definition

$$ \chi_C(x) = \begin{cases} 1 & \text{if }\, x \in C \\ 0 &\text{if }\, x \notin C \end{cases}

$$

In layman’s terms, $\chi_C(x)$ equals 1 if $x$ is in the set $C$, and 0 if $x$ is not in the set $C.$ (For those of you familiar with computer science, this function basically calculates a Boolean for whether $x$ is in $C$, then converts it to an integer.)

As one can see, this function relies on the set with which is it notated. In order to understand what we’re doing with this function, we therefore need to understand the properties involving the combination of multiple sets.

set properties

properties

In no particular order, the properties of characteristic functions are as follows:

1. $\chi_A=\chi_B$ if and only if $A = B$
2. Let $D^C = \{x: x \notin D\}$ (the set of all elements that aren’t in $D$). Then $\chi_{D^C} = 1-\chi_D$.
3. $\chi_{A \cap B} = \chi_A \chi_B = \min(\chi_A, \chi_B)$
4. $\chi_{A \cup B} = \chi_A + \chi_B - \chi_A\chi_B = \max(\chi_A, \chi_B)$
5. $\chi_{A \Delta B} = \chi_A + \chi_B - 2\chi_A\chi_B$
6. Idempotency property: $(\chi_A)^2 = \chi_A$