Category:Graph invariants: Difference between revisions

Template:No footnotes In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function, and one can freely convert between the two, but the characteristic function as defined below is better-suited to the methods of convex analysis.

Definition

Let ${\displaystyle X}$ be a set, and let ${\displaystyle A}$ be a subset of ${\displaystyle X}$. The characteristic function of ${\displaystyle A}$ is the function

${\displaystyle {\chi }_A:X\to \mathbb{ℝ}\cup \{+\mathrm{\infty }\}}$

taking values in the extended real number line defined by

${\displaystyle {\chi }_A(x):=\{\begin{array}{ll}0,& x\in A;\\ +\mathrm{\infty },& x\in ̸A.\end{array}}$

Relationship with the indicator function

Let ${\displaystyle {\mathbf{1}}_A:X\to \mathbb{ℝ}}$ denote the usual indicator function:

${\displaystyle {\mathbf{1}}_A(x):=\{\begin{array}{ll}1,& x\in A;\\ 0,& x\in ̸A.\end{array}}$

If one adopts the conventions that

• for any ${\displaystyle a\in \mathbb{ℝ}\cup \{+\mathrm{\infty }\}}$, ${\displaystyle a+(+\mathrm{\infty })=+\mathrm{\infty }}$ and ${\displaystyle a(+\mathrm{\infty })=+\mathrm{\infty }}$;
• ${\displaystyle \frac{1}{0}=+\mathrm{\infty }}$; and
• ${\displaystyle \frac{1}{+\mathrm{\infty }}=0}$;

then the indicator and characteristic functions are related by the equations

${\displaystyle {\mathbf{1}}_A(x)=\frac{1}{1+{\chi }_A(x)}}$

and

${\displaystyle {\chi }_A(x)=(+\mathrm{\infty })(1-{\mathbf{1}}_A(x)).}$

Bibliography

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