Characteristic Functions

Why

We want to indicate membership in a set by a function.¹

Definition

The characteristic function (or indicator function) of a set $X$ is a function from $X$ to $\set{0,1}$ which is $1$ if the argument is in $A$ and 0 otherwise.

The function which assigns to each subset $A$ of $X$ to characteristic function of $A$ is a one-to-one function from $\powerset{X}$ onto $2^{X}$.

Notation

Let $A$ be a non-empty set and $B \subset A$. We denote the characteristic function of $B$ in $A$ by $\chi _{B}: A \to R$. The Greek letter $\chi $ is a mnemonic for “characteristic”.

The subscript indicates the set on which the function is one. In other words, for all $B \subset A$, $\chi _{B}^{-1}(\set{1}) = B$.²

1. Future editions will elaborate, perhaps with forward-looking connections to Rectangular Functions. ↩︎
2. Another notation used is $1_{B}: A \to \set{0, 1}$ or $\mathbf{1}_B$. In the second notation here, the 1 is bold. ↩︎