Characteristic vector of a set ($1_A$)

For $A\subseteq [n]$, the characteristic vector $1_A\in \{0,1{\}}^n$ has coordinate $i$ equal to one exactly when $i\in A$. Under this identification, set union becomes coordinatewise Boolean OR.

Over ${\mathbb{F}}_2$, characteristic vectors satisfy $1_A\cdot 1_B=|A\cap B|\mathrm{mod}2$. Parity restrictions on sizes and intersections therefore become diagonal and off-diagonal conditions on a Gram matrix, allowing rank and linear-independence arguments.