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)

Explanations: 1 2
Proofs: 3 4
Propositions: 5

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References

Bibliography

1. Flachsmeyer, Jürgen: "Kombinatorik", VEB Deutscher Verlag der Wissenschaften, 1972, 3rd Edition