Characteristic Function of Union

Theorem

Let $A, B \subseteq S$.

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

Variant 1

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

Variant 2

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

Variant 3

$\chi_{A \mathop \cup B} = \max \set {\chi_A, \chi_B}$