and $C_{B}$. Explain why your answer works.
My problem is that we know what $C_{A}$ is, but how do you find the value of $C_{B}$?
The definition is not restricted for a particular subset $A$. It says, given any subset, which we call $A$ for the sake of this definition, we define a function which depends on that subset...
So $C_B(x)=1$ whenever $x\in B$, and $C_B(x)=0$ whenever $x\notin B$.