Characteristic Function of Intersection/Variant 2

Theorem
Let $A, B \subseteq S$.

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

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

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


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

By definition of characteristic function, we have:
 * $\min \set {\map {\chi_A} s, \map {\chi_B} s} = 1$


 * $\map {\chi_A} s = \map {\chi_B} s = 1$
 * $\map {\chi_A} s = \map {\chi_B} s = 1$

because $\chi_A, \chi_B$ take values in $\set {0, 1}$.

By definition of $\chi_A$ and $\chi_B$, this is equivalent to the statement that both $s \in A$ and $s \in B$.

That is:
 * $s \in A \cap B$

by definition of set intersection.