◀ ▲ ▶Branches / Set-theory / Definition: Indicator (Characteristic) Function, Carrier
Definition: Indicator (Characteristic) Function, Carrier
Let $S$ be a set. An indicator function (or characteristic function) $\chi$ on $S$ is a function $\chi:S\to\{0,1\},$ i.e. a function mapping each element of $S$ to exactly one of the two values $1$ or $0.$
For a given characteristic function $\chi$, the fiber of $1$ under $\chi$ is called its carrier.
Notes
- Every non-empty set $S$ can have many different indicator functions, in particular the two $\chi(x)=1,$ and $\chi'(x)=0$ for all $x\in S.$
- The carrier of a given $\chi$ defines a unique subset $C\subseteq S$ by $C=\{x\in S\mid \chi(x)=1\}.$
- Vice versa, for a given subset $A\subseteq S,$ there is a unique characteristic function $\chi_A$ such that its carrier equals $A,$ i.e. this characteristic function is defined by $$\chi_A(x):=\begin{cases}1&x\in S\wedge x\in A,\\0&x\in S\wedge x\not\in A.\\\end{cases}$$
Example
The carrier of the set of integers in the set of real numbers:
def carrier(N):
if N==floor(N):
return 1
else:
return 0
points= [(i, carrier(i)) for i in range(-5,10)]
list_plot(points)
Mentioned in:
Explanations: 1 2
Proofs: 3 4
Propositions: 5
Thank you to the contributors under CC BY-SA 4.0! 
- Github:
-
References
Bibliography
- Flachsmeyer, Jürgen: "Kombinatorik", VEB Deutscher Verlag der Wissenschaften, 1972, 3rd Edition
