Definition:Identity Arithmetic Function

Definition

The identity arithmetic function $\iota: S \to \Z$ is defined for $n \geq 1$ by:

$\forall n \in S: \iota \left({n}\right) = \delta_{n1}$

where:

$S$ is (in theory) any set, but in this context is usually one of the standard number sets $\Z, \Q, \R, \C$.

$\delta$ is the Kronecker delta.

That is:

$\forall n \in S: \iota \left({n}\right) = \begin{cases} 1 & : n = 1\\ 0 & : n \ne 1 \end{cases}$

Also see

The identity arithmetic function can be expressed in terms of the characteristic function $\chi_E: S \to \left\{{0, 1}\right\}$ where $E = \left\{{1}\right\}$:

$\forall n \in S: \iota \left({n}\right) = \chi_{\left\{{1}\right\}} \left({n}\right)$

but strictly speaking $\iota$ does not equal $\chi_{\left\{{1}\right\}}$ because the codomains are different:

$\operatorname{Cdm} \left({\iota}\right) = \Z$

$\operatorname{Cdm} \left({\chi_{\left\{{1}\right\}}}\right) = \left\{{0, 1}\right\}$

Note on Name

The name of this function can be confusing. It is clearly not an identity function as such.

In fact it is a function which returns an answer to the question:

Is $n$ equal to the identity element for multiplication? If so, return $1$, otherwise return $0$.