Characteristic Function of Union

Theorem
Let $A, B \subseteq S$.

Then:


 * $(1):\quad \chi_{A \cup B} = \min \left\{{\chi_A + \chi_B, 1}\right\}$
 * $(2):\quad \chi_{A \cup B} = \chi_A + \chi_B - \chi_{A \cap B}$
 * $(3):\quad \chi_{A \cup B} = \max \left\{{\chi_A, \chi_B}\right\}$

where $\chi$ denotes characteristic function.

Proof of $(1)$
By Characteristic Function Determined by 1-Fiber, it suffices to show:


 * $\min \left\{{\chi_A \left({s}\right) + \chi_B \left({s}\right), 1}\right\} = 1 \iff s \in A \cup B$

By the nature of the minimum, this amounts to showing that:


 * $\chi_A \left({s}\right) + \chi_B \left({s}\right) \ge 1 \iff s \in A \cup B$

As $\chi_A, \chi_B$ are characteristic functions, the left-hand side amounts to:


 * $s \in A \lor s \in B$

which is precisely the definition of $s \in A \cup B$.