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.