# Characteristic Function of Intersection

## Theorem

Let $A, B \subseteq S$.

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

### Variant 1

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

### Variant 2

$\chi_{A \cap B} = \min \left\{{\chi_A, \chi_B}\right\}$