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# Characteristic functions

(Redirected from Characteristic function)

${\displaystyle \chi _{A}(x)\equiv \chi _{p(A)}(x)\equiv \mathbf {I} _{A}(x)\equiv \mathbf {I} _{p(A)}(x)\,}$
returns 1 if ${\displaystyle \scriptstyle x\,}$ has the defining property ${\displaystyle \scriptstyle p(A)\,}$ of a given set ${\displaystyle \scriptstyle A\,}$ or 0 otherwise.
For example, the characteristic function of the primes ${\displaystyle \scriptstyle \chi _{\rm {prime}}(n)\,=\,[\Omega (n)\,=\,1]\,=\,\pi (n)-\pi (n-1)\,}$ yields 1 only when ${\displaystyle \scriptstyle n\,}$ is prime (see A010051). Another example is the characteristic function of squarefree numbers ${\displaystyle \scriptstyle q(n)\,=\,|\mu (n)|\,}$ (see A008966).