Characteristic Function of Union/Variant 1

Theorem
Let $A, B \subseteq S$.

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

Then:
 * $\chi_{A \mathop \cup B} = \min \set {\chi_A + \chi_B, 1}$

where $\min$ denotes the min operation.

Proof
By Characteristic Function Determined by 1-Fiber, it suffices to show:


 * $\min \set {\map {\chi_A} s + \map {\chi_B} s, 1} = 1 \iff s \in A \cup B$

By the nature of the min operation, this amounts to showing that:


 * $\map {\chi_A} s + \map {\chi_B} s \ge 1 \iff s \in A \cup B$

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 $s \in A \cup B$.