indicator function

The indicator function of a set $A$ is a function that maps members of the set to true, and maps non-members to false.

Formally,

Φ_{A} : X \to {0, 1}

where

Φ_{A} (x) := {\begin{cases} 1 & if x \in A, \\ 0 & if x \notin A \end{cases}

A set can thus be defined by its indicator function using set builder notation as

A = {x : Φ_{A} (x)}

If $A$ and $B$ are two subsets of $X$ , then

\begin{aligned} Φ_{A \cap B} & = min {Φ_{A}, Φ_{B}} = Φ_{A} \cdot Φ_{B} \\ Φ_{A \cup B} & = max {Φ_{A}, Φ_{B}} = Φ_{A} + Φ_{B} - Φ_{A} \cdot Φ_{B} \end{aligned}

The indicator function of the set complement of $A$ i.e. $A^{∁}$ is

Φ_{A^{∁}} = 1 - Φ_{A}