Characteristic Function of Union/Variant 2

Theorem
Let $A, B \subseteq S$.

Let $\chi_{A \mathop \cup B}$ be the characteristic function of their union $A \cup B$.

Then:
 * $\chi_{A \mathop \cup B} = \chi_A + \chi_B - \chi_{A \mathop \cap B}$

Proof
From Subset of Union:
 * $A, B \subseteq A \cup B$

From Intersection is Subset of Union:
 * $A \cap B \subseteq A \cup B$

Thus from Characteristic Function of Subset:


 * $\chi_{A \mathop \cup B} \left({s}\right) = 0 \implies \chi_A \left({s}\right) = \chi_B \left({s}\right) = \chi_{A \mathop \cap B} \left({s}\right) = 0$

Now suppose that $\chi_A \left({s}\right) + \chi_B \left({s}\right) - \chi_{A \mathop \cap B} \left({s}\right) = 0$.

That is, $\chi_A \left({s}\right) + \chi_B \left({s}\right) = \chi_{A \mathop \cap B} \left({s}\right)$.

Suppose the latter equals $1$, i.e. $s \in A \cap B$.

Then both $s \in A$ and $s \in B$, so by definition of characteristic function:


 * $\chi_A \left({s}\right) + \chi_B \left({s}\right) = 1 + 1 = 2$

Since $2 \ne 1$, it follows that $\chi_{A \mathop \cap B} \left({s}\right) \ne 1$, i.e. it equals $0$.

Thence, it follows that $\chi_A \left({s}\right) + \chi_B \left({s}\right) = 0$.

This only happens when $\chi_A \left({s}\right) = \chi_B \left({s}\right) = 0$.

Thus, $s \notin A$ and $s \notin B$, so $s \notin A \cup B$.

It finally follows that $\chi_{A \mathop \cup B} \left({s}\right) = 0$.

It is now established that:


 * $\chi_{A \cup B} \left({s}\right) = 0 \iff \chi_A \left({s}\right) + \chi_B \left({s}\right) - \chi_{A \mathop \cap B} \left({s}\right) = 0$

and from Characteristic Function Determined by 0-Fiber, the result follows.