Characteristic Function of Intersection

Theorem

Let $A, B \subseteq S$.

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

Variant 1

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

Variant 2

$\chi_{A \mathop \cap B} = \min \set {\chi_A, \chi_B}$